View source for Testwiki:School and university projects/Discrete and numerical mathematics/Learning plan

{{ambox
|nocat=true
| type  = notice
| text  =
<ul>
<li>To date, this educational and learning project has had four editions (2017, 2018, 2019, 2020).</li>
<li>There is an [[:es:Wikipedia:Proyecto_educativo/Matemática_discreta_y_numérica|equivalent project]] for contributing to the [[Spanish Wikipedia]] from the same starting date to present.</li>
</ul>
}}
<br />
{| style="margin:auto; width:55%; color:black;"
 | style="width:0%; text-align:right; background-color:White;" | [[File:Gftp.svg|center|55px]]
 | style="width:100%; text-align:center; background-color:#f1fadf; padding:1em 1em 1em 1em; border-style: solid; border-width: 0px 0px 0px 0px" | <!--<hr style="border-width: 1px;">--><span style="font-family:cursive;">Please '''contribute to the protection of the environment''': print this document only if you consider it absolutely necessary.</span><!--<hr style="border-width: 1px;">-->
|}
<br />
{| style="margin:auto; width:55%; color:black;"
 | style="width:0%; text-align:right; background-color:White;" | [[File:Adaptive-cycle.png|91px]]
 | style="width:100%; text-align:center; background-color:#f2eecb; padding:1em 1em 1em 1em; border-style: solid; border-width: 0px 0px 0px 0px" | <!--<hr style="border-width: 1px;">--><span style="font-family:cursive;">'''Warning:''' This page contains '''likely dynamic content''' (i.e., resources in a wider sense).</span><!--<hr style="border-width: 1px;">-->
|}
<br />
{{ambox
|nocat=true
| type  = notice
| text  =
<ul>
<li>Read all of this web page carefully because it contains important information for you.</li>
<li>Keep the link to it in safe custody, you may need to read this web page again.</li>
<li>Do not forget the [[Wikipedia:School and university projects/Discrete and numerical mathematics|<span style="background-color:SteelBlue; color:Gold; border:1px solid #000000; text-align:center;">&nbsp;'''associate learning project'''&nbsp;</span>]]<!-- /* Also using color box template: [[Wikipedia:School and university projects/Discrete and numerical mathematics|{{color box|SteelBlue|'''associate learning project'''|Gold}}]] */ -->, an optional continuous evaluation out-of-class practical activity such that if you are thinking of grading with distinction (''matrícula de honor'', in Spanish), your participation in it is strongly recommended.</li>
<li>Do not forget either the corresponding talk pages.</li>
</ul>
}}
<br />
{| style="margin:auto; width:90%; color:black; padding:1em 1em 1em 1em; text-align:left; background-color:#f2eecb;"
 | <div style="font-family:cursive;">
<p>This '''university learning plan''' consists of a primer on '''<u>[[discrete mathematics]]</u>''' and its applications including a brief introduction to a few '''<u>[[numerical analysis]]</u>'''.</p>
<p>It has a special focus on '''[[dialogic learning|dialogic learning (learning through argumentation)]]''' and '''[[computational thinking]]''', promoting  the development and enhancement of:</p>
* '''[[problem solving]]''' and '''[[decision making]]''' skills;
* positive attitudes towards '''[[How Not to Be Wrong: The Power of Mathematical Thinking|mathematical]]''', '''[[analytic reasoning|analytic]]''', '''[[reflective thinking|reflective]]''', '''[[creative thinking|creative]]''', '''[[lateral thinking|lateral]]''', '''[[divergent thinking|divergent]]''' and '''concrete [[critical thinking]]'''.
<p>My personal experience is the foundation of it. Wikipedia and other information sources, in English and Spanish, support it.</p>
<p>Course educational materials — [[File:Multimedia-icon.svg|32px|link=]] including video — by others, help to improve this plan.</p>
<p>Using Wikipedia, bibliography, multimedia and others, stimulate and enhance learning through '''[[Nonformal_learning#Contested_definitions|crossover learning]]''', '''[[Learning#Incidental_learning|incidental learning]]''', '''[[learning by doing]]''', '''[[learning by teaching]]''' and '''[[microlearning]]''', leaving renewed flavours of '''[[blended learning]]''' such as '''[[flipped learning]]'''.</p>
<p>Please, be free for suggesting improvements (see [[#To_keep_track,_know_more_or_write_a_comment|this section]]).</p>
</div>
 | <!--BEGIN EN CONSTRUCCIÓN style="width:30%; text-align:center; background-color:#f1fadf; border-style: solid; border-width: 0px 0px 0px 0px" | [[File:Ambox warning green construction.svg|center|46px]]<p><hr style="border-width: 1px;">This learning plan, which corresponds to the [https://zurbaran.unex.es/cloud/index.php/s/RqzqR04EI0UwnXX approved course program (ficha12a)] is '''under construction''' and therefore '''subject to change''', possibly until a few days before the beginning of the second academic semester. Please excuse the inconvenience.<hr style="border-width: 1px;"></p> END EN CONSTRUCCIÓN-->
|}
<br />
{| style="background: #eee; text-align: center; font-size:88%; border: 1px solid #aaa" align=center width=100%
| '''[[Wikipedia:School and university projects/Discrete and numerical mathematics/Learning plan#Introduction|Ex ante]]'''
| '''[[Wikipedia:School and university projects/Discrete and numerical mathematics/Learning plan#Background information|Background information]]'''
| '''[[Wikipedia:School and university projects/Discrete and numerical mathematics/Learning plan#Specific information|Specific information]]'''
| '''[[Wikipedia:School and university projects/Discrete and numerical mathematics/Learning plan#University project on the English Wikipedia|Project on Wikipedia]]'''
| '''[[Wikipedia:School and university projects/Discrete and numerical mathematics/Learning plan#Course outline|Course outline]]'''
| '''[[Wikipedia:School and university projects/Discrete and numerical mathematics/Learning plan#WP+: Paths on Wikipedia, bibliography (theory and proposed and solved exercises), multimedia and even more|Paths on Wikipedia]]'''
| '''[[Wikipedia:School and university projects/Discrete and numerical mathematics/Learning plan#Sample exam questions, instrumental and relational%5B%6e%6f%74%65%20%31%5D, and some answers|Sample exams]]'''
| '''[[Wikipedia:School and university projects/Discrete and numerical mathematics/Learning plan#Past qualifying activities and real exams with some solutions|Real exams]]'''
| '''[[Wikipedia:School and university projects/Discrete and numerical mathematics/Learning plan#Tentative course outline (chronogram for the 2019-2020 academic year)|Study programme]]'''
| '''[[Wikipedia:School and university projects/Discrete and numerical mathematics/Learning plan#Coda|Ex post]]'''
|}
<br />
{{TOC limit|6}}
<!--
   Transformar con pandoc, de html a mediawiki,
   en https://cloud.sagemath.com/projects/9941889f-c8f4-4b62-8d3f-dddb5e810269/files/

   pandoc ficheroorigen.html -f html -t mediawiki -s -o ficherodestino.txt
→
<!--Rojo = color: rgb(0, 0, 0);-->
<!-- <span style="font-family: liberation sans,sans-serif;"> -->
<!--          EL CURSO / THE COURSE          -->
{| id="mp-upper" style="width: 100%; margin:4px 0 0 0; background:none; border-spacing: 0px;"
<!--          PRELIMINARES, CONTENIDOS Y MÁS / PRELIMINARS, CONTENTS AND MORE          -->
| class="MainPageBG" style="width:80%; border:1px solid #deecc8; background:#f7fffa; vertical-align:top; color:#000;" |
{| id="mp-left" style="width:100%; vertical-align:top; background:#f7fffa;"
<!--          INTRODUCCIÓN / INTRODUCTION          -->
| style="padding:2px;" | <h2 id="pamdan-intro-h2" style="margin:3px; background:#deecc8; font-family:inherit; font-size:110%; font-weight:bold; border:1px solid #deecc8; text-align:left; color:rgb(0, 0, 0); padding:0.2em 0.4em;">Introduction<hr style="width: 100%; height: 4px; background-color: rgb(0, 102, 0);" /></h2>
|-
| style="color:#000; padding:2px 5px 5px;" | <div id="pamdan-intro"></div>
<div style="margin-left: 48%; text-align: right; color: rgb(0, 102, 0); background-color: rgb(255, 255, 223); background-image: none; padding: 0px; border-top: 4px solid rgb(0, 102, 0); border-bottom: 4px solid rgb(0, 102, 0); vertical-align: middle;">
{|
| 'I was speaking one day to a chemical expert about Avogrado's hypothesis concerning the number of molecules of the gases in equal volume, and its relation with the so-called Mariotte's law and its consequences in modern chemistry, and he came to answer: "Theories, theories! All of that does not matter to me ... That is for those who do science, I just apply it." I kept silent, torturing my mind to finding out how science can be applied without doing it, and finally, when after some time I knew why our expert had been close to dying, I understood it finally.'
|-
| <small>[[Miguel de Unamuno|Miguel de Unamuno]] (1864-1936): De la enseñanza superior en España [On higher education in Spain], Madrid, Revista Nueva, 1899, http://www.liburuklik.euskadi.eus/handle/10771/24524, p. 45. Also, in: ''Obras Completas de Miguel de Unamuno, Vol. VIII (Ensayos)'', pp. 1-58 (the quotation, on page 32).</small>
|}
</div>

=== <span style="color: rgb(0, 0, 0); font-style: italic; font-size:small;">Glossary of abbreviations</span> ===
{{smalldiv|1=
* CC BY, CC BY-SA, CC BY-ND, CC BY-NC, CC BY-NC-SA, CC BY-NC-ND: Creative Commons public licenses (https://creativecommons.org/licenses/).
* GratisOA: Gratis Open Access (http://legacy.earlham.edu/~peters/fos/overview.htm) (https://cyber.harvard.edu/hoap/Open_Access_(the_book)).
* ARR: [[All rights reserved|All Rights Reserved]].}}
=== <span style="color: rgb(0, 0, 0); font-style: italic;">Ex ante I: Mathematics and Computing</span> ===
<span style="color: rgb(0, 0, 0); font-style: italic;">(Just as an appetiser)</span>

==== <span style="color: rgb(0, 0, 0);">— Harangues</span> ====
* [http://www.dodgycoder.net/2012/09/q-with-nine-great-programmers.html Q&amp;A With Nine Great Programmers; 2006, by Jaroslaw &quot;sztywny&quot; Rzeszótko (AKA &quot;Stiff&quot;)] (in English) (just one Spanish translation: [http://www.dosideas.com/noticias/actualidad/247-10-preguntas-a-los-mas-grandes-programadores.html 10 preguntas a los más grandes programadores.])
* [http://www.skorks.com/2010/03/you-dont-need-math-skills-to-be-a-good-developer-but-you-do-need-them-to-be-a-great-one/ You Don't Need Math Skills To Be A Good Developer But You Do Need Them To Be A Great One; March 24, 2010, by Alan Skorkin.]

==== <span style="color: rgb(0, 0, 0); font-style: italic;">— Discrete mathematics</span> ====
* [https://www.quora.com/What-is-discrete-mathematics-and-why-is-it-so-important-for-computer-science What is discrete mathematics and why is it so important for computer science?; ''Quora''.]

==== <span style="color: rgb(0, 0, 0); font-style: italic;">— Algorithms</span> ====
* [https://www.quora.com/What-algorithms-should-I-know-to-become-a-good-programmer What algorithms should I know to become a good programmer?; ''Quora''.]

=== <span style="color: rgb(0, 0, 0); font-style: italic;">Ex ante II: Theme wikis</span> ===

==== <span style="color: rgb(0, 0, 0); font-style: italic;">— Wikipedia (Starting points)</span> ====

* [[Discrete mathematics]] (© CC BY-SA)
* [[Portal:Mathematics]] (© CC BY-SA)
* [[Portal:Technology]] (© CC BY-SA)

==== <span style="color: rgb(0, 0, 0); font-style: italic;">— Others:</span> ====

* [http://progopedia.com/ Progopedia (© GFDL &amp; BSD-style).]

=== <span style="color: rgb(0, 0, 0); font-style: italic;">Ex ante III: Graduates in computing (Spain)</span> ===

* [http://www.cpiiex.es/ Colegio Profesional de Ingenieros en Informática de Extremadura (CPIIEX) (Gratis Access)] (In Spanish).
** [http://www.cpiiex.es/images/archivos/cdigo%20deontolgico.pdf Código deontológico (Gratis Access)] (In Spanish).
* [http://www.ccii.es/ Consejo General de Colegios Profesionales de Ingeniería en Informática (CCII) (Gratis Access)] (In Spanish).

=== <span style="color: rgb(0, 0, 0); font-style: italic;">Ex ante IIII: Pre-university mathematical literacy</span> ===

* [http://matematicas.unex.es/actividades-y-noticias/actividades/hedima/ Calvo Jurado, Carmen, Cristina Gutiérrez, Concepción Marín, Pedro Martín, Rodrigo Martínez, Pablo Monfort, José Navarro e Ignacio Ojeda; (2012) HEDIMA: Temas básicos de Análisis Matemático, Álgebra Lineal y Geometría. Grupo HEDIMA, Departamento de Matemáticas, UEX. (© ARR).] (In Spanish).
* [https://www.unicoos.com/asignatura/matematicas Calle David, Unicoos. (© gratis OA).] (In Spanish).
* [http://ocw.upm.es/apoyo-para-la-preparacion-de-los-estudios-de-ingenieria-y-arquitectura/matematicas-preparacion-para-la-universidad Casaravilla Gil, A. et al. (2010) Apoyo para la preparación de los estudios de Ingeniería y Arquitectura: Matemáticas (Preparación para la Universidad). OCW UPM. (© CC BY-NC-SA).] (In Spanish).
* [http://ocw.innova.uned.es/matematicas-industriales/contenidos/presentacion.htm Díaz Hernández, A. M. et al. (2010) Curso 0 de Matemáticas. OCW UNED. (© CC BY-NC-ND).] (In Spanish).
* [http://www.edu365.cat/batxillerat/ciencies_tecnologia/matematiques.htm# EDU365CAT (2013) (Inici &gt; Batxillera &gt; Ciència i tecnologia &gt; Matemàtiques). Generalitat de Catalunya. (© gratis OA).] (In Catalan).
* [https://sites.google.com/site/redmatematicaantioquia/home/libros-de-la-red Gobernación de Antioquia (2015) 100 Problemas que todo bachiller debe entender y resolver. (© gratis OA).] (In Spanish).
* [http://personales.unican.es/gonzaleof/ González Ortíz, F. J. (2006) Proyecto MATEX. Gobierno de Cantabria. (© gratis OA).] (In Spanish).
* [http://www.lemat.unican.es/lemat/ Grupo Lemat (2009) Proyecto Lemat. Universidad de Cantabria, Gobierno de Cantabria y Gobierno de España. (© gratis OA).] (In Spanish).
* [http://recursostic.educacion.es/descartes/web/ MECD. Proyecto Descartes. (© CC BY-NC-ND).] (In Spanish).
* [http://recursostic.educacion.es/multidisciplinar/wikididactica/index.php/Categor%C3%ADas_de_Matem%C3%A1ticas_en_el_Bachillerato WikiDidácTICa (2013) Matemáticas en Bachillerato. Gobierno de España (Intef). (© gratis OA).] (In Spanish).

=== <span style="color: rgb(0, 0, 0); font-style: italic;">Ex ante V: General motivation</span> ===
<div style="margin-left: 48%; text-align: right; color: rgb(0, 0, 0); background-color: none; background-image: none; padding: 0px; border-top: 0px solid rgb(0, 102, 0); border-bottom: 0px solid rgb(0, 102, 0); vertical-align: middle;">'Listening to my father during those early years, I began to realise how important it was to be an enthusiast in life. He taught me that if you are interested in something, no matter what it is, go at it full speed ahead. Embrace it with both arms, hug it, love it and above all become passionate about it. Lukewarm is no good. Hot is no good, either. White hot and passionate is the only thing to be.'<br />
[[Roald Dahl|<small>Roald Dahl</small>]]: [[My Uncle Oswald]]. London, England (GB-ENG), UK: Penguin Books Limited, 1980, p. 37.
</div>

* [http://medialab-prado.es/article/apuntes_para_una_historia_social_de_la_computacion Andoni Alonso Puelles: '''Apuntes para una historia social de la computación'''. 2008. (© CC BY-SA). (In Spanish; the debate starts at 57:43).]
* [http://www.ted.com/talks/arthur_benjamin_s_formula_for_changing_math_education Arthur Benjamin: '''Teach statistics before calculus!''' 2009. (© CC BY-NC-SA). (In English, with Spanish subtitles).]
* [https://www.youtube.com/watch?v=Jj798r05Ls0 Pedro Cavadas:'''Address by Dr. Pedro Cavadas on the occasion of his investiture as Doctor 'honoris causa'''' of the Valencian International University (VIU), 00:26:00-01:01:33. (© ARR, gratisOA). (In Spanish).]
* [https://comunes.org/hack-for-your-rights/ Comunes Collective: '''Hack for your rights'''. © CC BY-SA.]
* {{cite web
 |url=https://www.medialab-prado.es/en/videos/javier-de-la-cueva-free-legal-ontology-move-commons-and-licenses-live
 |title='''Free Legal Ontology, Licenses to Live or Move Commons'''
 |last=Cueva
 |first=Javier de la
|author-link=Javier de la Cueva
 |date=
 |website=
 |publisher=
 |access-date=
 |quote= }} © CC BY-SA.
* [http://jafma.net/ana/theweekendarchaeologist/?p=733 Ana María Cruz Martín: <strong>Los ingenieros y Cervantes</strong>. <i>The Weekend Archaeologist</i>. 2017. (© ARR, gratisOA). (In Spanish).]
* [https://www.brainpickings.org/2012/08/10/10-rules-for-students-and-teachers-john-cage-corita-kent/ Maria Popova: '''10 Rules for Students, Teachers, and Life by John Cage and Sister Corita Kent'''. 2012. ''Brain Picking''. (© ARR, gratisOA).] (In Spanish: [https://www.deviolines.com/10-reglas-para-estudiantes-y-profesores-corita-kent-john-cage/ Jesús Fernández: '10 Reglas para estudiantes y profesores | Corita Kent | John Cage'. 2017. ''Deviolines''. (© ARR, gratisOA).])
* [https://www.youtube.com/watch?v=Zh3Yz3PiXZw David Maddox (Director): '''Alternative Math'''. 2017. (Short film). Produced by Ideaman Studios. (© ARR, gratisOA). (In English, with subtitles in several languages, including English and Spanish).]
* [http://www.news.uwa.edu.au/201309176069/alumni/tim-minchin-stars-uwa-graduation-ceremony Tim Minchin: '''Tim Minchin UWA Address 2013''' (© ARR, gratisOA). (In English, with English subtitles).] (Another video on YouTube [https://www.youtube.com/watch?v=-1QnC8Ii8Bg] includes an independent translation into Spanish). (In addition, there exists a [http://web.uwa.edu.au/?a=2386333 transcript] [in English] of Tim Minchin's address is available).
* {{cite web |url=https://www.youtube.com/watch?v=4ic7RNS4Dfo |others= |title='''KatelynOhashi - 10.0 Floor (1-12-19)'''. [The meaning of a 10: effort, sacrifice, discipline, work, perseverance, technique, security, confidence, innovation, improvisation, happiness, emotion, passion, perfection, excellence]. |last=[[UCLA Bruins|UCLA Athletics]] |first= |date= |website= |publisher= |access-date= |quote= }} (© ARR, gratisOA). (To watch a video about the sports life of [[Katelyn Ohashi|Katelyn ]][[Katelyn Ohashi|Ohashi]], [https://www.youtube.com/watch?v=z9b1KAoYz9s click here] (© ARR, gratisOA)).
:---
* We keep picking berries... —{{cite web |url=https://pages.gseis.ucla.edu/faculty/bates/berrypicking.html |others= |title='''The design of browsing and berrypicking techniques for the online search interface''' |last=[[Marcia J. Bates|Bates]] |first=[[Marcia J. Bates|Marcia J.]]<sup>([[:en:Marcia J. Bates|en]])</sup> |date=2017 |website= |publisher= |access-date= |quote= }} (© ARR, gratisOA).—.

=== <span style="color: rgb(0, 0, 0); font-style: italic;">Ex ante VI: Specific motivation</span> ===
* [http://www.ted.com/talks/hannah_fry_the_mathematics_of_love Hanna Fry: '''The mathematics of love'''. 2014. (© CC BY-NC-SA). (In English, with Spanish subtitles).]
* [http://medialab-prado.es/article/jugando_con_numeros Medialab-Prado, Madrid: '''Jugando con números'''. 2010-2011. (Mainly in Spanish).]
* [https://www.ted.com/talks/eduardo_saenz_de_cabezon_math_is_forever Eduardo Sáenz de Cabezón: '''Math is forever'''. 2014. (© CC BY-NC-SA). (In Spanish, with English subtitles).]
* [http://www.etereaestudios.com/docs_html/nbyn_htm/intro.htm Cristóbal Vila: '''Nature by numbers''', 2010, (© ARR, gratisOA)] — a short movie inspired on numbers, geometry and nature —. The theory behind the movie: [http://www.etereaestudios.com/docs_html/nbyn_htm/about_index.htm http://www.etereaestudios.com/docs_html/nbyn_htm/about_index.htm (© ARR, gratisOA)]. (More work by Cristóbal Vila, at his gallery online: [http://www.etereaestudios.com/index.html Etérea]).
* [https://www.washingtonpost.com/posteverything/wp/2015/07/20/why-i-give-my-students-a-tragedy-of-the-commons-extra-credit-challenge/?utm_term=.b53c18b6f4c1 Dylan Selterman: Why I give my students a 'tragedy of the commons' extra credit challenge. (''The Washington Post'', 2015). (© gratisOA). (En inglés / In English).] Relativo a esta noticia, en español, por ejemplo, en / Related to this news, in Spanish, for example, at: [http://www.europapress.es/desconecta/curiosity/noticia-pregunta-extra-ganar-puntos-examen-profesor-te-hara-pensar-20150716104226.html Europa Press: La pregunta extra para ganar puntos en un examen de este profesor te hará pensar.]
* {{cite web
 |url=ftp://ftp.aduni.org/videos/02_lect_01.rm
 
 |title='''What kinds of problems are solved in discrete math?'''
 |last=Simonson
 |first=Shai
 |date=
 |website=
 |publisher=
 |access-date=
 |quote= }} ([https://www.youtube.com/watch?v=h_9WjWENWV8 Alternate URL])
<strong>— Films</strong>
* [[X+Y|Morgan Matthews (dir.), '''X+Y''' (distributed in the United States as ''A Brilliant Young Mind''), 2014]] —[https://www.youtube.com/watch?v=mYAahN1G8Y8 X+Y (''Nathan solves math problem''; scene in English)]; [https://www.youtube.com/watch?v=A_-gDrXy6V0 X+Y (the same scene, subtitled in Spanish)].
* [[21 (2008 film)|Robert Luketic (dir.), '''21''', 2008]] —[https://www.youtube.com/watch?v=8DMnAAvakh0 scene (in English)] in which Ben Campbell resolves the [[Monty Hall Problem]] (the same scene [https://www.youtube.com/watch?v=BzAhrFrnpGM in Spanish]).
* [[:de:Die Vermessung der Welt (Film)|Detlev Buck (dir.), '''Die Vermessung der Welt''', 2012]], based on [[:de:Die Vermessung der Welt|homonymous novel]] by [[Daniel Kehlmann]] (translated into English by [[:w:Carol Brown Janeway|Carol Brown Janeway]], with the title 'Measuring the World'), on the life of [[Carl Friedrich Gauss]], [[Alexander von Humboldt]] and [[Aimé Bonpland]], [https://www.youtube.com/watch?v=bKZJVcNManE&t=4m00s scene of interest, in Spanish from minute 4:00 to 7:35] (the same scene, [https://www.youtube.com/watch?v=n2E_lfc2Bvc in German]).
* [[The Oxford Murders (film)|Álex de la Iglesia (dir.), '''The Oxford Murders''', 2008]], [https://www.youtube.com/watch?v=n2E_lfc2Bvc&t=14m15s scene of interest, in Spanish (14:15-18:33)] (the same scene [https://www.youtube.com/watch?v=z5T31UjmPME in English]).
* [https://vimeo.com/115677771 Sam Buntrock (dir.), '''ReCURSION''', 2014]
* [http://www.math.harvard.edu/~knill/mathmovies/ ''Mathematics in Movies'' (web page by Oliver Knill, Department of Mathematics, Harvard University)]
<!--
<strong>— Videos
* [https://www.youtube.com/channel/UC2AXB2-VLUGx3HMTMKbgomQ MatemáticasTV, vídeos de matemáticas]
-->
<strong>— Prove why it is so</strong>
* [https://www.youtube.com/watch?v=KSsl355pbsU&t=4m25s «La tabla del 9», fragment of «Un día en el colegio», ''Las Aventuras de los Payasos'', 1973 (from 4:25 to 7:25)] (in Spanish).
* [https://www.youtube.com/watch?v=CS_OwinLvyg  Multiplying by 6, 7, 8 and 9 with your hands. ''Guiainfantil''] ([https://www.youtube.com/watch?v=x2Nr-f02AUY click here]).
* [https://www.youtube.com/watch?v=SSaLk8M55bs ''(11-15)Multiplication Using hands'' (Explanation in the comments)].
|-
<!--          INFORMACIÓN GENERAL / BACKGROUND INFORMATION          -->
| style="padding:2px;" | <h2 id="pamdan-ede-h2" style="margin:3px; background:#deecc8; font-family:inherit; font-size:110%; font-weight:bold; border:1px solid #deecc8; text-align:left; color:rgb(0, 0, 0); padding:0.2em 0.4em;">Background information<hr style="width: 100%; height: 4px; background-color: rgb(0, 102, 0);" /></h2>

==== <span style="color: rgb(0, 0, 0); font-style: italic;">Universities</span> ====

=====— International =====
* [[European University Association|European University Association (EUA)]]
* [[International Association of Universities|International Association of Universities (IAU)]]
* [[AEGEE|''Association des États Généraux des Étudiants de l'Europe'' (AEGEE)]]
* [[Board of European Students of Technology|Board of European Students of Technology (BEST)]] <sup>([[:es:Consejo de Estudiantes Europeos de Tecnología|es]])</sup>
* [http://iasas.global/ International Association of Student Affairs and Services (IASAS)] (Also: [https://twitter.com/iasasglobal  IASAS-Global(@IASASGlobal)|Twitter])
* [[Erasmus Student Network|Erasmus Student Network (ESN)]]
* [[International Union of Students|International Union of Students (IUS)]]
* More: [[:Category:Student organizations|Student organizations]] &nbsp; · &nbsp; [[:Category:University associations and consortia|University associations and consortia]]

=====— Spain: Institutions, organizations, associations =====
* [http://atuspain.es/ Asociación para la Transparencia Universitaria (ATU). © gratisOA.] (In Spanish).
* [http://www.canae.org/ Confederación Estatal de Asociaciones de Estudiantes (CANAE). © gratisOA.] (In Spanish).
* [http://www.crue.org/ Crue Universidades Españolas. © gratis OA.] (In Spanish).
* [http://www.cje.org/ Consejo de la Juventud de España (CJE). © gratisOA.] (In Spanish).
* [http://www.creup.es/ Coordinadora de Representantes de Universidades Públicas (CREUP). © gratisOA.] (In Spanish).
* [http://faest.org/ Federación de Asociaciones de Estudiantes Progresistas del Estado Español, Ginés de los Ríos (FAEST). © gratisOA.] (In Spanish).
* [http://www.educacionyfp.gob.es/educacion-mecd/ Ministerio de Educación y Formación Profesional - Ministerio de Ciencia, Innovación y Universidades. © gratisOA.] (In Spanish).
* [http://www.runae.org/Runae/ Red Universitaria de Asuntos Estudiantiles (RUNAE). © gratisOA.] (In Spanish).
* [http://www.sindicatodeestudiantes.net/ Sindicato de Estudiantes (SE). © gratisOA.] (In Spanish).

=====— Spain: Legislation =====
* [https://www.boe.es/buscar/doc.php?id=BOE-A-2010-20147 University Student Statute. © gratisOA.] (In Spanish).
* [http://www.boe.es/buscar/act.php?id=BOE-A-2007-7786 Universities Organic Law. © gratisOA.] (In Spanish).
* [https://www.boe.es/buscar/act.php?id=BOE-A-1985-11578 University Professors Regime. © gratisOA.]([https://web.ua.es/es/ugt/documentos/informacion-pdi/borrador-estatuto.pdf University Professors Statute (Draft). © gratisOA.]) (In Spanish).

==== <span style="color: rgb(0, 0, 0); font-style: italic;">University of Extremadura (Spain)</span> ====
* [http://www.unex.es/organizacion/gobierno/vicerrectorados/vicealumn/normativas/NORMATIVAS General regulations of the University of Extremadura. © gratis OA.] (In Spanish).
* [http://www.unex.es/organizacion/defensor_universitario University Defence Office of the University of Extremadura. © gratisOA.] (In Spanish).
* [http://www.unex.es/organizacion/gobierno/vicerrectorados/vicealumn/funciones/consejo_estudiantes/index__html Student Council of the University of Extremadura. © gratisOA.] (In Spanish).
* [https://voluntariadouex.wordpress.com/entidades/ Volunteerism UEX] (In Spanish).

==== <span style="color: rgb(0, 0, 0); font-style: italic;">School of Technology (EPCC)</span> ====
* [https://www.unex.es/conoce-la-uex/centros/epcc Official website. © gratisOA.] (In Spanish).
* [http://escuelapolitecnicacc.blogspot.com.es/ Official blog. © gratisOA.] (In Spanish).
* [https://twitter.com/EPCC_Unex Official Twitter. © gratisOA.] (In Spanish).
* [http://www.unex.es/conoce-la-uex/centros/epcc/consejo_alumnos Student Council of the EPCC. © gratisOA.] (In Spanish).
|-
<!--          INFORMACIÓN ESPECÍFICA / SPECIFIC INFORMATION          -->
| style="padding:2px;" | <h2 id="pamdan-ede-h2" style="margin:3px; background:#deecc8; font-family:inherit; font-size:110%; font-weight:bold; border:1px solid #deecc8; text-align:left; color:rgb(0, 0, 0); padding:0.2em 0.4em;">Specific information<hr style="width: 100%; height: 4px; background-color: rgb(0, 102, 0);" /></h2>

==== <span style="color: rgb(0, 0, 0); font-style: italic;">— Professor</span> ====
<blockquote style="margin-top: 0em; margin-bottom: 0em;">
Juan Miguel León Rojas

Office: 1904/1/9 (according to the planimetry of Cáceres campus facilities and services: building [Civil Engineering premises]/floor/office) (you may consult the [https://zurbaran.unex.es/cloud/index.php/s/9ZYJep76byeHsKB course programme (ficha12a)] to find out where it is).

E-mail: <kbd>jm<!--despam-->leon&#x40;<span>unex</span>.es</kbd>.

[http://www.unex.es/conoce-la-uex/centros/epcc/centro/profesores/info/profesor?id_pro=jmleon Office hours].
</blockquote>

==== <span style="color: rgb(0, 0, 0); font-style: italic;">— Course description</span> ====
<blockquote style="margin-top: 0em; margin-bottom: 0em;">
This course is a primer on discrete mathematics and its applications including a very short introduction to a few numerical methods.

UEX code: 501272.
</blockquote>

==== <span style="color: rgb(0, 0, 0); font-style: italic;">— Rationale</span> ====
<blockquote style="margin-top: 0em; margin-bottom: 0em;">
The recommendations included in the ''Computer Engineering Curricula 2016''<sup>*</sup> and in the ''Computer Science Curricula 2013''<sup>†</sup>, among others, have been considered.

Regarding '''Discrete Mathematics''', the latter report identifies the following topics as the knowledge base for discrete structures (pp.76-81):
* (DS1) Functions, relations and sets,
* (DS2) Basic logic,
* (DS3) Proof techniques,
* (DS4) Basics of counting,
* (DS5) Graphs and trees, and
* (DS6) Discrete probability,
to which we would add:
* (DM1) Algebraic structures,
* (DM2) Matrices,
* (DM3) Algorithms and complexity, and
* (DM4) Basic number theory.
On the other hand, we have to keep in mind that some of these topics are studied in other courses taught at the School of Technology: DS6, in Statistics (UEX 501270); DM2, in Linear Algebra (UEX 502382); DM3, in Introduction to Programming (UEX 502304) and in Analysis and Design of Algorithms (UEX 501273); DS5, in Analysis and Design of Algorithms (UEX 501273) and in Data Structures and Information (UEX 501271), although from an algorithmic point of view.

With respect to '''Numerical Calculus''' and in order to provide students with a sufficient introduction to the algorithms and methods for computing discrete approximations used to solving continuous problems, in terms of linear and non linear approaches to a problem, we identify as essential contents:
* (NC1) Roots of Equations,
* (NC2) Linear Algebraic Equations, and
* (NC3) Curve Fitting (regression and interpolation).
On the other hand, again, we have to keep in mind that some of these topics are studied in other courses taught at the School of Technology: NC2, in Linear Algebra (UEX 502382); NC3, in Statistics (UEX 501270) (with regard to regression).

With all this in mind and meeting all the essential requirements of the [https://zurbaran.unex.es/cloud/index.php/s/9ZYJep76byeHsKB academic program (ficha12a)], 60 hours are programmed as can be seen in a synthetic way in the [[Wikipedia:School and university projects/Discrete and numerical mathematics/Learning plan#Course outline|course outline]] and scheduled in the [[Wikipedia:School and university projects/Discrete and numerical mathematics/Learning plan#Tentative course outline (chronogram for the 2019-2020 academic year)|tentative course outline (chronogram for the 2019-2020 academic year)]].
<hr style="width:25%;">
<sup>*</sup> <small>https://www.computer.org/cms/Computer.org/professional-education/curricula/ComputerEngineeringCurricula2016.pdf</small>
<br />
<sup>†</sup> <small>https://www.acm.org/education/CS2013-final-report.pdf</small>
</blockquote>

==== <span style="color: rgb(0, 0, 0); font-style: italic;">— Course objectives</span> ====
<blockquote style="margin-top: 0em; margin-bottom: 0em;">
After taking this course students should have reached the following objectives:

* '''Targets''': Representation, formulation, abstraction, modelling, verification and generalization.

* '''General''': Acquire scientific culture and mathematical culture in particular. Enhance reflective and creative attitudes. Enhance skills and abilities of analysis, search, discovery, verification and generalization. Promote the development and enhancement of problem-solving skills and of positive attitudes towards mathematical, analytical and concrete critical thinking. Be prepared for independent, critical study and assessment of elementary academic and informative publications about the topics covered in the course. Develop the capacity for lifelong learning.

* '''Common''': Enhance the ability to develop strategies for problem solving and decision making. Increase the ability to interpret the results obtained. Increase the rigor in the arguments and develop the reading and writing skills, the ability to use information and the capacity to make written or oral presentation of ideas and reasoning.

* '''Specific for themes 1 (Fundamentals) and 2 (Number Theory)''': Enhance the ability to understand and use the logical-mathematical language. Develop the capacity for abstraction through the construction of logical-mathematical arguments. Enhance the capacity of logical-mathematical reasoning in its deductive, inductive, abductive and algorithmic types.

* '''Specific for themes 3 (Combinatorics) and 4 (Difference Equations)''': Enhance the capacity of logical-mathematical reasoning in its inductive, algorithmic and recursive types. Enhance the ability to count.

</blockquote>

==== <span style="color: rgb(0, 0, 0); font-style: italic;">— Prerequisites</span> ====
<blockquote style="margin-top: 0em; margin-bottom: 0em;">
Although in respect of scientific knowledge, '''it has no particular prerequisites''', some prior background in maths (mainly in algebra, calculus and probability) and computing (mainly in programming) is welcomed but in no way presupposed. Regarding '''English language''', it may be desirable that you are at a intermediate conversational level, e.g. at least as skilled as an independent (self-reliant) user ('''level B''') according to the [[Common European Framework of Reference for Languages]]<sup>*</sup>. You might '''find out your English level''' taking this [https://learnenglish.britishcouncil.org/en/content free online English test] and then you might '''improve your knowledge of the English language''', for instance, [https://learnenglish.britishcouncil.org/en/skills practising your English skills at your level], and many more things available on these pages by the British Council ([http://www.fpa.es/en/princess-of-asturias-awards/laureates/2005-alliance-franaise-societa-dante-alighieri-british-council-goethe-institut-instituto-cervantes-e-instituto-cames.html Prince of Asturias Award for Communication and Humanities 2005]). 

<hr style="width:25%;">
<sup>*</sup> <small>Please keep in mind that it is enough to know the English language at a CEFR B1 level [https://www.gov.uk/english-language to apply for British citizenship or to settle in the UK] and at a CEFR B2 level [https://www.gov.uk/tier-4-general-visa/knowledge-of-english to study in the UK at a degree level or above].</small>
</blockquote>

==== <span style="color: rgb(0, 0, 0); font-style: italic;">— Course program</span> ====
<blockquote style="margin-top: 0em; margin-bottom: 0em;">
===== Academic year 2019-2020 =====
* [https://zurbaran.unex.es/cloud/index.php/s/kRSjAXdJyFdADtg In Spanish.]
* [https://zurbaran.unex.es/cloud/index.php/s/9ZYJep76byeHsKB In English.]
===== Academic year 2018-2019 =====
* [https://zurbaran.unex.es/cloud/index.php/s/48EteTlt44s3JTa In Spanish.]
* [https://zurbaran.unex.es/cloud/index.php/s/RqzqR04EI0UwnXX In English.]
<!--* [[:Archivo:POLI1617 GII 2 AMPMAT-FURMATH 501272-BILINGUAL 2x1.pdf|Versión bilingüe, en español e inglés.]]-->
</blockquote>

==== <span style="color: rgb(0, 0, 0); font-style: italic;">— School hours</span> ====
<blockquote style="margin-top: 0em; margin-bottom: 0em;">
* [https://www.unex.es/conoce-la-uex/centros/epcc/informacion-academica/horarios Class meetings and planners.]
</blockquote>

==== <span style="color: rgb(0, 0, 0); font-style: italic;">— Textbooks</span> ====

===== <span style="color: rgb(0, 0, 0); font-style: italic;">— Discrete mathematics</span> =====

====== <span style="color: rgb(0, 0, 0); font-style: italic;">Textbook</span> ======
For the discrete mathematics part of the course, students are encouraged to use the following book 
as a <strong>textbook</strong>:

* Rosen, Kenneth H. (2012). <i>Discrete Mathematics and its Applications</i> (7th  edition) (USA edition). New York: McGrawHill. {{ISBN|978-0-07-338309-5|0-07-338309-0|invalid1=yes|invalid2=yes}} © ARR. http://www.mheducation.com/highered/product.M0073383090.html?searchContext=discrete+mathematics

(However, its eighth edition is already available — 2019, http://highered.mheducation.com/sites/125967651x/information_center_view0/index.html).

As this book cover the vast majority of the material of the course — which, incidentally, corresponds to what is currently taught in hundreds of universities in the field of discrete mathematics —, students are encouraged to adopt and study it. Rosen's book is both a <strong>textbook</strong> and a <strong>workbook</strong> with lots of exercises and practical cases (computer projects, computations and explorations). It is even a <strong>guidebook</strong> including suggested readings, Despite its encyclopaedic spirit, it is also a handbook including lists of key terms and results and review questions.

<div style="margin-left: 2.5%; text-align: left; color: rgb(0, 0, 0); background-color: none; background-image: none; padding: 0px; border-top: 0px solid rgb(0, 102, 0); border-bottom: 0px solid rgb(0, 102, 0); vertical-align: middle;">
====== Companion website ======
In addition, it has a <strong>companion website</strong>: http://www.mhhe.com/rosen.

For instance, you can download a complete set of '''lecture slides''': http://highered.mheducation.com/sites/0073383090/student_view0/lecture_powerpoint_slides.html

Please be aware that:
:* There is a further international edition, the Global edition (2013, adapted by Kamala Krithivasan, {{ISBN|978-0-07-1315012|0-07-242434-6|invalid1=yes|invalid2=yes}}, © ARR, companion website: http://www.mhhe.com/rosenGE), that although it is also a 7th edition, it includes new topics and the exercises are in different order.
:* As you know, the content of newer editions are usually updated and improved versions of the content of older editions, occasionally including new content, so it is highly recommendable that, as far as possible, you read and study the new versions of the sections and exercises. However, within that aim of making continuous improvements, some contents may have been removed so it is also important to take a look to the older and newer editions, therefore keep mainly in mind these editions (even though only the seventh is highlighted):
:** fifth (2003, {{ISBN|978-0-07-242434-8|0-07-242434-6|invalid1=yes|invalid2=yes}}, © ARR, companion website: http://www.mhhe.com/math/advmath/rosen/r5/) — the fifth has been the last edition translated into Spanish: (2010, <i>Matemática discreta y sus aplicaciones</i> (5th ed.), Madrid: McGraw-Hill/Interamericana de España, S. A. U. {{ISBN|84-481-4073-7|invalid1=yes}} —,
:** sixth (2006, {{ISBN|978-0-07-288008-3|0-07-288008-2|invalid1=yes|invalid2=yes}}, © ARR, companion website: http://highered.mheducation.com/sites/0072880082/information_center_view0/index.html),
:** '''seventh''' (2012, {{ISBN|978-0-07-338309-5|0-07-338309-0|invalid1=yes|invalid2=yes}}, © ARR, companion website: http://www.mhhe.com/rosen),
:** seventh global (2013, {{ISBN|978-0-07-1315012|invalid1=yes|invalid2=yes}}, © ARR, companion website: http://www.mhhe.com/rosenGE), and
:** eighth (2019, {{ISBN|978-1-259-67651-2|invalid1=yes|invalid2=yes}}, © ARR, companion website: http://highered.mheducation.com/sites/125967651x/information_center_view0/index.html).
All these companion websites include, among other material and resources, '''interactive demos''', '''self assessments''' and '''extra examples'''.

====== <span style="color: rgb(0, 0, 0); font-style: italic;">Companion books describing solutions for each of the proposed exercises</span> ======
On the other side, this book is accompanied by books describing solutions for each of the proposed exercises, for instance, for the 5th and 7th US editions:
* Kenneth H. Rosen, Jerrold Grossman, <i>Student's Solutions Guide</i> (odd exercises) (5th ed., 2003, {{ISBN|0-07-247477-7|invalid1=yes}}) (7th ed., 2012, {{ISBN|978-0-07-735350-6|0-07-735350-1|invalid1=yes|invalid2=yes}}, © ARR.
* Kenneth H. Rosen, Jerrold Grossman, <i>Instructor's Resource Guide</i> (even exercises) (5th ed., 2003, {{ISBN|0-07-247480-7|invalid1=yes}}) (7th ed., 2012, {{ISBN|978-0-07-735349-0|0-07-735349-8|invalid1=yes|invalid2=yes}}), © ARR.

====== <span style="color: rgb(0, 0, 0); font-style: italic;">Companion books exploring and discussing contents and solutions to the proposed 'computer projects' and 'computations and explorations'</span> ======
And also by the supplementary books exploring and discussing contents and solutions to the 'computer projects' and 'computations and explorations' sections, from the 7th US edition:
* Daniel R. Jordan, <i>Exploring Discrete Mathematics using Maple</i>, 2nd edition, http://highered.mheducation.com/sites/0073383090/student_view0/exploring_discrete_mathematics_using_maple.html, © ARR.
* Daniel R. Jordan, <i>Exploring Discrete Mathematics using Mathematica</i>, 1st edition, http://highered.mheducation.com/sites/0073383090/student_view0/exploring_discrete_mathematics_using_mathematica.html, © ARR.

====== <span style="color: rgb(0, 0, 0); font-style: italic;">Companion book about applications of discrete mathematics</span> ======
Finally, you can download another supplement, one book about applications of discrete mathematics, last edition, paired with Rosen's book 6th edition, in any case for you to study it once you finish the course, except for the chapters that are of interest to it:
* John G. Michaels, Kenneth H. Rosen <i>Applications of Discrete Mathematics</i>, 2007, http://highered.mheducation.com/sites/0072880082/student_view0/applications_of_discrete_mathematics.html, http://highered.mheducation.com/sites/0073383090/student_view0/applications_of_discrete_mathematics.html, http://highered.mheducation.com/sites/0071315012/student_view0/applications_of_discrete_mathematics.html, {{ISBN|978-0-07-041823-3}}.
<!--
El siguiente cuadro le será de ayuda para emparejar las correspondientes secciones entre algunos de los textos recomendados en el plan docente de la asignatura, incluyendo cuatro de las diferentes ediciones del libro de Rosen, desde la quinta a la séptima y la séptima global / The following table will help you match up corresponding sections among some of the recommended texts in the course program, including four of the different editions of Rosen's book, from the fifth to the seventh and the seventh global ([[:Archivo:KDEM.DA.HD.703-0A.pdf|descargar / download]]).
-->

</div>

===== <span style="color: rgb(0, 0, 0); font-style: italic;">— Numerical calculus</span> =====
<blockquote style="margin-top: 0em; margin-bottom: 0em;">
For the short numerical calculus part of the course, students are encouraged to use the 
following book as a <strong>textbook</strong>:

* Chapra, Steven C., & Canale, Raymond P. (2006) <i>Numerical Methods for Engineers</i> (5th international edition). New York: The McGrawHill Companies, Inc. {{ISBN|0-07-124429-8|invalid1=yes}}. © ARR.

Companion website: http://www.mhhe.com/engcs/general/chapra/ 

Please be aware that:
* Although we use the fifth international edition, note that the seventh edition of this book has been published (http://www.mheducation.com/highered/product.M007339792X.html).

At the UEX library, you have electronic access to the 6th edition, in Spanish: http://0-www.ingebook.com.lope.unex.es/ib/NPcd/IB_BooksVis?cod_primaria=1000187&codigo_libro=4250

<div style="clear: both;"></div>
</blockquote>


==== <span style="color: rgb(0, 0, 0); font-style: italic;">— To find out more, while course is running (or once it is finished)</span> ====
<blockquote style="margin-top: 0em; margin-bottom: 0em;">
In addition to the references that appear in the [[Wikipedia:School and university projects/Discrete and numerical mathematics/Learning plan#Course outline|course outline]] and in the [https://zurbaran.unex.es/cloud/index.php/s/RqzqR04EI0UwnXX academic program (ficha12a)], and to those that can be mentioned in the classroom (large group and seminar/lab meetings) or posted on the [[Wikipedia talk:School and university projects/Discrete and numerical mathematics/Learning plan|talk page of the learning plan]] or at the [https://campusvirtual.unex.es/zonauex/avuex/ UEX online campus] in the course private forum, and to those that are referenced in the [https://drive.google.com/open?id=1e7s9zI5NSBQm_qMf-nNBK7r4WVbTK7XI 13 question selections] that are used throughout the course, you should consider:
* as the course develops:
** [[#WP+:_Paths_on_Wikipedia,_bibliography_(theory_and_proposed_and_solved_exercises),_multimedia_and_even_more|WP+: Paths on Wikipedia, bibliography (theory and proposed and solved exercises), multimedia and even more]];
** {{cite book |last=Epp |first=Susanna S. |author-link= |date=2019 |title=Discrete Mathematics with Applications |language= |trans-title= |format= |location=Andover, Hampshire (GB-HAM), England (GB-ENG), UK |publisher=Cengage Learning |edition= 5th |isbn=978-1337694193 |url=}} © ARR;
* once the course is finished:
** John G. Michaels, Kenneth H. Rosen <i>Applications of Discrete Mathematics</i>, 2007, http://highered.mheducation.com/sites/0072880082/student_view0/applications_of_discrete_mathematics.html, http://highered.mheducation.com/sites/0073383090/student_view0/applications_of_discrete_mathematics.html, http://highered.mheducation.com/sites/0071315012/student_view0/applications_of_discrete_mathematics.html, {{ISBN|978-0-07-041823-3}}.
** {{cite book |last=Wolfram |first=Stephen |author-link=Stephen Wolfram |date=2002 |title=A New Kind of Science |format=HTML/PDF |location=Champaign, Illinois (US-IL), USA |publisher=Wolfram Media, Inc. |isbn=1-57955-008-8 |url=http://blog.stephenwolfram.com/2017/05/a-new-kind-of-science-a-15-year-view/}} © gratisOA.
<div style="clear: both;"></div>
</blockquote>

==== <span style="color: rgb(0, 0, 0); font-style: italic;">— Communicating</span> ====

* Course private forum at the [https://campusvirtual.unex.es/portal/ UEX online campus]
* [[Wikipedia talk:{{PAGENAME}}|Talk page of the learning plan]]
|-
<!--          PROYECTOS EDUCATIVOS EN WIKIPEDIA          -->
| style="padding:2px;" | <h2 id="pamdan-intro-h2" style="margin:3px; background:#deecc8; font-family:inherit; font-size:110%; font-weight:bold; border:1px solid #deecc8; text-align:left; color:rgb(0, 0, 0); padding:0.2em 0.4em;">University project on the English Wikipedia<hr style="width: 100%; height: 4px; background-color: rgb(0, 102, 0);" /></h2>
<span style="color: rgb(0, 0, 0); font-style: normal;">Participating in MATDIN is an <b>optional</b>&nbsp;continuous evaluation out-of-class practical activity which is worth a try for contributing to your <b>personal development</b>and because it might help you <b>boost your course grade</b>; furthermore, if you are thinking of grading with distinction ['matrícula de honor', in Spanish], your participation in this project&nbsp;<b>is strongly recommended</b>. Find out more on
[[Wikipedia:School and university projects/Discrete and numerical mathematics|its descriptive web page]] and in [[Wikipedia talk:School and university projects/Discrete and numerical mathematics/Learning plan#Welcome to the course and to its learning plan strengthened by the English Wikipedia (academic year 2019-2020)|the welcome message to the course]].
</span>

<div style="margin-left: 5%; margin-right: 5%; text-align: left; color: rgb(0, 0, 0); background-color: #c6dbf7; background-image: none; padding: 0px; border-top: 4px solid #c6dbf7; border-bottom: 4px solid #c6dbf7; vertical-align: middle;">
{|
| It is important that you become aware that joining the university project 'Discrete numerical mathematics' is optional. Therefore, it is entirely up to you to do it. But if you do it, remember, you are required to:
<ul style = "list-style: none; margin-left: 5; padding-left: 1.75em; text-indent: -1.75em;">
<li>(a) use your true identity on free, open and public access web pages (Wikipedia) — although you can use an alias as your username, you must report your real identity (first, middle and last name) on your user page on the English Wikipedia —;</li>
<li>(b) be polite and respect diversity (please remember, diversity is a wealth, neither a problem nor a threat);</li>
<li>(c) comply with the rules and obligations laid down by the project coordination for this project ([[Wikipedia:School and university projects/Discrete and numerical mathematics|click and read them here]]), in particular the dynamic commitments ([[Wikipedia:School_and_university_projects/Discrete_and_numerical_mathematics#Dynamic_commitments|click and read them here]]);</li>
<li>(d) help the individuals involved in the project as much as possible;</li>
<li>(e) above all, commit yourself to you.</li>
</ul>
|}
</div>

==== <span style="color: rgb(0, 0, 0); font-style: italic;">— On the English-language Wikipedia</span> ====
:* [[Wikipedia:School and university projects/Discrete and numerical mathematics]], and
:* [[Wikipedia talk:School and university projects/Discrete and numerical mathematics/Learning plan#Welcome to the course and to its learning plan strengthened by the English Wikipedia (academic year 2019-2020)|Welcome to the course and to its learning plan strengthened by the English Wikipedia (academic year 2019-2020)]] (above all regarding the assessment of your work in the course).

==== <span style="color: rgb(0, 0, 0); font-style: italic;">— Communicating</span> ====
<blockquote style="margin-top: 0em; margin-bottom: 0em;">
To keep track of the project you have joined to, please follow the recommendations on [[Wikipedia:School and university projects/Discrete and numerical mathematics|its descriptive page]], particularly on [[Wikipedia:School_and_university_projects/Discrete_and_numerical_mathematics#The_basics|'The basics']] subsection.
</blockquote>
==== <span style="color: rgb(0, 0, 0); font-style: italic;">— Equivalent project on the Spanish-language Wikipedia</span> ====
:<span style="color: rgb(0, 0, 0); font-style: italic;">(Only if you take the course in Spanish).</span>
:* [[:es:Wikipedia:Proyecto educativo/Matemática discreta y numérica|Wikipedia:Proyecto educativo/Matemática discreta y numérica]], and
:* [[:es:Wikipedia discusión:Proyecto educativo/Matemática discreta y numérica/Plan de aprendizaje#Bienvenida a la asignatura y a su plan de aprendizaje apoyado en la Wikipedia en español (año académico 2019-2020)|Bienvenida a la asignatura y a su plan de aprendizaje apoyado en la Wikipedia en español (año académico 2019-2020)]] (above all regarding the assessment of your work in the course).

|-
<!--          CONTENIDOS / CONTENTS          -->
| style="padding:2px;" | <h2 id="pamdan-ede-h2" style="margin:3px; background:#deecc8; font-family:inherit; font-size:110%; font-weight:bold; border:1px solid #deecc8; text-align:left; color:rgb(0, 0, 0); padding:0.2em 0.4em;">Contents and learning paths on Wikipedia<hr style="width: 100%; height: 4px; background-color: rgb(0, 102, 0);" /></h2>
|-
| 

<div style="margin-left: 48%; text-align: right; color: rgb(0, 102, 0); background-color: rgb(255, 255, 223); background-image: none; padding: 0px; border-top: 4px solid rgb(0, 102, 0); border-bottom: 4px solid rgb(0, 102, 0); vertical-align: middle;">
'Do I contradict myself?<br />Very well then I contradict myself,<br />(I am large, I contain multitudes.)'<br />
<small>[[Walt Whitman|Walt Whitmann]] (1819-1892): <em>Song of Myself</em> (in <em>Leaves of Grass</em>, 1855)</small>
</div>
|-
| style="color:#000; padding:2px 5px 5px;" | <div id="pamdan-c"></div>
<div style="clear: both;"></div>

=== <span style="color: rgb(0, 0, 0); font-style: italic;">Considerations</span> ===
* [[Help:Using Wikipedia for mathematics self-study|Using Wikipedia for mathematics self-study]]
* [[Wikipedia:WikiProject Mathematics/Proofs|Proofs]]

{| style="clear:both;"
|-
| colspan="2" style="padding:1em 1em 1em 1em; border:1px solid #dfdfdf; background-color:#C4D8E2" valign="top"|
=== <span style="color: rgb(0, 0, 0); font-style: italic;">Course outline</span> ===
<blockquote>

* <span style="color:SteelBlue"><u>'''Theme 1:'''</u> '''[[Wikipedia:{{PAGENAME}}#Theme 1.- Fundamentals|FUNDAMENTALS]]'''</span> (16 h LG and 5 h SL)<div role="note" class="hatnote">Must-study: Rosen's book 7th ed. USA (§ 1.1-1.8, 2.1-2.3, 2.5, 9.1, 9.3, 9.5-9.6); Rosen's book 7th Global ed. (§ 1.7, 12.1-12.4); and [https://drive.google.com/open?id=1urEb_6ln_bJ0qBqiFlSv9UuVHCiXoZ51 León-Rojas, J. M., ''Notas incompletas de clase'' (Incomplete class notes) (in Spanish)].</div><div role="note" class="hatnote">Must-do: Every recommended exercise in every section covered for this theme in the [[Wikipedia:School_and_university_projects/Discrete_and_numerical_mathematics/Learning_plan#Calendar_of_activities|calendar of activities]]; [https://drive.google.com/drive/folders/1e7s9zI5NSBQm_qMf-nNBK7r4WVbTK7XI Question selections no. 1 to 5]; [[Wikipedia:School_and_university_projects/Discrete_and_numerical_mathematics/Learning_plan#Theme_1.-_Fundamentals_2|Sample exam questions for the theme 1]]; [[Wikipedia:School_and_university_projects/Discrete_and_numerical_mathematics/Learning_plan#Sample_preparatory_exams|Preparatory exam questions for the theme 1]]; and [[Wikipedia:School_and_university_projects/Discrete_and_numerical_mathematics/Learning_plan#Past_qualifying_activities_and_real_exams_with_some_solutions|Real exam questions for the theme 1]].</div>{{see also|iarchive:KDEM.DA.HD.70301|l1=León-Rojas, J. M., KDEM DA/HD – 703-01|Wikipedia:School_and_university_projects/Discrete_and_numerical_mathematics/Learning_plan#Logic|l2=WP+ (Logic)|Wikipedia:School_and_university_projects/Discrete_and_numerical_mathematics/Learning_plan#Sets, relations and functions|l3=WP+ (Sets, relations and functions)|Wikipedia:School_and_university_projects/Discrete_and_numerical_mathematics/Learning_plan#Cardinality|l4=WP+ (Cardinality)|Wikipedia:School_and_university_projects/Discrete_and_numerical_mathematics/Learning_plan#Algebraic structures|l5=WP+ (Algebraic structures).}}{{Hatnote|More detail can be found in the [[Wikipedia:School_and_university_projects/Discrete_and_numerical_mathematics/Learning_plan#Calendar_of_activities|Calendar of activities]].}}
::<span style="color:Sienna">Contents: ► '''[[Wikipedia:{{PAGENAME}}#Logic|Logic]]''': propositions, propositional equivalences, predicates and quantifiers, nested quantifiers, translating English statements into the language of logic and vice versa, valid arguments and rules of inference; direct and indirect proofs, verification and refutation strategies (truth tables, proof by contraposition, proof by contradiction, normal forms, natural deduction, semantic tableaux). ► '''[[Wikipedia:{{PAGENAME}}#Sets, relations and functions|Sets]]''': concepts and definitions, cardinality and power set; relations (membership, inclusion and equality), operations (union, intersection, complement, difference, symmetric difference) and properties, partition, cardinality of the union, cartesian product. ► '''[[#Functions|Maps and functions]]''': types (injective, surjective and bijective), monotony, representation (cartesian, arrow-set, matrix-based and graph-based), composition, inverse; multiset. ► '''[[#Relations|Relations]]''': properties, representing relations using matrices and graphs; equivalence relations, equivalence classes and partitions; tolerance relations; orderings, Hasse diagrams; preference relations. ► '''[[Wikipedia:{{PAGENAME}}#Cardinality, induction and recursion|Cardinality]]''': infinite sets, countability, Cantor's diagonal argument, Cantor's theorem and the continuum hypothesis. ► '''[[#Induction| Induction]]''': weak, strong and structural; well ordering. ► '''[[Wikipedia:{{PAGENAME}}#Algebraic structures|Algebraic structures]]''': magma, semigroup, monoid, group, ring, integral domain, field; homomorphism.</span>
::<span style="color:DarkGreen">Seminars/Labs: ► '''[1]''': Proofs and refutations, I; ► '''[2]''': Proofs and refutations, II; ► '''[3]''': Proofs and refutations, III; ► '''[4]''': Induction and recursion; ► '''[5]''': Cardinality and algebraic structures.</span>
:<div role="note" class="hatnote">Connections: ...</div>
* <span style="color:SteelBlue"><u>'''Theme 2:'''</u> '''[[Wikipedia:{{PAGENAME}}#Number theory|NUMBER THEORY]]'''</span> (9h LG and 3 h SL){{Hatnote|Must-study: Rosen's book 7th ed. USA (§ 4.1, 4.3-4.6); [http://www2.uca.es/matematicas/Docencia/2005-2006/ESI/1711003/Apuntes/apuntes.htm (lessons 10-13)]; and [http://ocw.upm.es/matematica-aplicada/matematica-discreta/contenidos/material-de-clase UPM (theme 3)].}}<div role="note" class="hatnote">Must-do: Every recommended exercise in every section covered for this theme in the [[Wikipedia:School_and_university_projects/Discrete_and_numerical_mathematics/Learning_plan#Calendar_of_activities|calendar of activities]]; [https://drive.google.com/drive/folders/1e7s9zI5NSBQm_qMf-nNBK7r4WVbTK7XI Question selections no. 6 to 8]; [[Wikipedia:School_and_university_projects/Discrete_and_numerical_mathematics/Learning_plan#Theme_2.-_Number_theory_2|Sample exam questions for the theme 2]]; [[Wikipedia:School_and_university_projects/Discrete_and_numerical_mathematics/Learning_plan#Sample_preparatory_exams|Preparatory exam questions for the theme 2]]; and [[Wikipedia:School_and_university_projects/Discrete_and_numerical_mathematics/Learning_plan#Past_qualifying_activities_and_real_exams_with_some_solutions|Real exam questions for the theme 2]].</div>{{see also|Wikipedia:School_and_university_projects/Discrete_and_numerical_mathematics/Learning_plan#Number_theory|l1=WP+ (Number theory).}}{{Hatnote|More detail can be found in the [[Wikipedia:School_and_university_projects/Discrete_and_numerical_mathematics/Learning_plan#Calendar_of_activities|Calendar of activities]].}}
::<span style="color:Sienna">Contents: ► '''[[#Divisibilityandmodulararithmetic|Divisibility and modular arithmetic]]''': divisibility, division algorithm, modular arithmetic. ► '''[[#Primosandgreatestcommondivisor|Primes and greatest common divisor]]''': integer representations, prime numbers and their properties, the fundamental theorem of arithmetic, conjectures and open problems about primes, greatest common divisor and least common multiple, the Euclidean algorithm, Bézout's theorem and the extended Euclidean algorithm. ► '''[[#Solvingcongruences|Solving congruences]]''': linear congruences, Euler's φ function, the Chinese remainder theorem, Euler-Fermat's theorem, Fermat's little theorem, Wilson's theorem and Wolstenholme's theorem. ► '''[[Applicationsofcongruences|Applications of congruences]]''': cryptography. ► '''[[#Divisibilityrules|Divisibility rules]]''': power residues, divisibility rules. ► '''[[#Diophantineequations|Diophantine equations]]''': linear equations, systems.</span>
::<span style="color:DarkGreen">Seminars/Labs: ► '''[6]''': Divisibility, modular arithmetic, primes, gcd and congruences; ► '''[7]''': Diophantine and congruence equations, I; ► '''[8]''': Diophantine and congruence equations, II.</span>
:<div role="note" class="hatnote">Connections: ...</div>
* <span style="color:SteelBlue"><u>'''Theme 3:'''</u> '''[[Wikipedia:{{PAGENAME}}#Combinatorics|COMBINATORICS]]'''</span> (8h LG and 3 h SL)<div role="note" class="hatnote">Must-study: Rosen's book 7th ed. USA (§ 6.1-6.5) and [https://dialnet.unirioja.es/servlet/libro?codigo=250473 Franco Brañas, J. R., Espinel Febles, M. C., Almeida Benítez, P. R. ''Manual de Combinatoria'' (in Spanish)].</div><div role="note" class="hatnote">Must-do: Every recommended exercise in every section covered for this theme in the [[Wikipedia:School_and_university_projects/Discrete_and_numerical_mathematics/Learning_plan#Calendar_of_activities|calendar of activities]]; [https://drive.google.com/drive/folders/1e7s9zI5NSBQm_qMf-nNBK7r4WVbTK7XI Question selections no. 9 to 11]; [[Wikipedia:School_and_university_projects/Discrete_and_numerical_mathematics/Learning_plan#Theme_3.-_Combinatorics_2|Sample exam questions for the theme 3]]; [[Wikipedia:School_and_university_projects/Discrete_and_numerical_mathematics/Learning_plan#Sample_preparatory_exams|Preparatory exam questions for the theme 3]]; and [[Wikipedia:School_and_university_projects/Discrete_and_numerical_mathematics/Learning_plan#Past_qualifying_activities_and_real_exams_with_some_solutions|Real exam questions for the theme 3]].</div>{{see also|Wikipedia:School_and_university_projects/Discrete_and_numerical_mathematics/Learning_plan#Combinatorics|l1=WP+ (Combinatorics).}}{{Hatnote|More detail can be found in the [[Wikipedia:School_and_university_projects/Discrete_and_numerical_mathematics/Learning_plan#Calendar_of_activities|Calendar of activities]].}}
::<span style="color:Sienna">Contents: ► '''[[Thebasicsofcounting|The basics of counting]]''': the sum rule, the product rule, the subtraction rule (inclusion-exclusion principle) and the division rule; the pigeonhole principle and its generalization; binomial coefficients and identities; variations, permutations and combinations. ► '''[[#Combinatorialproofs|Combinatorial proofs]]''': bijective proofs and double counting proofs. ► '''[[#Combinatorialmodeling|Combinatorial modeling]]''': 1<sup>st</sup>, sample selection and unit labelling with and without repetition; 2<sup>nd</sup>, grouping units (distribution, storage or placement of objects into recipients); 3<sup>rd</sup>, partitions of sets, and 4<sup>th</sup>, partitions of numbers.</span>
::<span style="color:DarkGreen">Seminars/Labs: ► '''[9]''': Combinatorics, I; ► '''[10]''': Combinatorics, II; ► '''[11]''': Combinatorics, III.</span>
:<div role="note" class="hatnote">Connections: [[Game theory]]; [[Network science]]; [[Number theory]]; [[Topology]].</div>
* <span style="color:SteelBlue"><u>'''Theme 4:'''</u> '''[[Wikipedia:{{PAGENAME}}#Finite difference equations (recurrence relations)|DIFFERENCE EQUATIONS]]'''</span> (8 h LG and 2 h SL)<div role="note" class="hatnote">Must-study: Rosen's book 7th ed. USA (§ 2.4, 8.1-8.3) and [http://matema.ujaen.es/jnavas/web_modelos/pdf_mmb08_09/sistemas%20dinamicos.pdf Navas Ureña, J., Esteban Ruiz, F. J., Quesada Teruel, J. M. 'Tema 9: Ecuaciones y sistemas en diferencias'] (in: [http://matema.ujaen.es/jnavas/web_modelos/pdf_mmb08_09/texto%20completo.pdf ''Modelos matemáticos en biología. Teoría'']) (in Spanish).</div><div role="note" class="hatnote">Must-do: Every recommended exercise in every section covered for this theme in the [[Wikipedia:School_and_university_projects/Discrete_and_numerical_mathematics/Learning_plan#Calendar_of_activities|calendar of activities]] — also some exercises from [http://matema.ujaen.es/jnavas/web_modelos/problemas/ptema2.pdf Navas Ureña, J., Esteban Ruiz, F. J., Quesada Teruel, J. M. 'Tema 2: Modelos discretos II'] [in: [http://matema.ujaen.es/jnavas/web_modelos/problemas/problemas%20completo.pdf ''Modelos matemáticos en biología. Ejercicios resueltos y propuestos'']] [in Spanish] —; [https://drive.google.com/drive/folders/1e7s9zI5NSBQm_qMf-nNBK7r4WVbTK7XI Question selections no. 12 to 13]; [[Wikipedia:School_and_university_projects/Discrete_and_numerical_mathematics/Learning_plan#Theme_4.-_Finite_difference_equations_(recurrence_relations)|Sample exam questions for the theme 4]]; [[Wikipedia:School_and_university_projects/Discrete_and_numerical_mathematics/Learning_plan#Sample_preparatory_exams|Preparatory exam questions for the theme 4]]; and [[Wikipedia:School_and_university_projects/Discrete_and_numerical_mathematics/Learning_plan#Past_qualifying_activities_and_real_exams_with_some_solutions|Real exam questions for the theme 4]].</div>{{see also|Wikipedia:School_and_university_projects/Discrete_and_numerical_mathematics/Learning_plan#Finite_difference_equations_(recurrence_relations)|l1=WP+ (Finite difference equations [recurrence relations]).}}{{Hatnote|More detail can be found in the [[Wikipedia:School_and_university_projects/Discrete_and_numerical_mathematics/Learning_plan#Calendar_of_activities|Calendar of activities]].}}
::<span style="color:Sienna">Contents: ► '''[[#Lineardifferenceequations|Linear difference equations]]''': homogeneous and non-homogeneous; with constant coefficients; direct; simple or multiple; indirect: systems of linear difference equations. ► '''[[#Lineardiscretedynamicalsystems|Linear discrete dynamical systems]]''': population dynamics, linear discrete dynamical models, BIDE models, Markov chains. ► '''[[#Solvingequationsnumerically|Solving equations numerically]]''': method of successive approximations (fixed point iteration); secant method.</span>
::<span style="color:DarkGreen">Seminars/Labs: ► '''[12]''': Difference equations, I; ► '''[13]''': Difference equations, II.</span>
:<div role="note" class="hatnote">Connections: ...</div>
* [[File:High-contrast-input-keyboard.svg|17px|link=]] <span style="color:ForestGreen">Optional seminars/labs, two [[Wikipedia:How to run an edit-a-thon|editathons]] (whenever we have time, the first in the middle of the semester and the second at the end; anyway, the exact content of this personal-enhancement tasks will not be mandatory covered on any exam): ► '''[14]''': Start of the optional reading and writing of a mathematical and computational, reflective, critical and analytical short essay of the text [https://archive.org/details/tinkertoycomputer00dewd A. K. Dewdney (1993) ''The Tinkertoy Computer and other machinations. Chapter 17: Automated Math''. New York: W. H. Freeman]. ('Juegos de ordenador. De cómo un par de programas obtusos pasan por genios en los tests de inteligencia.' <i>Investigación y Ciencia</i>, No. 116, May 1986, pp. 94-98, Prensa Científica, S. A.<!--,https://issuu.com/joshidelhampfert/docs/niyc116--> [In Spanish]). ► '''[15]''': Start of the optional reading and writing of a mathematical and computational, reflective, critical and analytical short essay about the [[Collatz conjecture|Collatz conjecture]] and its 'visualization' (for example: [https://www.jasondavies.com/collatz-graph/ <i>Collatz Graph: All Numbers Lead to One</i>, de Jason Davies]). (See also the [[/Sandbox#Collatz conjecture|sandbox/workshop of this course]]).</span>
* <span style="color:SteelBlue">[[Wikipedia:{{PAGENAME}}#Appendices|Appendices (optional)]]</span> (some complimentary or curious facts apart from the present programme, although related to it)
::<span style="color:Sienna">Contents: ► [[Wikipedia:{{PAGENAME}}#Graphs|Graphs]]; [[Wikipedia:{{PAGENAME}}#Numerical calculus 2|Numerical calculus]]; [[Wikipedia:School_and_university_projects/Discrete_and_numerical_mathematics/Learning_plan#Complimentary_knowledge_pills|Complimentary knowledge pills]]; [[Wikipedia:School_and_university_projects/Discrete_and_numerical_mathematics/Learning_plan#Editathons (intensive collaborative learning meetings)|Editathons]].</span>
:<div role="note" class="hatnote">Connections: ...</div>

</blockquote>

|}

<div style="clear: both;"></div>

=== <span style="color: rgb(0, 0, 0); font-style: italic;">WP+: Paths on Wikipedia, bibliography (theory and proposed and solved exercises), multimedia and even more</span> ===
{| style="margin:auto; width:90%; color:black; padding:1em 1em 1em 1em; text-align:left; background-color:#f2eecb;"
 | <span style="font-family:cursive;">Using Wikipedia, bibliography, multimedia and others, stimulate and enhance learning through '''[[Nonformal_learning#Contested_definitions|crossover learning]]''', '''[[Learning#Incidental_learning|incidental learning]]''', '''[[learning by doing]]''', '''[[learning by teaching]]''' and '''[[microlearning]]''', leaving renewed flavours of '''[[blended learning]]''' such as '''[[flipped learning]]'''.</span>
|}
==== Very important warning ====

{{ambox
|nocat=true
| type  = notice
| text  = English-language Wikipedia is a collective and voluntary work that, like all works, '''contains inaccuracies'''<sup>(1)</sup>, while «perfection is a polished collection of mistakes» (phrase often attributed on the web to Mario Benedetti and to Mario Satz).

I encourage you to correct them, to contribute with everything you learn about the topics we deal with, to contrast with Wikipedia in other languages if you know them, to gather information from other texts, in short, to drink in good sources. And write new articles in the English-language Wikipedia. Ultimately, '''contribute to improve this free encyclopedia as you study and learn'''.

By way of example, some good sources on the web are:<br />
: in English:
:* [https://www.encyclopediaofmath.org/index.php/Main_Page Encyclopedia of Mathematics]
:* [http://www-history.mcs.st-and.ac.uk/ MacTutor History of Mathematics archive]
:* [https://plato.stanford.edu/index.html Stanford Encyclopedia of Philosophy]
:* [https://planetmath.org/ PlanetMath] (© CC BY-SA)
:* [https://www.britannica.com/ Encyclopædia Britannica]
:* [https://arxiv.org/ arXiv.org]
:* [https://philarchive.org/ PhilArchive]
:* [https://oeis.org/ The On-Line Encyclopedia of Integer Sequences® (OEIS®)] (© CC BY-NC)
: in Spanish:
:* [http://www.enciclonet.com/ Enciclonet]
:* [http://vps280516.ovh.net/divulgamat15/index.php Divulgamat]
:* {{cite web
 |url=https://www.ugr.es/~eaznar/matematicos.htm
 |title=Biografías de algunos matemáticos
 |last=Aznar
 |first=Enrique R.
 |language=
 |date=
 |website=
 |publisher=
 |access-date=
 |quote= }}
:* [http://www.filosofia.org/pcero.htm Proyecto Filosofía en español (Fundación Gustavo Bueno)]
: and projects such as:
:* [https://proofwiki.org ProofWiki] (© CC BY-SA)
:* [https://dlmf.nist.gov/ NIST Digital Library of Mathematical Functions]
:* [http://mathworld.wolfram.com/ MathWorld]
<!-- * [https://courses.csail.mit.edu/6.042/fall17/mcs.pdf Lehman, Eric; Leighton, F. Thomson; Meyer , Albert R. Mathematics for Computer Science. Rev. November 10, 2017. © CC BYSA.] -->

Basically, those recommended,
* on the Spanish Wikipedia by:
** '''''Wikiproyecto Matemáticas''''': '[[:es:Wikiproyecto:Matemáticas#Enlaces útiles|Useful links]]' and
** '''''Wikiproyecto Filosofía''''': '[[:es:Wikiproyecto:Filosofía/recursos|Resources]]', and
* on the English Wikipedia by:
** '''WikiProject Mathematics''': '[[Wikipedia:WikiProject_Mathematics/Reference_resources|Reference resources]]', and
** '''WikiProject Philosophy''': '[[Wikipedia:WikiProject_Philosophy/Reference_resources|Reference resources]]'.
<hr>
<sup>(1)</sup> <small>To find out more: [[Reliability of Wikipedia|Reliability of Wikipedia]] • [[:es:Confiabilidad de Wikipedia|Confiabilidad de Wikipedia]] • [[:pt:Confiabilidade da Wikipédia|Confiabilidade da Wikipédia]] • [[:fr:Fiabilité de Wikipédia en anglais|Fiabilité de Wikipédia en anglais]]</small>
}}

<br />
{| style="background: #eee; text-align: center; font-size:88%; border: 1px solid #aaa" align=center width=100%
| '''[[Wikipedia:School and university projects/Discrete and numerical mathematics/Learning plan#Logic|Logic]]'''
| '''[[Wikipedia:School and university projects/Discrete and numerical mathematics/Learning plan#Sets, relations and functions|Sets, relations and functions]]'''
| '''[[Wikipedia:School and university projects/Discrete and numerical mathematics/Learning plan#Cardinality, induction and recursion| Cardinality, induction and recursion]]'''
| '''[[Wikipedia:School and university projects/Discrete and numerical mathematics/Learning plan#Algebraic structures|Algebraic structures]]'''
| '''[[Wikipedia:School and university projects/Discrete and numerical mathematics/Learning plan#Number theory|Number theory]]'''
| '''[[Wikipedia:School and university projects/Discrete and numerical mathematics/Learning plan#Combinatorics|Combinatorics]]'''
| '''[[Wikipedia:School and university projects/Discrete and numerical mathematics/Learning plan#Finite difference equations (recurrence relations)|Difference equations]]'''
| '''[[Wikipedia:School and university projects/Discrete and numerical mathematics/Learning plan#Graphs|Appendix: Graphs]]'''
| '''[[Wikipedia:School and university projects/Discrete and numerical mathematics/Learning plan#Numerical calculus|Appendix: Numerical calculus]]'''
| '''[[Wikipedia:School and university projects/Discrete and numerical mathematics/Learning plan#More complimentary knowledge pills|Appendix: More knowledge pills]]'''
|}
<br />

<div style="column-count:1;-moz-column-count:1;-webkit-column-count:1; font-size:85%;">

==== <span style="color: rgb(0, 0, 0);">Theme 1.- Fundamentals</span> ====

{| class="mw-collapsible wikitable {{#ifeq: {{{plegada|no}}}|sí|mw-collapsed|}}" width="100%" style="text-align:left; padding:0px; background-color:#F9F9F9;"
| align="left"   |

===== Logic =====
|-
| style="background-color:#F9F9F9; text-align:left; font-size:100%; padding:6px" |
{|
|[[File:The-impossibility-of-intent.png|200px]]
|<div id="toc_Logic" class="toc">
<div class="toctitle">
'''Contents'''
</div>
# [[#Logic_Key_concepts|Key concepts]]
# [[#Logic_Connections|Connections]]
# [[#Logic_Bibliography|Bibliography]]
# [[#Logic_Software|Software]]
# [[#Logic_Multimedia|Multimedia]]
# [[#Logic_See_also|See also]]
# [[#Logic_To_find_out_more|To find out more]]
</div>
|}

====== <span id="Logic_Key_concepts"></span>Key concepts ======
<hr style="margin-left: 0px; width:50%;">
<!-- {{columns-list|colwidth=50%|style=width:100%;font-style:normal;| -->
{{main|Mathematical logic}}
{{Main category|Mathematical logic}}
{{See also|Classical logic|Logic|Formalism (philosophy of mathematics)|List of logic symbols|Category:Logic}}
'''— Propositional logic'''
{{main|Propositional logic}}
{{Main category|Propositional calculus}}
* [[Proposition]]
* [[Logical connective]]
* [[Well-formed formula]]
* [[Propositional formula]]
* [[Truth value]]
** [[Logical truth]]
** [[False (logic)]]
* [[Consistency]]
* [[Truth function]]
** [[Truth function#Table of binary truth functions|Table of binary truth functions]]
* [[Functional completeness]] (Functionally complete set of connectives; adequate set of connectives)
* [[Logical equivalence]]
<div role="note" class="hatnote">See also: [[Conditioned disjunction]], [[Indicative conditional]] and [[Logical constant]].</div>
'''— Verification and rebuttal strategies, I'''
{{see also|Category:Theorems in propositional logic}}
* [[Truth table]]
* [[Contraposition]]
* [[Reductio ad absurdum]] (see also: [[Proof by contradiction]])
<ul>
<li> Normal forms, I: <div class="hlist inline">
* [[Conjunctive normal form]]
* [[Disjunctive normal form]]
</div></li></ul>
<ul>
<li> Normal forms, II: <div class="hlist inline">
* [[Negation normal form]]
* [[Algebraic normal form]]
</div></li></ul>
'''— Predicate logic'''
{{Main|First-order logic}}
{{Main category|Predicate logic}}
* [[Quantifier (linguistics)]]
* [[Quantifier (logic)]]
* [[Free variables and bound variables]]
* [[Universal quantification]]
* [[Existential quantification]]
* [[Uniqueness quantification]]
* [[Quantifier elimination]]
{{See also|Completeness (logic)}}
'''— Translating English statements into the language of logic and vice versa'''
* [[Syllogism]]
* {{noredirect|Logical calculus|{{color|#BA0000|Logical calculus}}}} <sup>([[:es:Cálculo lógico|es]]; [[:pt:Cálculo lógico|pt]]; [[:ca:Càlcul lògic|ca]])</sup>
<!-- * {{interlanguage link|{{no redirect|Logical calculus|{{color|#BA0000|Logical calculus}}}}|es|Cálculo lógico|pt|Cálculo lógico|ca|Càlcul lògic|display=yes}} -->
* [[Square of opposition]]
* [[Donkey sentence]]
<!-- * [[Quantifier (logic)#Distribution of quantifiers over logic connectives|Distribution of quantifiers over logic connectives]] <span style="color:red;">(Somebody has to write it!)</span> -->
'''— Valid arguments and inference rules'''
* [[Law of identity]]
* [[Law of noncontradiction]]
* [[Law of excluded middle]]
* [[Argument]]
* [[Premise]]
* [[Logical consequence]]
* [[Validity (logic)]]
* [[Counterargument]]
* [[Soundness]]
* [[Logical reasoning]]
* [[Deductive reasoning]]
* [[Problem of deduction]] <sup>([[:es:Problema de la justificación de la deducción|es]]; [[:ca:Problema de la justificació de la deducció|ca]])</sup>
* [[Rule of inference]]
* [[Inductive reasoning]]
* [[Problem of induction]]
* [[Abductive reasoning]]
<div role="note" class="hatnote">See also:
* [[Reasoning methods]] <sup>([[:es:Métodos de razonamiento|es]])</sup> and [[Cogency]] <sup>([[:es:Cogencia|es]]; [[:pt:Convicção|pt]]; [[:ca:Qualitat de convincent|ca]])</sup><!--* {{interlanguage link|Reasoning methods|es|Métodos de razonamiento|display=yes}} and {{interlanguage link|Cogency|es|Cogencia|pt|Convicção|ca|Qualitat de convincent|display=yes}}''.-->
* [[Stylistic device]], [[Modes of persuasion]], [[Argumentation theory]], [[Fallacy]], [[Enthymeme]], {{noredirect|Paralogism|{{color|#BA0000|Paralogism}}}} <sup>([[:es:Paralogismo|es]]; [[:pt:Paralogismo|pt]]; [[:fr:Paralogisme|fr]]; [[:it:Paralogismo|it]]; [[:ca:Paralogisme|ca]])</sup><!--{{interlanguage link|{{no redirect|Paralogism|{{color|#BA0000|Paralogism}}}}|es|Paralogismo|pt|Paralogismo|fr|Paralogisme|it|Paralogismo|ca|Paralogisme|display=yes}}-->, {{noredirect|Sophism|{{color|#BA0000|Sophism}}}} <sup>([[:pt:Sofisma|pt]]; [[:fr:Sophisme|fr]]; [[:it:Sofisma|it]])</sup><!--{{interlanguage link|{{no redirect|Sophism|{{color|#BA0000|Sophism}}}}|pt|Sofisma|fr|Sophisme|it|Sofisma|display=yes}}--> and [[Antinomy]];
* [[Paradox]];
* [[:Category:Reasoning]].
</div>
'''— Direct and indirect proofs'''
* [[Proof theory]]
* [[Methods of proof]] (see also: [[Constructive proof]], [[Proof by exhaustion]], [[Counterexample]] and [[Proof by infinite descent]])
* [[Mathematical fallacy]] (i.e., invalid proof)
{{See also|Category:Theorem proving software systems|Problem solving|Theory of justification}}
'''— Verification and rebuttal strategies, II'''
* [[Natural deduction]]
* [[Logic puzzle]]
* [[MU puzzle]]
* [[Knights and knaves]] (Questions about truthful and deceitful people)
* [[Portia|Portia's caskets]] (external link: [https://www.theguardian.com/science/2017/feb/13/can-you-solve-it-the-mystery-of-portias-caskets Bellos, Alex (2017). Can you solve it? The mystery of Portia's caskets. ''The Guardian''])
'''— Verification and rebuttal strategies, III'''
<ul>
<li> Normal forms, III: <div class="hlist inline">
* [[Prenex normal form]]
* [[Skolem normal form]]
</div></li></ul>
* [[Method of analytic tableaux]] (Semantic tableaux)
'''— Some unusual situations in Logic'''
* [[Infinite regress]]
* [[Münchhausen trilemma]]
* [[Liar paradox]]
* [[Epimenides paradox]]
* [[Definite description]]
* [[Curry's paradox]]
* See also: [[Paradox]], [[List of paradoxes#Logic|List of paradoxes: Logic]] and [[Wikipedia:School and university projects/Discrete and numerical mathematics/Learning plan#More complimentary knowledge pills|More complimentary knowledge pills]]

====== <span id="Logic_Connections"></span>Connections ======
<hr style="margin-left: 0px; width:50%;">
'''— Automated reasoning'''
{{Main|Automated reasoning}}
'''— Boolean algebra'''
{{Main|Boolean algebra}}
* [[Canonical normal form]]
* Simplifying Boolean functions
** [[Karnaugh map]]
** [[Quine–McCluskey algorithm]]
'''— Diagrammatic reasoning'''
{{main|Diagrammatic reasoning}}
'''— Logic gates'''
{{Main|Logic gate}}
<!--
{| border=0 align=center cellspacing=20
|-
| <center>{{pp-book-cover</center>
 |title=
 |subtitle=
 |cover-image=The-impossibility-of-intent.png
 |cover-color=#ffffff
}} Libro wiki (itinerario en Wikipedia)<br />¡Pulse en el título, no en la imagen de portada!</center>
|}
-->

====== <span id="Logic_Bibliography"></span>Bibliography: theory and proposed and solved exercises ======
<hr style="margin-left: 0px; width:50%;">
{| style="width: 100%;"
|In English: || In Spanish:
|- style="vertical-align:top;"
| style="width: 50%;" |
* —¤— Kenneth H. Rosen. <i>Discrete mathematics and its applications</i>. New York, New York State (US-NY), United States: McGraw-Hill, 7th edition, 2012. {{ISBN|978-0-07-338309-5}}. (Chapter 1 and related exercises).
| style="width: 50%;" |
* —¤— Amador Antón y Pascual Casañ, <i>Lógica Matemática. Ejercicios. I. Lógica de enunciados</i>. Valencia, Valencian Community (ES-VC), Spain: NAU llibres, 3rd edition, 1987. {{ISBN|84-85630-42-4}}
* —¤— María Manzano y Antonia Huertas, <i>Lógica para principiantes</i>. Humanes de Madrid, Madrid, Community of Madrid (ES-MD), Spain: Alianza, 2006. {{ISBN|84-206-4570-2}}.
* —¤— Kenneth A. Rosen. <i>Matemática discreta y sus aplicaciones</i>. Aravaca, Madrid, Community of Madrid (ES-MD), Spain: McGraw-Hill/Interamericana de España, S.A.U., 5th edition, 2004. {{ISBN|84-481-4073-7}}. (Sections 1.1, 1.2, 1.3, 1.4, 1.5, 3.1 and related exercises).
|}

====== <span id="Logic_Software"></span>Software ======
<hr style="margin-left: 0px; width:50%;">
{| style="width: 100%;"
|In English: || In Spanish:
|- style="vertical-align:top;"
| style="width: 50%;" |
* <s>—¤— [http://turner.faculty.swau.edu/mathematics/materialslibrary/truth/ Truth Tables (truth table calculator)]. © gratisOA.</s>
* [https://mrieppel.net/prog/truthtable.html Michael Rieppel's '''Truth Table Generator''']. © gratisOA.
* [https://github.com/LonamiWebs/Py-Utils/tree/master/logicmind Logicmind (command line model calculator)]. © MIT License.
* [http://formal.cs.utah.edu:8080/pbl/PBL.php Tyler Sorensen's PBL, a Boolean algebra/propositional logic library written in Python]. © BSD License.
* [https://www.joachim-breitner.de/blog/682-The_Incredible_Proof_Machine '''The Incredible Proof Machine''', by Joachim Breitner]
* —¤— Wolfgang Schwarz's '''Tree Proof Generator''' (© gratisOA):
** [http://www.umsu.de/logik/trees/ Tree proof generator for closed semantic tableaux (it provides with a countermodel in the case of an open tableau)];
** [https://www.umsu.de/logik/oldtrees/ Tree proof generator for open and closed semantic tableaux].
* —¤— Christian Gottschall's '''Gateway to Logic''' (© gratisOA) ([https://www.erpelstolz.at/christian/logic.html Gateway to Logic]):
** [https://www.erpelstolz.at/gateway/formular-uk-zentral.html Logic calculator (for classic bivalent propositional logic)] (see also: [https://www.erpelstolz.at/gateway/server-taskhelp.html about what the calculator can do] and [https://www.erpelstolz.at/gateway/server-languagehelp.html about the calculator syntax]);
** [https://www.erpelstolz.at/gateway/formular-uk-ableitung.html Automated Theorem Prover for Classical Predicate Logic] (see also: [https://www.erpelstolz.at/gateway/pl-examples.html examples of prover syntax]);
** [https://www.erpelstolz.at/gateway/formular-uk-beweis.html Proof Checker] (see also: [https://www.erpelstolz.at/gateway/proof-syntax.html about the checker syntax]).<!-- Ejemplo en el segundo: (P and (P -> (Q or ~P))) -> (Q or ~P) ; EN EL PRIMERO, CON «Prove the proposition» -->
* [https://www.wolframalpha.com/examples/LogicSetTheory.html WolframAlpha: Examples for Logic & Set theory (Boole algebra, set theory and transfinite numbers)]. © gratisOA.
| style="width: 50%;" |
* —¤— [http://www.cburch.com/logisim/ Logisim] (a graphical tool for designing and simulating logic circuits)] (in Spanish, English and more languages). © GNU GPL.
|}

====== <span id="Logic_Multimedia"></span>[[File:Multimedia-icon.svg|32px|link=]] Multimedia ======
<hr style="margin-left: 0px; width:50%;">
{| style="width: 100%;"
|In English: || In Spanish:
|- style="vertical-align:top;"
| style="width: 50%;" |
* {{cite web
 |url=https://www.youtube.com/watch?v=gX1bh7A96DE
 |title=Section 1.1 Intro to Propositional Logic
 |last=Baugher
 |first=Greg A.
 |department=Department of Math/Science/Informatics, Penfield College, Mercer University
 |location=Georgia (US-GA), USA
 |format=Video
 |publisher=
 |date=
 |website=
 |access-date=
 |quote= }}
* {{cite web
 |url=https://www.youtube.com/watch?v=rMV6pMntwaw
 |title=Section 1.2 Applications of Propositional Logic
 |last=Baugher
 |first=Greg A.
 |department=Department of Math/Science/Informatics, Penfield College, Mercer University
 |location=Georgia (US-GA), USA
 |format=Video
 |publisher=
 |date=
 |website=
 |access-date=
 |quote= }}
* {{cite web
 |url=https://www.youtube.com/watch?v=KfQ0U1WJ7sM
 |title=Sections 1.3-1.4 Prop Equivalences and Quantifiers
 |last=Baugher
 |first=Greg A.
 |department=Department of Math/Science/Informatics, Penfield College, Mercer University
 |location=Georgia (US-GA), USA
 |format=Video
 |publisher=
 |date=
 |website=
 |access-date=
 |quote= }}
* {{cite web
 |url=https://www.youtube.com/watch?v=V2I5Ak9z3aE
 |title=Section 1.6 Rules of Inference
 |last=Baugher
 |first=Greg A.
 |department=Department of Math/Science/Informatics, Penfield College, Mercer University
 |location=Georgia (US-GA), USA
 |format=Video
 |publisher=
 |date=
 |website=
 |access-date=
 |quote= }}
* {{cite web
 |url=https://www.youtube.com/watch?v=k-NgRZ0nHKk
 |title=Section 1.7 Introduction to Proofs
 |last=Baugher
 |first=Greg A.
 |department=Department of Math/Science/Informatics, Penfield College, Mercer University
 |location=Georgia (US-GA), USA
 |format=Video
 |publisher=
 |date=
 |website=
 |access-date=
 |quote= }}
* {{cite web
 |url=https://www.youtube.com/watch?v=K71pGliDL1g
 |title=Section 1.8 Proof Methods and Strategy
 |last=Baugher
 |first=Greg A.
 |department=Department of Math/Science/Informatics, Penfield College, Mercer University
 |location=Georgia (US-GA), USA
 |format=Video
 |publisher=
 |date=
 |website=
 |access-date=
 |quote= }}
* {{cite web
 |url=https://www.youtube.com/watch?v=xlUFkMKSB3Y
 |title=Lecture 1 - Propositional Logic
 |last=Krithivasan
 |first=Kamala
 |format=Video
 |publisher=Indian Institute of Technology Madras
 |date=
 |website=
 |access-date=
 |quote= }}
* {{cite web
 |url=https://www.youtube.com/watch?v=0uTE24o3q-o
 |title=Lecture 2 - Propositional Logic (Contd.)
 |last=Krithivasan
 |first=Kamala
 |format=Video
 |publisher=Indian Institute of Technology Madras
 |date=
 |website=
 |access-date=
 |quote= }}
* {{cite web
 |url=https://www.youtube.com/watch?v=DmCltf8ypks
 |title=Lecture 3 - Predicates & Quantifiers
 |last=Krithivasan
 |first=Kamala
 |format=Video
 |publisher=Indian Institute of Technology Madras
 |date=
 |website=
 |access-date=
 |quote= }}
* {{cite web
 |url=https://www.youtube.com/watch?v=jNeISigUCo0
 |title=Lecture 4 - Predicates & Quantifiers (Contd.)
 |last=Krithivasan
 |first=Kamala
 |format=Video
 |publisher=Indian Institute of Technology Madras
 |date=
 |website=
 |access-date=
 |quote= }}
* {{cite web
 |url=https://www.youtube.com/watch?v=kZ6UqFm8lnw
 |title=Lecture 5 - Logical Inference
 |last=Krithivasan
 |first=Kamala
 |format=Video
 |publisher=Indian Institute of Technology Madras
 |date=
 |website=
 |access-date=
 |quote= }}
* {{cite web
 |url=https://www.youtube.com/watch?v=ruwZxR2YRpE
 |title=Lecture 6 - Resolution Principles & Application to PROLOG
 |last=Krithivasan
 |first=Kamala
 |format=Video
 |publisher=Indian Institute of Technology Madras
 |date=
 |website=
 |access-date=
 |quote= }}
* {{cite web
 |url=https://www.youtube.com/watch?v=0gflLmuhHOg
 |title=Lecture 7 - Methods of Proof
 |last=Krithivasan
 |first=Kamala
 |format=Video
 |publisher=Indian Institute of Technology Madras
 |date=
 |website=
 |access-date=
 |quote= }}
* {{cite web
 |url=https://www.youtube.com/watch?v=rzxJ1nN_J4g
 |title=Lecture 8 - Normal Forms
 |last=Krithivasan
 |first=Kamala
 |format=Video
 |publisher=Indian Institute of Technology Madras
 |date=
 |website=
 |access-date=
 |quote= }}
* {{cite web
 |url=https://www.youtube.com/watch?v=WoPuWLNdtOY
 |title=Lecture 9 - Proving programs correct
 |last=Krithivasan
 |first=Kamala
 |format=Video
 |publisher=Indian Institute of Technology Madras
 |date=
 |website=
 |access-date=
 |quote= }}
* {{cite web
 |url=ftp://ftp.aduni.org/videos/02_lect_02.rm
 
 |title=Lecture 2 - Boolean Algebra and formal logic
 |last=Simonson
 |first=Shai
 |format=Video
 |publisher=ArsDigita University
 |date=
 |website=
 |access-date=
 |quote= }} ([https://www.youtube.com/watch?v=BrDto0heyaQ Alternate URL])
| style="width: 50%;" |
* {{cite web |last=Jordán Lluch |first=Cristina |title=¿Usamos la Lógica en la vida cotidiana? |format=Video |publisher=Universidad Politécnica de Valencia (UPV) |url=https://www.youtube.com/watch?v=7qCGT0DDEpE |access-date=}}
* {{cite web |last=Jordán Lluch |first=Cristina |title=Lógica, ¿para qué? |format=Video |publisher=Universidad Politécnica de Valencia (UPV) |url=https://www.youtube.com/watch?v=YHM51NfUJBU |access-date=}}
* {{cite web |last=Hervás Jorge |first=Antonio |title=Lógica proposicional |format=Video |publisher=Universidad Politécnica de Valencia (UPV) |url=https://www.youtube.com/watch?v=qg_TKXAiuGc |access-date=}}
* {{cite web |last=Bustince Sola |first=Nicanor Humberto |title=Lógica proposicional (Máster universitario en investigación en inteligencia artificial: Técnicas avanzadas de representación del conocimiento y razonamiento, Módulo 2,  Introducción, Tema 1) |format=Video |publisher=Universidad Pública de Navarra (UPNA) |url=https://www.youtube.com/watch?v=6Q7evuBaP1Q |access-date=}}
* {{cite web |last=Jordán Lluch |first=Cristina |title=Si y sólo si |format=Video |publisher=Universidad Politécnica de Valencia (UPV) |url=https://www.youtube.com/watch?v=Tt5FFtU1OMk |access-date=}}
* {{cite web |last=Jordán Lluch |first=Cristina |title=Condición necesaria y suficiente |format=Video |publisher=Universidad Politécnica de Valencia (UPV) |url=https://www.youtube.com/watch?v=ETkN5UtrcPc |access-date=}}
* {{cite web |last=Jordán Lluch |first=Cristina |title=¿Cuándo un razonamiento es correcto? |format=Video |publisher=Universidad Politécnica de Valencia (UPV) |url=https://www.youtube.com/watch?v=qzPfqpN00b8 |access-date=}}
* {{cite web |last=Hervás Jorge |first=Antonio |title=Demostración por contradicción aplicando tablas de verdad |format=Video |publisher=Universidad Politécnica de Valencia (UPV) |url=https://www.youtube.com/watch?v=em97usrGIHk |access-date=}}
* {{cite web |last=Hervás Jorge |first=Antonio |title=Proposiciones: métodos de demostración |format=Video |publisher=Universidad Politécnica de Valencia (UPV) |url=https://www.youtube.com/watch?v=O7LOh4Sa5qE |access-date=}}
* {{cite web |last=Bustince Sola |first=Nicanor Humberto |title=Lógica de primer orden (Máster universitario en investigación en inteligencia artificial: Técnicas avanzadas de representación del conocimiento y razonamiento, Módulo 2,  Introducción, Tema 2) |format=Video |publisher=Universidad Pública de Navarra (UPNA) |url=https://www.youtube.com/watch?v=fvO6HSL6up0 |access-date=}}
|}

====== <span id="Logic_See_also"></span>See also ======
<hr style="margin-left: 0px; width:50%;">
{| style="width: 100%;"
|In English: || In Spanish:
|- style="vertical-align:top;"
| style="width: 50%;" |
* {{cite techreport |first=Juan Miguel |last=León-Rojas |title=Matching tables for corresponding exercises from the 5th, 6th, 7th and 7th global editions of Rosen's book Discrete Mathematics and its Applications, Chapter 1 on The Foundations: Logic and Proofs (Bilingual edition, Spanish/English) |number=DA/HD – 703-01 |institution=KDEM (Knowledge Discovery Engineering and Management) |date=2017-01-31  |url=https://archive.org/details/KDEM.DA.HD.70301}}
* [https://www.nsa.gov/news-features/puzzles-activities/puzzle-periodical/ NSA's Puzzle Periodical]
| style="width: 50%;" |
* {{cite techreport |first=Juan Miguel |last=León-Rojas |title=Cuadros de emparejamiento para los ejercicios de las ediciones 5ª, 6ª, 7ª y 7ª global del libro Matemáticas discretas y sus aplicaciones, de Rosen, Capítulo 1 sobre Los Fundamentos: Lógica y Demostraciones (Edición bilingüe, español/inglés) |number=DA/HD – 703-01 |institution=KDEM (Knowledge Discovery Engineering and Management) |date=2017-01-31  |url=https://archive.org/details/KDEM.DA.HD.70301}}
* [https://zurbaran.unex.es/cloud/index.php/s/AWs67FyPv9P7z0v León-Rojas, J. M. (2009): Lógica proposicional (Notas incompletas de clase), v. 1.6.3-alpha2, rev. 17-02-2014. © CGL, CC0, CC BY.]
* [https://zurbaran.unex.es/cloud/index.php/s/Odtm7IIFElOLsmK León-Rojas, J. M. (2009): Lógica de primer orden (Notas incompletas de clase), v. 0.3.1-alpha0, rev. 17-02-2014. © CGL, CC0, CC BY.]
* [http://logicae.usal.es/mambo/index.php?option=com_summalogicaexxi&menu_task=Principal Summa Logicae] ([[:es:María Manzano Arjona|María Manzano]], Gustavo Santos, Belén Pérez Lancho, José Meseguer, Enrique Alonso, Huberto Marraud, Antonia Huertas, Dick de Jongh, Giovanna D'Agostino, Alberto Policriti)
* [http://sigma-fmo.blogspot.com.es/search?q=sofismas Sigma (2008) Los sofismas. In: Sigma, blog of the Department of Mathematics of the Facultad Multidisciplinaria de Occidente at the University of El Salvador. (© gratisOA).]
|}

====== <span id="Logic_To_find_out_more"></span>To find out more ======
<hr style="margin-left: 0px; width:50%;">
<ol style="list-style-type:circle;">
  <li>[[Portal:Mathematics]]</li>
  <li>[[Portal:Philosophy]]</li>
  <li>And more:</li>
  <ol style="list-style-type:circle;">
    <li>[[Outline of logic]]</li>
    <li>[[:Category:Concepts in logic]]</li>
    <li>[[Wikipedia:WikiProject Logic|WikiProject Logic]]</li>
    <li>[[Logic alphabet]]</li>
    <li>[http://us.metamath.org/ Metamath]. [http://us.metamath.org/copyright.html#pd © Public domain (with some exceptions)]</li>
    <li>[[Equational logic]]; for instance, chapter 5 (Equational Logic: Part 1) from Backhouse, Roland, ''Program Construction. The Correct Way'', 2002.</li>
  </ol>
  <li>And even more:</li>
  <ol style="list-style-type:circle;">
    <li>[[Index of logic articles]]</li>
    <li>[[List of logicians]]</li>
  </ol>
</ol>
{{Logic|state=collapsed}}
{{Classical logic|state=collapsed}}
{{Mathematical logic|state=collapsed}}
{{Metalogic|state=collapsed}}
{{Non-classical logic|state=collapsed}}
{{Philosophical logic|state=collapsed}}
{{Common logical symbols|state=collapsed}}
{{Logical connectives|state=collapsed}}
{{Logical paradoxes|state=collapsed}}
{{Logical truth|state=collapsed}}
|}
<!--
{|class="wikitable" style="border-style: solid; border-width: 1px 1px 1px 1px; width=100%;"
|
===== Logic =====
<hr/>
* [[Logic]]
* [[Mathematical logic]]
* [[Propositional calculus]]
* [[Truth table|Truth tables]]
* [[Inference rules]]
* [[Conjunctive normal form]]
* [[Disjunctive normal form]]
* [[First-order logic]]
* [[Method of analytic tableaux|Semantic tableaux]]
<hr width="50%"/>
To find out more:
* [[Outline of logic]]
<hr />
====== External links ======
* [http://cala.unex.es/cala/epistemowikia/index.php/Archivo:KDEM.DA.HD.703-01.pdf KDEM.DA.HD.703-01.pdf (descargar / download)] (emparejamiento de ejercicios de las ediciones 5ª, 6ª, 7ª y 7ª Global del Rosen: Los fundamentos: lógica y demostraciones / The foundations: logic and proofs
* [http://cala.unex.es/cala/epistemowikia/index.php/Archivo:AM.01-01-Logica-proposicional-v1.6.3alpha2-17-02-2014-CGL-CC0-CCBY.pdf León-Rojas, J. M. (2009): Lógica proposicional (Notas incompletas de clase), v. 1.6.3-alpha2, rev. 17-02-2014. © CGL, CC0, CC BY.]
* [http://cala.unex.es/cala/epistemowikia/index.php/Archivo:AM.01-02-Logica-de-primer-orden-v0.3.1alpha0-17-02-2014-CGL-CC0-CCBY.pdf León-Rojas, J. M. (2009): Lógica de primer orden (Notas incompletas de clase), v. 0.3.1-alpha0, rev. 17-02-2014. © CGL, CC0, CC BY.]
* [http://turner.faculty.swau.edu/mathematics/materialslibrary/truth/ Truth Tables (calculadora de tablas de verdad). © gratisOA.]
* [https://github.com/LonamiWebs/Py-Utils/tree/master/logicmind Logicmind (calculadora de modelos, en línea de comandos). © MIT License.]
* [http://formal.cs.utah.edu:8080/pbl/PBL.php Tyler Sorensen's PBL, a Boolean algebra/propositional logic library written in Python]
* [http://www.umsu.de/logik/trees/ Tree Proof Generator (calculadora de tablas semánticas). © gratisOA.]
* [https://www.washingtonpost.com/posteverything/wp/2015/07/20/why-i-give-my-students-a-tragedy-of-the-commons-extra-credit-challenge/ Dylan Selterman: '''Why I give my students a 'tragedy of the commons' extra credit challenge.''' (''The Washington Post'', 2015). (© gratisOA). (En inglés / In English).] Relativo a esta noticia, en español, por ejemplo, en / Related to this news, in Spanish, for example, at: [http://www.europapress.es/desconecta/curiosity/noticia-pregunta-extra-ganar-puntos-examen-profesor-te-hara-pensar-20150716104226.html Europa Press: La pregunta extra para ganar puntos en un examen de este profesor te hará pensar.]
* [http://sigma-fmo.blogspot.com.es/search?q=sofismas Sigma (2008) Los sofismas. En: Sigma, blog del Departamento de Matemática de la Facultad Multidisciplinaria de Occidente de la Universidad de El Salvador. (© gratisOA).]
* [http://us.metamath.org/ Metamath]. [http://us.metamath.org/copyright.html#pd © Dominio público (con excepciones) / Public domain (with some exceptions)]
{{Logic|collapsed}}
{{Mathematical logic|collapsed}}
|}
-->
{| class="mw-collapsible wikitable {{#ifeq: {{{plegada|no}}}|sí|mw-collapsed|}}" width="100%" style="text-align:left; padding:0px; background-color:#F9F9F9;"
| align="left"   |

===== Sets, relations and functions =====
|-
| style="background-color:#F9F9F9; text-align:left; font-size:100%; padding:6px" |
{|
|[[File:Set partitions 4; Hasse; matrices.svg|200px]]
|<div id="toc_SetsRelationsFunctions" class="toc">
<div class="toctitle">
'''Contents'''
</div>
# [[#SetsRelationsFunctions_Key_concepts|Key concepts]]
# [[#SetsRelationsFunctions_Connections|Connections]]
# [[#SetsRelationsFunctions_Bibliography|Bibliography]]
# [[#SetsRelationsFunctions_Software|Software]]
# [[#SetsRelationsFunctions_Multimedia|Multimedia]]
# [[#SetsRelationsFunctions_To_find_out_more|To find out more]]
</div>
|}

====== <span id="SetsRelationsFunctions_Key_concepts"></span>Key concepts ======
<hr style="margin-left: 0px; width:50%;">

'''<div id="Sets">— Sets</div>'''
{{Main|Set (mathematics)}}
{{Main category|Set theory}}

<div style="margin-left: 1.5em; font-style: normal;">
'''- Concepts and definitions'''
* [[Naive set theory]]
* [[Element (mathematics)]]
* [[Extensional and intensional definitions]]
* [[Universal set]]
* [[Empty set]]
* [[Cardinal number]]
* [[Power set]]
</div>

<div style="margin-left: 1.5em; font-style: normal;">
'''- Relations'''
* [[Element (mathematics)#Notation and terminology|Set membership]]
* [[Indicator function]]
* [[Subset]] (inclusion relation)
* [[Equality (mathematics)#Equality in set theory|Equality in set theory]]
</div>

<div style="margin-left: 1.5em; font-style: normal;">
'''- Set operations and representation'''
* [[Algebra of sets]]
* [[Carroll diagram]]
* [[Euler diagram]]
* [[Venn diagram]]
* [[Union (set theory)]]
* [[Intersection (set theory)]]
* [[Disjoint sets]]
* [[Disjoint union]]
* [[Complement (set theory)]]
* [[Complement (set theory)#Relative complement|Set difference]]
* [[Symmetric difference]]
</div>

<div style="margin-left: 1.5em; font-style: normal;">
'''- Properties'''
* [[Union (set theory)#Algebraic properties|Of set union]]
* [[Intersection (set theory)#PAlgebraic properties|Of set intersection]]
* [[Union (set theory)#Cardinality|Cardinality of set union]] (see also: [[Inclusion–exclusion principle]])
* [[Complement (set theory)#Properties|Of absolute complement]] and [[Complement (set theory)#Properties 2|Of relative complement]]
* [[Complement (set theory)#Properties 2|Of set difference]]
* [[Symmetric difference#Properties|Of symmetric difference]]
* [[De Morgan's laws#Set theory and Boolean algebra|De Morgan's laws]]
</div>

<div style="margin-left: 1.5em; font-style: normal;">
'''- Partition and cover'''
* [[Partition of a set]]
* [[Cover (topology)]]
</div>

<div style="margin-left: 1.5em; font-style: normal;">
'''- Cartesian product'''
* [[Cartesian product]]
* [[Cartesian product#n-ary Cartesian power|n-ary Cartesian power]]
* [[Ordered pair]]
* [[Tuple]]
</div>

'''<div id="Relations">— Relations</div>'''
{{main|Binary relation}}
{{main category|Mathematical relations}}
{{see also|Finitary relation}}

<div style="margin-left: 1.5em; font-style: normal;">
'''- Concepts and definitions'''
* [[Arity]]
* [[n-ary relation]]
* [[Logical matrix]]
* [[Converse relation]]
* [[Complement (set theory)#Complementary relation|Complementary relation]]
</div>

<div style="margin-left: 1.5em; font-style: normal;">
'''- Representation'''
* [[Cartesian coordinate system]]
* [[Mathematical correspondence]] <sup>([[:es:Correspondencia matemática|es]]; [https://www.encyclopediaofmath.org/index.php/Correspondence ref])</sup> (graphical representation using sets and arrows)
* [[Adjacency matrix]]
* [[Incidence matrix]]
* [[Directed graph]]
</div>

<div style="margin-left: 1.5em; font-style: normal;">
'''- Outstanding properties'''
* [[Reflexive relation]]
* [[Irreflexive relation]]
* [[Coreflexive relation]]
* [[Quasi-reflexive relation]]
* [[Symmetric relation]]
* [[Asymmetric relation]]
* [[Antisymmetric relation]]
* [[Transitive relation]]
* {{noredirect|Intransitive relation|{{color|#BA0000|Intransitive relation}}}} <sup>([[:es:Relación intransitiva|es]])</sup>
* {{noredirect|Antitransitive relation|{{color|#BA0000|Antitransitive relation}}}}
* [[Quasitransitive relation]]
* [[Euclidean relation]]
* [[Serial relation]]
* [[Trichotomy (mathematics)|Trichotomous relation]]
* [[Connex relation|Semiconnex relation]] (Complete relation)
* [[Connex relation]] (Total, linear, or strong complete relation)
* [[Circular relation]]
</div>

<div style="margin-left: 1.5em; font-style: normal;">
'''- Outstanding closures'''
* [[Closure (mathematics)]]
* [[Closure (mathematics)#Binary relation closures|Binary relation closures]]
* [[Reflexive closure]]
* [[Symmetric closure]]
* [[Transitive closure]]
</div>

<div style="margin-left: 1.5em; font-style: normal;">
'''- Operations'''
* [[Union of relations]]
* [[Intersection of relations]]
* [[Composition of relations]]
</div>

<div style="margin-left: 1.5em; font-style: normal;">
'''- Outstanding types'''

<div style="margin-left: 2.0em; font-style: normal;">
'''<div id="Equivalencerelations">- Equivalence relations</div>'''
* [[Partial equivalence relation]]
* [[Equivalence relation]]
</div>

<div style="margin-left: 2.0em; font-style: normal;">
'''<div id="Tolerancerelations">- Tolerance relations</div>'''
* [[Dependency relation]]
* [[Tolerance relation]]
</div>

<div style="margin-left: 2.0em; font-style: normal;">
'''<div id="Orderings">- Orderings</div>'''
* [[Preorder]]
* [[Weak ordering#Total preorders|Total preorders]]
* [[Partially ordered set]]
* [[Chain (poset)]]
* [[Ascending chain condition|Ascending and descending chain conditions]]
* [[Antichain]]
* [[Lexicographical order]]
* [[Product order]]
* [[Covering relation]]
* [[Transitive reduction]]
* [[Hasse diagram]]
* [[Total order]]
* [[Dense order]]

<div style="margin-left: 2.5em; font-style: normal;">
'''<div id="Outstandingelements">- Outstanding elements</div>'''
* [[Greatest and least elements]]
* [[Maximal and minimal elements]]
* [[Upper and lower bounds]]
* [[Infimum and supremum]]
* [[Maxima and minima#In relation to sets|Minimum and maximum]]
* See also: ''[[:es:Anexo:Extremos de conjunto acotado|Anexo:Extremos de conjunto acotado]]'' (some examples of extrema of bounded sets, on the Spanish Wikipedia)
</div>
* [[Contour set]]
* [[Bounded set#Boundedness in order theory|Boundedness in order theory]]
* [[Lattice (order)]]
</div>

<div style="margin-left: 2.0em; font-style: normal;">
'''<div id="Indiferenceandpreferencerelations">- Indiference and preference relations</div>'''
* [[Preference]]
</div>

<div style="margin-left: 2.0em; font-style: normal;">
'''<div id="Wellorder">- Well order</div>'''
* [[Well-order]]
* [[Well-founded relation]]
* [[Axiom of choice]]
* [[Zermelo's well-ordering theorem]] (Note: The fact that every set may be well ordered, is what most people apparently also call 'Well-ordering principle' (WOP); ref., e.g.: Bagaria, Joan, 'Set Theory', The Stanford Encyclopedia of Philosophy (Fall 2019 Edition), Edward N. Zalta (ed.), URL = <https://plato.stanford.edu/archives/fall2019/entries/set-theory/>, and so many others; what Wikipedia and Wolfram MathWorld and others call WOP is the particular case of being <math>\mathbb{N}</math> a well-ordered set by <; in short, the current situation in the literature is that WOP has two meanings, either Zermelo's Theorem or <math>\mathbb{N}</math> is well-orderd by <).
* [[Zorn's lemma]]
</div>

</div>

'''<div id="Functions">— Functions</div>'''
{{Main|Function (mathematics)|Map (mathematics)}}
{{Main category|Functions and mappings}}
<div style="margin-left: 1.5em; font-style: normal;">
'''- Concepts and definitions'''
* [[Mathematical correspondence]] <sup>([[:es:Correspondencia matemática|es]]; [https://www.encyclopediaofmath.org/index.php/Correspondence ref])</sup> (graphical representation using sets and arrows)
* [[Domain of a function]]
* [[Partial function]]
* [[Partial function|Total function]] (mapping or map [''aplicación'', in Spanish]; see: [[Map (mathematics)]])
* [[Codomain]]
* [[Image (mathematics)]]
* [[Range (mathematics)]]
* [[Restriction (mathematics)]]
* [[Identity function]]
* {{no redirect|Discrete function|{{color|#BA0000|Discrete function}}}} <sup>([[:es:Función discreta|es]])</sup>
* [[Floor and ceiling functions]]
* [[Graph of a function]]
</div>

<div style="margin-left: 1.5em; font-style: normal;">
'''- Outstanding types'''
* [[Bijection, injection and surjection]]
* [[Injective function]]
* [[Surjective function]]
* [[Bijection]]
* [[Gallery of mathematical correspondences]] <sup>([[:es:Galería de correspondencias matemáticas|es]])</sup>
</div>

<div style="margin-left: 1.5em; font-style: normal;">
'''- Operations'''
* [[Function composition]]
* [[Inverse function]]
* {{no redirect|Inverse image|{{color|#BA0000|Inverse image}}}} <sup>([[:es:Imagen inversa|es]]; [[:pt:Imagem recíproca|pt]]; [[:fr:Image réciproque|fr]]; [[:it:Controimmagine|it]])</sup> (see also: [[Image (mathematics)#Inverse image]] and [[Inverse function#Preimages]])
</div>

<div style="margin-left: 1.5em; font-style: normal;">
'''- Multisets'''
* [[Multiset]]
* [[Multiplicity (mathematics)]]
</div>

'''<div id="Paradoxes">— Paradoxes</div>'''
* [[Russell's paradox]] (see also: [[Class (set theory)]])
* [[Interesting number paradox]]

====== <span id="SetsRelationsFunctions_Connections"></span>Connections ======
<hr style="margin-left: 0px; width:50%;">
'''— Extensive systems'''
{{Main|Extensive system}}
{{see also|Comparative system|Ordinal scale|Proportional scale}}
{{Hatnote|Worth-read article: [[Jesús Mosterín|Mosterín, Jesús]] (1978, January). [https://www.investigacionyciencia.es/revistas/investigacion-y-ciencia/riego-gota-a-gota-16/la-estructura-de-los-conceptos-cientficos-747 La estructura de los conceptos científicos. Investigación y Ciencia, 16, 82-93].}}
{{Hatnote|Worth-read chapter: [[Jesús Mosterín|Mosterín, Jesús]] (1993). Los conceptos científicos. In: Ulises Moulines, Carlos (ed.). La ciencia: estructura y desarrollo. Madrid:Trota, Consejo Superior de Investigaciones Científicas and Sociedad Estatal Quinto Centenario. Pages 15-30. ISBN 84-87699-72-3}}
'''— Entity-relationship model'''
{{Main|Entity-relationship model}}
{{see also|Relational database|Relational model|Relational algebra|Relational calculus|Codd's theorem}}
<!--
{| border=0 align=center cellspacing=20
|-
| <center>{{pp-book-cover</center>
 |title=
 |subtitle=
 |cover-image=Set partitions 4; Hasse; matrices.svg
 |cover-color=#76b3df
}} Libro wiki (itinerario en Wikipedia)<br />¡Pulse en el título, no en la imagen de portada!</center>
|}
-->

====== <span id="SetsRelationsFunctions_Bibliography"></span>Bibliography: theory and proposed and solved exercises ======
<hr style="margin-left: 0px; width:50%;">
{| style="width: 100%;"
|In English: || In Spanish:
|- style="vertical-align:top;"
| style="width: 50%;" |
* —¤— Kenneth A. Rosen. <i>Discrete mathematics and its applications</i>. 7th edition. (Sections 2.1, 2.2, 2.3, Chapter 9 and related exercises). McGraw-Hill, New York, New York, United States, 2012, {{ISBN|978-0-07-338309-5}}
| style="width: 50%;" |
* A. M. Díaz Hernández. Material from «Curso 0: Matemáticas». Part: Conjuntos, correspondencias y relaciones. OCW UNED. (Theory and exercises). 2010. © CC BY-NC-ND. http://ocw.innova.uned.es/matematicas-industriales/contenidos/pdf/tema6.pdf
* P. J. Herrero Piñeyro. Material from «Topología de Espacios Métricos». Part: Conjuntos, aplicaciones y números. OCW UM. 2010. © CC BY-NC-SA. http://ocw.um.es/ciencias/topologia-de-espacios-metricos/material-de-clase-1/cap-0-conj-apl.pdf
* —¤— [https://zurbaran.unex.es/cloud/index.php/s/1AXsF0VpnJre8XL J. M. León Rojas (2009): Conjuntos, aplicaciones y relaciones (Notas incompletas de clase), v. 1.4.1-alpha2, rev. 23-02-2014. © CGL, CC0, CC BY.]
* —¤— Kenneth A. Rosen. <i>Matemática discreta y sus aplicaciones</i>. 5th edition. (Sections 1.6, 1.7, 1.8, Chapter 7 and related exercises). McGraw-Hill/Interamericana de España, S.A.U., Aravaca (Madrid), Madrid, Spain, 2004, {{ISBN|84-481-4073-7}}
|}

====== <span id="SetsRelationsFunctions_Software"></span>Software ======
<hr style="margin-left: 0px; width:50%;">
{| style="width: 100%;"
|In English: || In Spanish:
|- style="vertical-align:top;"
| style="width: 50%;" |
* [https://www.wolframalpha.com/examples/SetTheory.html WolframAlpha (Examples, Mathematics, Set Theory)]
| style="width: 50%;" |
|}

====== <span id="SetsRelationsFunctions_Multimedia"></span>[[File:Multimedia-icon.svg|32px|link=]] Multimedia ======
<hr style="margin-left: 0px; width:50%;">
{| style="width: 100%;"
|In English: || In Spanish:
|- style="vertical-align:top;"
| style="width: 50%;" |
'''— Sets'''
* {{cite web
 |url=https://www.youtube.com/watch?v=fdm2e0loOfI
 |title=Section 2.1 The Basics of Sets
 |last=Baugher
 |first=Greg A.
 |department=Department of Math/Science/Informatics, Penfield College, Mercer University
 |location=Georgia (US-GA), USA
 |format=Video
 |publisher=
 |date=
 |website=
 |access-date=
 |quote= }}
* {{cite web
 |url=https://www.youtube.com/watch?v=E4AtYIM8HB0
 |title=Section 2.2 Set Operations
 |last=Baugher
 |first=Greg A.
 |department=Department of Math/Science/Informatics, Penfield College, Mercer University
 |location=Georgia (US-GA), USA
 |format=Video
 |publisher=
 |date=
 |website=
 |access-date=
 |quote= }}
* {{cite web
 |url=https://www.youtube.com/watch?v=9AUCdsmBGmA
 |title=Lecture 10 - Sets
 |last=Krithivasan
 |first=Kamala
 |format=Video
 |publisher=Indian Institute of Technology Madras
 |date=
 |website=
 |access-date=
 |quote= }}
* {{cite web
 |url=https://www.youtube.com/watch?v=7k4Di5u-oUU
 |title=Lecture 12 - Set operations on strings over an alphabet
 |last=Krithivasan
 |first=Kamala
 |format=Video
 |publisher=Indian Institute of Technology Madras
 |date=
 |website=
 |access-date=
 |quote= }}
* {{cite web
 |url=ftp://ftp.aduni.org/videos/02_lect_03.rm
 
 |title=Sets
 |last=Simonson
 |first=Shai
 |format=Video
 |publisher=ArsDigita University
 |date=
 |website=
 |access-date=
 |quote= }} ([https://www.youtube.com/watch?v=t3XdRbPNtdg Alternate URL])
'''— Relations'''
* {{cite web
 |url=https://www.youtube.com/watch?v=_BIKq9Xo_5A
 |title=Lecture 13 - Relations
 |last=Krithivasan
 |first=Kamala
 |format=Video
 |publisher=Indian Institute of Technology Madras
 |date=
 |website=
 |access-date=
 |quote= }}
* {{cite web
 |url=https://www.youtube.com/watch?v=6WGWFwgXhd4
 |title=Lecture 18 - Special properties of relations
 |last=Krithivasan
 |first=Kamala
 |format=Video
 |publisher=Indian Institute of Technology Madras
 |date=
 |website=
 |access-date=
 |quote= }}
* {{cite web
 |url=https://www.youtube.com/watch?v=xuhZSt4Rw0k
 |title=Lecture 19 - Closure of relations
 |last=Krithivasan
 |first=Kamala
 |format=Video
 |publisher=Indian Institute of Technology Madras
 |date=
 |website=
 |access-date=
 |quote= }}
* {{cite web
 |url=https://www.youtube.com/watch?v=EN0JhX7phZo
 |title=Lecture 20 - Closure of relations (contd.)
 |last=Krithivasan
 |first=Kamala
 |format=Video
 |publisher=Indian Institute of Technology Madras
 |date=
 |website=
 |access-date=
 |quote= }}
* {{cite web
 |url=https://www.youtube.com/watch?v=buXrxLfaF94
 |title=Lecture 21 - Order relations
 |last=Krithivasan
 |first=Kamala
 |format=Video
 |publisher=Indian Institute of Technology Madras
 |date=
 |website=
 |access-date=
 |quote= }}
* {{cite web
 |url=https://www.youtube.com/watch?v=Fk8nJjzohr8
 |title=Lecture 22 - Order relations and equivalence relations
 |last=Krithivasan
 |first=Kamala
 |format=Video
 |publisher=Indian Institute of Technology Madras
 |date=
 |website=
 |access-date=
 |quote= }}
* {{cite web
 |url=https://www.youtube.com/watch?v=uOE7rSKTDbo
 |title=Lecture 23 - Equivalence relations and partitions
 |last=Krithivasan
 |first=Kamala
 |format=Video
 |publisher=Indian Institute of Technology Madras
 |date=
 |website=
 |access-date=
 |quote= }}
* {{cite web
 |url=ftp://ftp.aduni.org/videos/02_lect_17.rm
 
 |title=Equivalence Relations and Partial Orders
 |last=Simonson
 |first=Shai
 |format=Video
 |publisher=ArsDigita University
 |date=
 |website=
 |access-date=
 |quote= }} ([https://www.youtube.com/watch?v=xiwFOcGGFPs&index=17&list=PL__5ik6vMY50HNfWppaGxQydlSMTGQWzC Otra URL])
'''— Functions'''
* {{cite web
 |url=https://www.youtube.com/watch?v=dmRoKI3R9ic
 |title=Section 2.3 Functions
 |last=Baugher
 |first=Greg A.
 |department=Department of Math/Science/Informatics, Penfield College, Mercer University
 |location=Georgia (US-GA), EUA
 |format=Vídeo
 |publisher=
 |date=
 |website=
 |access-date=
 |quote= }}
* {{cite web
 |url=https://www.youtube.com/watch?v=7cTWea9YAJE
 |title=Lecture 24 - Functions
 |last=Krithivasan
 |first=Kamala
 |format=Video
 |publisher=Indian Institute of Technology Madras
 |date=
 |website=
 |access-date=
 |quote= }}
* {{cite web
 |url=https://www.youtube.com/watch?v=695iAm935cY
 |title=Lecture 25 - Functions (Contd.)
 |last=Krithivasan
 |first=Kamala
 |format=Video
 |publisher=Indian Institute of Technology Madras
 |date=
 |website=
 |access-date=
 |quote= }}
* {{cite web
 |url=https://www.youtube.com/watch?v=ZECJHfsf4Vs
 |title=Lecture 26 - Functions (Contd.)
 |last=Krithivasan
 |first=Kamala
 |format=Video
 |publisher=Indian Institute of Technology Madras
 |date=
 |website=
 |access-date=
 |quote= }}
| style="width: 50%;" |
'''— Sets'''
* {{cite web |last=Jordán Lluch |first=Cristina |title=Conjuntos y notación |format=Video |publisher=Universidad Politécnica de Valencia (UPV) |url=https://www.youtube.com/watch?v=bVFRSpBrkAU |access-date=}}
* {{cite web |last=Jordán Lluch |first=Cristina |title=Diagramas de Venn e igualdad de conjuntos |format=Video |publisher=Universidad Politécnica de Valencia (UPV) |url=https://www.youtube.com/watch?v=mKg6AkUJD7s |access-date=}}
* {{cite web |last=Jordán Lluch |first=Cristina |title=Subconjuntos de conjuntos |format=Video |publisher=Universidad Politécnica de Valencia (UPV) |url=https://www.youtube.com/watch?v=DiVLOZe7aFk |access-date=}}
* {{cite web |last=Jordán Lluch |first=Cristina |title=Conjunto de partes de un conjunto |format=Video |publisher=Universidad Politécnica de Valencia (UPV) |url=https://www.youtube.com/watch?v=_--0y8lWheY |access-date=}}
* {{cite web |last=Jordán Lluch |first=Cristina |title=Unión de conjuntos |format=Video |publisher=Universidad Politécnica de Valencia (UPV) |url=https://www.youtube.com/watch?v=7eTEeQ3eA-s |access-date=}}
* {{cite web |last=Jordán Lluch |first=Cristina |title=Intersección de conjuntos |format=Video |publisher=Universidad Politécnica de Valencia (UPV) |url=https://www.youtube.com/watch?v=O3fa1DqknQQ |access-date=}}
* {{cite web |last=Jordán Lluch |first=Cristina |title=Diferencia de conjuntos |format=Video |publisher=Universidad Politécnica de Valencia (UPV) |url=https://www.youtube.com/watch?v=HCwXfCGV7TI |access-date=}}
* {{cite web |last=Jordán Lluch |first=Cristina |title=Complementario de un conjunto |format=Video |publisher=Universidad Politécnica de Valencia (UPV) |url=https://www.youtube.com/watch?v=e9SjaS_Wgq8 |access-date=}}
* {{cite web |last=Jordán Lluch |first=Cristina |title=Diferencia simétrica de conjuntos |format=Video |publisher=Universidad Politécnica de Valencia (UPV) |url=https://www.youtube.com/watch?v=414ZK_Lf12w |access-date=}}
* {{cite web |last=Jordán Lluch |first=Cristina |title=Propiedades booleanas de la teoría de conjuntos |format=Video |publisher=Universidad Politécnica de Valencia (UPV) |url=https://www.youtube.com/watch?v=d8PMEjjwFxg |access-date=}}
* {{cite web |last=Jordán Lluch |first=Cristina |title=Teorema de inclusión - exclusión en teoría de conjuntos |format=Video |publisher=Universidad Politécnica de Valencia (UPV) |url=https://www.youtube.com/watch?v=FXsnnS7KgvQ |access-date=}}
* {{cite web |last=Jordán Lluch |first=Cristina |title=Simplificación de expresiones en teoría de conjuntos |format=Video |publisher=Universidad Politécnica de Valencia (UPV) |url=https://www.youtube.com/watch?v=5tA8zHlsor4 |access-date=}}
* {{cite web |last=Jordán Lluch |first=Cristina |title=¿Cómo justifico que una igualdad de conjuntos no es valida? |format=Video |publisher=Universidad Politécnica de Valencia (UPV) |url=https://www.youtube.com/watch?v=eHWOfjm2slw |access-date=}}
* {{cite web |last=Jordán Lluch |first=Cristina |title=Recubrimiento y partición de un conjunto |format=Video |publisher=Universidad Politécnica de Valencia (UPV) |url=https://www.youtube.com/watch?v=cNus6wILCvQ |access-date=}}
* {{cite web |last=Jordán Lluch |first=Cristina |title=Producto cartesiano de conjuntos |format=Video |publisher=Universidad Politécnica de Valencia (UPV) |url=https://www.youtube.com/watch?v=4roKSnLxIgc |access-date=}}
* {{cite web |last=Soto Espinosa |first=Jesús |title=Conjuntos finitos |format=Vídeo |publisher=Escuela Politécnica Superior, Universidad Católica San Antonio de Murcia (UCAM) |location=Guadalupe, Murcia, Región de Murcia (ES-MC), España |url=https://www.youtube.com/watch?v=wrpAuwh8wuo&index=59&list=PL2qtqrBO8xujlH2bo2UPMQ3m9Hj9yEBur |access-date=}}
* {{cite web |last=Soto Espinosa |first=Jesús |title=Ejercicios |format=Vídeo |publisher=Escuela Politécnica Superior, Universidad Católica San Antonio de Murcia (UCAM) |location=Guadalupe, Murcia, Región de Murcia (ES-MC), España |url=https://www.youtube.com/watch?v=GnF0-N--ckI&index=67&list=PL2qtqrBO8xujlH2bo2UPMQ3m9Hj9yEBur |access-date=}} (Ejercicio 1).
'''— Relations'''
* {{cite web |last=Jordán Lluch |first=Cristina |title=¿Por qué estudiar relaciones binarias? |format=Video |publisher=Universidad Politécnica de Valencia (UPV) |url=https://www.youtube.com/watch?v=eZfytGhk5kY |access-date=}}
* {{cite web |last=Jordán Lluch |first=Cristina |title=Relaciones binarias. Notación |format=Video |publisher=Universidad Politécnica de Valencia (UPV) |url=https://www.youtube.com/watch?v=xch45chcp7s |access-date=}}
* {{cite web |last=Jordán Lluch |first=Cristina |title=Propiedades reflexiva y simétrica de relaciones binarias. Definiciones y caracterización |format=Video |publisher=Universidad Politécnica de Valencia (UPV) |url=https://www.youtube.com/watch?v=BLoPz12Jq_0 |access-date=}}
* {{cite web |last=Jordán Lluch |first=Cristina |title=Operaciones entre relaciones binarias: Unión, intersección, diferencia, complementario |format=Video |publisher=Universidad Politécnica de Valencia (UPV) |url=https://www.youtube.com/watch?v=brTNWYGSlaE |access-date=}}
* {{cite web |last=Jordán Lluch |first=Cristina |title=Operaciones entre relaciones binarias. Ejemplo en forma implícita |format=Video |publisher=Universidad Politécnica de Valencia (UPV) |url=https://www.youtube.com/watch?v=uA5o-IXKckg |access-date=}}
* {{cite web |last=Jordán Lluch |first=Cristina |title=Relaciones binarias. Representación gráfica |format=Video |publisher=Universidad Politécnica de Valencia (UPV) |url=https://www.youtube.com/watch?v=d3gw322Dj0g |access-date=}}
* {{cite web |last=Jordán Lluch |first=Cristina |title=Relaciones binarias. Representación matricial |format=Video |publisher=Universidad Politécnica de Valencia (UPV) |url=https://www.youtube.com/watch?v=eceQOOOhQZY |access-date=}}
* {{cite web |last=Jordán Lluch |first=Cristina |title=Inversa de una relacion binaria |format=Video |publisher=Universidad Politécnica de Valencia (UPV) |url=https://www.youtube.com/watch?v=w_5KS7okLEs |access-date=}}
* {{cite web |last=Jordán Lluch |first=Cristina |title=Relaciones binarias de equivalencia |format=Video |publisher=Universidad Politécnica de Valencia (UPV) |url=https://www.youtube.com/watch?v=IJFhuwNWzCM |access-date=}}
* {{cite web |last=Jordán Lluch |first=Cristina |title=Ejemplos sencillos de relaciones binarias de equivalencia |format=Video |publisher=Universidad Politécnica de Valencia (UPV) |url=https://www.youtube.com/watch?v=zY8Lhm09T2A |access-date=}}
* {{cite web |last=Jordán Lluch |first=Cristina |title=Cotas superiores e inferiores de una relacion binaria |format=Video |publisher=Universidad Politécnica de Valencia (UPV) |url=https://www.youtube.com/watch?v=MKOC0pT-FOU |access-date=}}
* {{cite web |last=Jordán Lluch |first=Cristina |title=Maximos, minimos, maximales, minimales de una relacion binaria de orden |format=Video |publisher=Universidad Politécnica de Valencia (UPV) |url=https://www.youtube.com/watch?v=u3kkrV53s7Q |access-date=}}
* {{cite web |last=Jordán Lluch |first=Cristina |title=Operaciones entre relaciones binarias: Ejemplos en contexto real |format=Video |publisher=Universidad Politécnica de Valencia (UPV) |url=https://www.youtube.com/watch?v=14jU_veyJ4E |access-date=}}
'''— Functions'''
* {{cite web |last=Soto Espinosa |first=Jesús |title=Aplicaciones entre conjuntos finitos |format=Vídeo |publisher=Escuela Politécnica Superior, Universidad Católica San Antonio de Murcia (UCAM) |location=Guadalupe, Murcia, Región de Murcia (ES-MC), España |url=https://www.youtube.com/watch?v=HPynhpbEzjw&index=58&list=PL2qtqrBO8xujlH2bo2UPMQ3m9Hj9yEBur |access-date=}}
* {{cite web |last=Soto Espinosa |first=Jesús |title=Ejercicios |format=Vídeo |publisher=Escuela Politécnica Superior, Universidad Católica San Antonio de Murcia (UCAM) |location=Guadalupe, Murcia, Región de Murcia (ES-MC), España |url=https://www.youtube.com/watch?v=GnF0-N--ckI&index=67&list=PL2qtqrBO8xujlH2bo2UPMQ3m9Hj9yEBur |access-date=}} (Ejercicio 2).
* {{cite web |last=Soto Espinosa |first=Jesús |title=Aplicaciones. Ejercicio 1 |format=Vídeo |publisher=Escuela Politécnica Superior, Universidad Católica San Antonio de Murcia (UCAM) |location=Guadalupe, Murcia, Región de Murcia (ES-MC), España |url=https://www.youtube.com/watch?v=5fvoXlavFSY |access-date=}}
* {{cite web |last=Soto Espinosa |first=Jesús |title=Aplicaciones. Ejercicio 2 |format=Vídeo |publisher=Escuela Politécnica Superior, Universidad Católica San Antonio de Murcia (UCAM) |location=Guadalupe, Murcia, Región de Murcia (ES-MC), España |url=https://www.youtube.com/watch?v=86wGtnflQBk |access-date=}}
* {{cite web |last=Soto Espinosa |first=Jesús |title=Aplicaciones. Ejercicio 3 |format=Vídeo |publisher=Escuela Politécnica Superior, Universidad Católica San Antonio de Murcia (UCAM) |location=Guadalupe, Murcia, Región de Murcia (ES-MC), España |url=https://www.youtube.com/watch?v=z8Ojz1j_hL8 |access-date=}}
|}

====== <span id="SetsRelationsFunctions_To_find_out_more"></span>To find out more ======
<hr style="margin-left: 0px; width:50%;">
<ol style="list-style-type:circle;">
  <li>[[Portal:Mathematics]]</li>
</ol>
{{Set theory|collapsed}}
{{Binary relations|collapsed}}
|}
{| class="mw-collapsible wikitable {{#ifeq: {{{plegada|no}}}|sí|mw-collapsed|}}" width="100%" style="text-align:left; padding:0px; background-color:#F9F9F9;"
| align="left"   |

===== Cardinality, induction and recursion =====
|-
| style="background-color:#F9F9F9; text-align:left; font-size:100%; padding:6px" |
{|
|[[File:Von Neumann Hierarchy.svg|200px]]
|<div id="toc_Cardinality" class="toc">
<div class="toctitle">
'''Contents'''
</div>
# [[#Cardinality_Key_concepts|Key concepts]]
# [[#Cardinality_Bibliography|Bibliography]]
# [[#Cardinality_Software|Software]]
# [[#Cardinality_Multimedia|Multimedia]]
# [[#Cardinality_See_also|See also]]
# [[#Cardinality_To_find_out_more|To find out more]]
</div>
|}

====== <span id="Cardinality_Key_concepts"></span>Key concepts ======
<hr style="margin-left: 0px; width:50%;">
'''— Cardinality'''
{{Main|Cardinality|Cardinal number}}
{{Main category|Cardinal numbers}}

<div style="margin-left: 1.5em; font-style: normal;">
'''- Infinite sets'''
* [[Actual infinity|Potential infinity and actual infinity]]
* [[Infinity]]
* [[List of paradoxes#Infinity and infinitesimals|Paradoxes about infinity and infinitesimals]]
<div style="margin-left: 2em; font-style: normal;">
* [[Galileo's paradox]]
* [[Hilbert's paradox of the Grand Hotel]]
* [[Ross–Littlewood paradox]] (see also: [[Supertask]] and [[#Cardinality_Connections|Connections: Hypercomputability]])
</div>
* [[Equinumerosity]]
* [[Finite set]]
* [[Infinite set]]
</div>

<div style="margin-left: 1.5em; font-style: normal;">
'''- <math>\mathbb{N}</math>, <math>\mathbb{Z}</math> and <math>\mathbb{Q}</math> are countable sets'''
* [[Peano axioms]]
* [[Countable set|Countably infinite set]] (Denumerable set)
* [[Enumeration]]
* [[Aleph_number#Aleph-naught|Aleph-sub-zero]]
</div>

<div style="margin-left: 1.5em; font-style: normal;">
'''- <math>\mathbb{R}</math> is an uncountable set'''
* [[Cantor's diagonal argument]]
* [[Uncountable set]] (Non-denumerable set)
</div>

<div style="margin-left: 1.5em; font-style: normal;">
'''- Cantor's Theorem and the Continuum Hypothesis'''
* [[Cantor's theorem]]
* [[Aleph number]]
* (see also: ''[[:es:Número cardinal (teoría de conjuntos)|Número cardinal]]'' [Cardinal number (set theory)] on the Spanish Wikipedia)
* [[Transfinite number]]
* [[Continuum hypothesis]]
</div>

'''<div id="Induction">— Induction</div>'''
* [[Mathematical induction#Description|Weak induction]]
* [[Mathematical_induction#Complete_induction|Strong induction]]
* [[Structural induction]]

'''<div id="Recursion">— Recursion</div>'''
* [[Recursion]]
* [[Recursion (computer science)]]
* [[Structural induction and recursive definitions]] <sup>([[:es:Inducción estructural y definiciones recursivas|es]])</sup>

====== <span id="Cardinality_Connections"></span>Connections ======
<hr style="margin-left: 0px; width:50%;">
'''— Hypercomputability'''
{{Main|Hypercomputation}}
* [[Supertask]]
* [[Thomson's lamp]]
* [[Zeno machine]]
{{see also|Transcomputational problem|Quantum computing|Matrioshka brain}}
<!--
{| border=0 align=center cellspacing=20
|-
| <center>{{pp-book-cover</center>
 |title=
 |subtitle=
 |cover-image=Von Neumann Hierarchy.svg
 |cover-color=#76b3df
}} Libro wiki (itinerario en Wikipedia)<br />¡Pulse en el título, no en la imagen de portada!</center>
|}
-->

====== <span id="Cardinality_Bibliography"></span>Bibliography: theory and proposed and solved exercises ======
<hr style="margin-left: 0px; width:50%;">
{| style="width: 100%;"
|In English: || In Spanish:
|- style="vertical-align:top;"
| style="width: 50%;" |
* —¤— Kenneth A. Rosen. <i>Discrete mathematics and its applications</i>. 7th edition. (Sections 2.5, 5.1, 5.2, 5.3, Chapter 9 and related exercises). McGraw-Hill, New York, New York, United States, 2012, {{ISBN|978-0-07-338309-5}}
| style="width: 50%;" |
* A. M. Díaz Hernández. Material from «Curso 0: Matemáticas». Part: Conjuntos, correspondencias y relaciones. OCW UNED. (Theory and exercises). 2010. © CC BY-NC-ND. http://ocw.innova.uned.es/matematicas-industriales/contenidos/pdf/tema6.pdf
* P. J. Herrero Piñeyro. Material from «Topología de Espacios Métricos». Part: Conjuntos, aplicaciones y números. OCW UM. 2010. © CC BY-NC-SA. http://ocw.um.es/ciencias/topologia-de-espacios-metricos/material-de-clase-1/cap-0-conj-apl.pdf
* —¤— [https://zurbaran.unex.es/cloud/index.php/s/zuVfX2tZ9Vq30OK J. M. León Rojas (2001): Cardinalidad (Notas incompletas de clase), v. 1.0-alpha2, rev. 24-02-2014. © CGL, CC0, CC BY.]
* —¤— Kenneth A. Rosen. <i>Matemática discreta y sus aplicaciones</i>. 5th edition. (Sections 3.2.5, 3.3, 3.4, Chapter 7 and related exercises). McGraw-Hill/Interamericana de España, S.A.U., Aravaca (Madrid), Madrid, Spain, 2004, {{ISBN|84-481-4073-7}}
|}

====== <span id="Cardinality_Software"></span>Software ======
<hr style="margin-left: 0px; width:50%;">
{| style="width: 100%;"
|In English: || In Spanish:
|- style="vertical-align:top;"
| style="width: 50%;" |
* [https://www.wolframalpha.com/examples/TransfiniteNumbers.html WolframAlpha (Examples, Mathematics, Transfinite Numbers))]
| style="width: 50%;" |
|}

====== <span id="Cardinality_Multimedia"></span>[[File:Multimedia-icon.svg|32px|link=]] Multimedia ======
<hr style="margin-left: 0px; width:50%;">
{| style="width: 100%;"
|In English: || In Spanish:
|- style="vertical-align:top;"
| style="width: 50%;" |
* {{cite web
 |url=https://www.youtube.com/watch?v=Z-QJfMV2a94
 |title=Sections 5.1-5.2 Mathematical Induction and Strong Induction
 |last=Baugher
 |first=Greg A.
 |department=Department of Math/Science/Informatics, Penfield College, Mercer University
 |location=Georgia (US-GA), EUA
 |format=Vídeo
 |publisher=
 |date=
 |website=
 |access-date=
 |quote= }}
* {{cite web
 |url=https://www.youtube.com/watch?v=vTjUPrkD6_8
 |title=Sections 5.3-5.4 Recursive Definitions and Recursive Algorithms
 |last=Baugher
 |first=Greg A.
 |department=Department of Math/Science/Informatics, Penfield College, Mercer University
 |location=Georgia (US-GA), EUA
 |format=Vídeo
 |publisher=
 |date=
 |website=
 |access-date=
 |quote= }}
* {{cite web
 |url=https://www.youtube.com/watch?v=oU60TuGHxe0
 |title=Lecture 11 - Induction
 |last=Krithivasan
 |first=Kamala
 |format=Video
 |publisher=Indian Institute of Technology Madras
 |date=
 |website=
 |access-date=
 |quote= }}
* {{cite web
 |url=ftp://ftp.aduni.org/videos/02_lect_05.rm
 |title=Lecture 5 - Diagonalization, functions and sums review
 |last=Simonson
 |first=Shai
 |format=Video
 |publisher=ArsDigita University
 |date=
 |website=
 |access-date=
 |quote= }} ([https://www.youtube.com/watch?v=ZnGnuyRQU-M Another URL])
* {{cite web
 |url=ftp://ftp.aduni.org/videos/02_lect_10.rm
 |title=Lecture 10 - Mathematical induction
 |last=Simonson
 |first=Shai
 |format=Video
 |publisher=ArsDigita University
 |date=
 |website=
 |access-date=
 |quote= }} ([https://www.youtube.com/watch?v=QuOq50pB_yI&index=10&list=PL__5ik6vMY50HNfWppaGxQydlSMTGQWzC Another URL])
| style="width: 50%;" |
* {{cite web |last=Jordán Lluch |first=Cristina |title=¿Qué se entiende por cardinal de un conjunto? |format=Video |publisher=Universidad Politécnica de Valencia (UPV) |url=https://www.youtube.com/watch?v=R3CDl4eIZjE |access-date=}}
* {{cite web |last=Jordán Lluch |first=Cristina |title=Principio de inducción |format=Video |publisher=Universidad Politécnica de Valencia (UPV) |url=https://www.youtube.com/watch?v=jKFSivPZQYo |access-date=}}
* {{cite web |last=Jordán Lluch |first=Cristina |title=Ejercicios de inducción |format=Video |publisher=Universidad Politécnica de Valencia (UPV) |url=https://www.youtube.com/watch?v=g7kJEnKo-IQ |access-date=}}
* {{cite web |last=Jordán Lluch |first=Cristina |title=La inducción como método para demostrar desigualdades |format=Video |publisher=Universidad Politécnica de Valencia (UPV) |url=https://www.youtube.com/watch?v=u4UvxQvnkKU |access-date=}}
* {{cite web |last=Jordán Lluch |first=Cristina |title=Ejemplo de desigualdad demostrada por inducción |format=Video |publisher=Universidad Politécnica de Valencia (UPV) |url=https://www.youtube.com/watch?v=YtfXYDac_To |access-date=}}
* {{cite web |last=Jordán Lluch |first=Cristina |title=Una conocida igualdad demostrada por inducción |format=Video |publisher=Universidad Politécnica de Valencia (UPV) |url=https://www.youtube.com/watch?v=n7G_IiH6iz0 |access-date=}}
|}

<!-- ====== <span id="Cardinality_See_also"></span>See also ======
<hr style="margin-left: 0px; width:50%;">
* <del>[http://cala.unex.es/cala/epistemowikia/index.php/Ejercicios_propuestos_sobre_cardinalidad_y_algunas_soluciones Ejercicios propuestos sobre cardinalidad y algunas soluciones] (Proposed exercises about cardinality and some solutions) (On Epistemowikia and in Spanish, for the time being)</del> -->

====== <span id="Cardinality_To_find_out_more"></span>To find out more ======
<hr style="margin-left: 0px; width:50%;">
<ol style="list-style-type:circle;">
  <li> Manuel José González Ortiz (2000). La hipótesis del continuo. ''Números 43-44'', artículo n. 63 (pp. 315-318). Sociedad Canaria "Isaac Newton" de Profesores de Matemáticas y Nivola Libros y Ediciones S.L. Disponible en: http://www.sinewton.org/numeros/index.php?option=com_content&view=article&id=72:volumen-43-septiembre-2000&catid=35:sumarios-webs&Itemid=66</li>
<!--  <li>Gaussianos. La hipótesis del continuo: del susto de Cantor a la prueba de Cohen. https://www.gaussianos.com/la-hipotesis-del-continuo-del-susto-de-cantor-a-la-prueba-de-cohen/</li> -->
  <li>Continuum hypothesis. ''Encyclopedia of Mathematics''. Disponible en: http://www.encyclopediaofmath.org/index.php?title=Continuum_hypothesis</li>
  <li>Koellner, Peter, "The Continuum Hypothesis", ''The Stanford Encyclopedia of Philosophy'' (Winter 2016 Edition), Edward N. Zalta (ed.). Disponible en: https://plato.stanford.edu/archives/win2016/entries/continuum-hypothesis/.</li>
  <li>The Continuum Hypothesis (la página web «oficial» de la hipótesis del continuo, en ''Infinity Ink'' [Nancy McGough, 1992]). Disponible en: http://www.ii.com/math/ch/</li>
  <li>[[Portal:Mathematics]]</li>
  <li>And more:</li>
  <ol style="list-style-type:circle;">
    * [[:Category:Set theory]]
  </ol>
</ol>
{{Set theory|collapsed}}
{{Binary relations|collapsed}}
|}
{| class="mw-collapsible wikitable {{#ifeq: {{{plegada|no}}}|sí|mw-collapsed|}}" width="100%" style="text-align:left; padding:0px; background-color:#F9F9F9;"
| align="left"   |

===== Algebraic structures =====
|-
| style="background-color:#F9F9F9; text-align:left; font-size:100%; padding:6px" |
{|
|[[File:Associativity of binary operations (without question marks).svg|200px]]
|<div id="toc_AlgebraicStructures" class="toc">
<div class="toctitle">
'''Contents'''
</div>
# [[#AlgebraicStructures_Key_concepts|Key concepts]]
# [[#AlgebraicStructures_Bibliography|Bibliography]]
# [[#AlgebraicStructures_Software|Software]]
# [[#AlgebraicStructures_Multimedia|Multimedia]]
# [[#AlgebraicStructures_To_find_out_more|To find out more]]
</div>
|}

====== <span id="AlgebraicStructures_Key_concepts"></span>Key concepts ======
<hr style="margin-left: 0px; width:50%;">
'''— Algebraic structures'''
{{Main|Algebraic structure}}
{{Main category|Algebraic structures}}
* [[Law of composition]] <sup>([[:es:Ley de composición|es]])</sup>
* [[Binary operation]]
* [[Binary operation#External binary operations|External binary operations]] (see also: [[External (mathematics)]])
* [[Mathematical structure]]
* [[Identity element]] (neutral element)
* [[Absorbing element]] (annihilating element)
* [[Inverse element]]
** [[Additive inverse]] (opposite number, sign change,  negation)
** [[Multiplicative inverse]] (reciprocal)

'''— Magma, semigroup and monoid'''
* [[Magma (algebra)|Magma]]
* [[Semigroup]] and [[Monoid]]
* [[Quasigroup]] and [[Quasigroup#Loops|Loop]]

'''— Group'''
{{Main|Group theory}}
* [[Group (mathematics)|Group]]
* [[Abelian group]] (commutative group)
<div role="note" class="hatnote">See also: [[Cyclic permutation]] (Cycle), [[Symmetric group]], [[Permutation group]], [[Isometry group]], [[Stirling numbers of the first kind]] and [[List of small groups]]</div>

'''— Ring, integral domain and field'''
* [[Ring (mathematics)|Ring]]
* [[Unit ring]] <sup>([[:es:Anillo unitario|es]])</sup>
* [[Commutative ring]]
* [[Ring of sets]]
* [[Prime ring]]
* [[Boolean ring]]
* [[Ordered ring]]
* [[Domain (ring theory)]]
* [[Integral domain]]
* [[Unique factorization domain]]
* [[Field (mathematics)|Field]]
* [[Ordered field]]
* [[Vector space]], [[Module (mathematics)|Module]] and [[Algebra over a field]]

'''— Homomorphisms'''
* [[Homomorphism]]
* [[Group homomorphism]]
* [[Isomorphism]]
* [[Group isomorphism]]
* [[Ring homomorphism]]

====== <span id="AlgebraicStructures_Connections"></span>Connections ======
<hr style="margin-left: 0px; width:50%;">
'''— Cryptography'''
{{main|Cryptography}}
{{Main category|Cryptography}}
* [[Pairing-based cryptography]]
'''— Category theory'''
{{main|Category theory}}
<div role="note" class="hatnote">Live article: [[John Baez|Baez, John]] (November 8, 2019). [https://elpais.com/elpais/2019/11/06/ciencia/1573042148_224789.html 'Qué es la teoría de categorías y cómo se ha convertido en tendencia'. ''El País'' (In: Café y Teoremas)] (In Spanish).</div>
* [[Olog]]
'''— Coding theory'''
{{Main|Coding theory}}
* [[Linear code]]
<div role="note" class="hatnote">See also: [[Block code]], [[Group code]] and [[Hamming code]]</div>
<!--
{| border=0 align=center cellspacing=20
|-
| <center>{{pp-book-cover</center>
 |title=
 |subtitle=
 |cover-image=Associativity of binary operations (without question marks).svg
 |cover-color=#76b3df
}} Libro wiki (itinerario en Wikipedia)<br />¡Pulse en el título, no en la imagen de portada!</center>
|}
-->

====== <span id="AlgebraicStructures_Bibliography"></span>Bibliography: theory and proposed and solved exercises  ======
<hr style="margin-left: 0px; width:50%;">
{| style="width: 100%;"
|In English: || In Spanish:
|- style="vertical-align:top;"
| style="width: 50%;" |
* —¤— Ben Brubaker. <i>The Very Basics of Groups, Rings, and Fields</i>. 2006. http://www-users.math.umn.edu/~brubaker/docs/152/152groups.pdf
* —¤— Peter J. Cameron. <i>Introduction to Algebra</i>. 2nd edition. Oxford University Press, New York, United States. 2008. ISBN: 978–0–19–856913–8.
* —¤— Peter J. Cameron. <i>Notes on Algebraic Structures</i>. 2006. http://www.maths.qmul.ac.uk/~pjc/notes/algstr.pdf
* Kenneth H. Rosen. <i>Discrete Mathematics and its Applications</i>. English Global Edition, 2013, 7th edition., ISBN-13: 978-0-07-131501-2 (adapted by: Kamala Krithivasan). (Chapter 12: «Algebraic Structures and Coding Theory»; slides: <http://highered.mheducation.com/sites/0071315012/student_view0/lecture_powerpoint_slides.html>).
| style="width: 50%;" |
* —¤— Máximo Anzola and José Caruncho. <i>Problemas de Álgebra. Tomo 1. Conjuntos-Grupos</i>. 3rd edition. Primer Ciclo, Madrid, Spain. (Chapter 9 «Operaciones», 24 solved exercises; Chapter 10 «Grupos-Estructura», 154 solved exercises), 1981. ISBN: 84-300-4073-0.
* —¤— Máximo Anzola and José Caruncho. <i>Problemas de Álgebra. Tomo 2. Anillos - Polinomios - Ecuaciones</i>. 3ªrd edition. Primer Ciclo, Madrid, Spain. (Chapter 1 «Estructura de anillo», 40 solved exercises; Chapter 4 «Estructura de cuerpo», 32 solved exercises), 1982. ISBN: 84-300-6417-6.
* Department of Algebra, University of Seville. <i>Álgebra básica. Notas de teoría</i>. http://www.algebra.us.es/Documentos/AB_1011_Teoria
* —¤— José García García and Manuel López Pellicer. <i>Álgebra lineal y geometría. Curso teórico-práctico</i>. 7th edition. Marfil, Alcoy, Spain. ISBN: 84-268-0269-9.
* [https://zurbaran.unex.es/cloud/index.php/s/f0CBUCmh1YQ9Qxf J. M. León Rojas (2009): Estructuras algebraicas (Notas incompletas de clase), v. 1.4.2-alpha2, rev. 25-04-2014. © CGL, CC0, CC BY.]
* [https://zurbaran.unex.es/cloud/index.php/s/Mt8LnWnr5hKrBJ0 J. M. León Rojas (2014): Estructuras mixtas (Notas incompletas de clase), v. 0.1.1-alpha0. © CGL, CC0, CC BY.]
|}

====== <span id="AlgebraicStructures_Software"></span>Software ======
<hr style="margin-left: 0px; width:50%;">
{| style="width: 100%;"
|In English: || In Spanish:
|- style="vertical-align:top;"
| style="width: 50%;" |
* [https://www.wolframalpha.com/examples/FiniteGroups.html WolframAlpha (Examples, Mathematics, Finite Groups)]
| style="width: 50%;" |
|}

====== <span id="AlgebraicStructures_Multimedia"></span>[[File:Multimedia-icon.svg|32px|link=]] Multimedia ======
<hr style="margin-left: 0px; width:50%;">
{| style="width: 100%;"
|In English: || In Spanish:
|- style="vertical-align:top;"
| style="width: 50%;" |
'''— Algebras'''
* {{cite web
 |url=https://www.youtube.com/watch?v=dQ4wU0k7JKI
 |title=Algebras
 |last=Krithivasan
 |first=Kamala
 |format=Video
 |publisher=Indian Institute of Technology Madras
 |date=
 |website=
 |access-date=
 |quote= }}
* {{cite web
 |url=https://www.youtube.com/watch?v=urd468CJCcU
 |title=Algebras (contd.)
 |last=Krithivasan
 |first=Kamala
 |format=Video
 |publisher=Indian Institute of Technology Madras
 |date=
 |website=
 |access-date=
 |quote= }}
* {{cite web
 |url=https://www.youtube.com/watch?v=YB6CP1RUvgk
 |title=Algebras (contd.)
 |last=Krithivasan
 |first=Kamala
 |format=Video
 |publisher=Indian Institute of Technology Madras
 |date=
 |website=
 |access-date=
 |quote= }}
* {{cite web
 |url=https://www.youtube.com/watch?v=qPtGlrb_sXg
 |title=Lecture 40 - Lattices
 |last=Krithivasan
 |first=Kamala
 |format=Video
 |publisher=Indian Institute of Technology Madras
 |date=
 |website=
 |access-date=
 |quote= }}
| style="width: 50%;" |
'''— Groups'''
* {{cite web |last=Soto Espinosa |first=Jesús |title=Grupos |format=Video |publisher=Universidad Católica de Murcia (UCAM) |url=https://www.youtube.com/watch?v=s1dawM-Bc0E |access-date=}}
* {{cite web |last=Ovalle Rodríguez |first=Janinna |title=El grupo simétrico |format=Video |publisher=Universidad Abierta y a Distancia de México (UnADM) |url=https://www.youtube.com/watch?v=s1dawM-Bc0E |access-date=}}
* {{cite web |author=RTVE |title=«La aventura del saber». Serie: «Más por menos». Capítulo: «La geometría se hace arte» (Script + presentation = Antonio Pérez) (Minute 5,22, the Alhambra, its mosaics and group theory; recommendation: see entire chapter) |format=Video |publisher=Radio televisión española (RTVE) |url=http://www.rtve.es/alacarta/videos/mas-por-menos/aventura-del-saber-serie-mas-menos-geometria-se-hace-arte/1291007/ |access-date=}} © gratisOA.

'''— Examples of groups'''
* {{cite web |last=Soto Espinosa |first=Jesús |title=Grupos. Ejemplo 1 |format=Video |publisher=Universidad Católica de Murcia (UCAM) |url=https://www.youtube.com/watch?v=y1QsPdVlNO0 |access-date=}}
* {{cite web |last=Soto Espinosa |first=Jesús |title=Grupos. Ejercicio 2 |format=Video |publisher=Universidad Católica de Murcia (UCAM) |url=https://www.youtube.com/watch?v=_o8ET5sTiog |access-date=}}
* {{cite web |last=Soto Espinosa |first=Jesús |title=Grupos. Ejercicio 3 |format=Video |publisher=Universidad Católica de Murcia (UCAM) |url=https://www.youtube.com/watch?v=h2odbGPBDKc |access-date=}}
* {{cite web |last=Ovalle Rodríguez |first=Janinna |title=El grupo aditivo de los enteros módulo n |format=Video |publisher=Universidad Abierta y a Distancia de México (UnADM) |url=https://www.youtube.com/watch?v=Z2B7AeMzmY0 |access-date=}}
* {{cite web |last=Ovalle Rodríguez |first=Janinna |title=Grupo de simetrías del cuadrado I. El grupo simétrico |format=Video |publisher=Universidad Abierta y a Distancia de México (UnADM) |url=https://www.youtube.com/watch?v=tM7lCNp_-1Y |access-date=}}
* {{cite web |last=Ovalle Rodríguez |first=Janinna |title=Grupo de simetrías del cuadrado II. El grupo simétrico |format=Video |publisher=Universidad Abierta y a Distancia de México (UnADM) |url=https://www.youtube.com/watch?v=onT343PebWc |access-date=}}
* {{cite web |last=Ovalle Rodríguez |first=Janinna |title=Ciclos y órbitas |format=Video |publisher=Universidad Abierta y a Distancia de México (UnADM) |url=https://www.youtube.com/watch?v=OQ-QrFCNWSQ |access-date=}}
* {{cite web |last=Ovalle Rodríguez |first=Janinna |title=Grupos de permutaciones. El grupo simétrico |format=Video |publisher=Universidad Abierta y a Distancia de México (UnADM) |url=https://www.youtube.com/watch?v=4yLEW3neP2s |access-date=}}

'''— Homomorphism of groups'''
* {{cite web |last=Soto Espinosa |first=Jesús |title=Homomorfismo de grupos. Inyectividad. Sobreyectividad |format=Video |publisher=Universidad Católica de Murcia (UCAM) |url=https://www.youtube.com/watch?v=_vU4K5iEjDA |access-date=}}

'''— Rings'''
* {{cite web |last=Soto Espinosa |first=Jesús |title=Anillo |format=Video |publisher=Universidad Católica de Murcia (UCAM) |url=https://www.youtube.com/watch?v=7xFE_-LM7iA |access-date=}}
* {{cite web |last=Soto Espinosa |first=Jesús |title=El anillo <math>\mathbb{Z}_n</math>. Congruencias |format=Video |publisher=Universidad Católica de Murcia (UCAM) |url=https://www.youtube.com/watch?v=nQrF0YayFlM |access-date=}}

'''— Integral domains'''
* {{cite web |last=Soto Espinosa |first=Jesús |title=Dominio de integridad |format=Video |publisher=Universidad Católica de Murcia (UCAM) |url=https://www.youtube.com/watch?v=lX2dejAl0j8 |access-date=}}
* {{cite web |last=Soto Espinosa |first=Jesús |title=Unidad en un dominio de integridad |format=Video |publisher=Universidad Católica de Murcia (UCAM) |url=https://www.youtube.com/watch?v=_seRzMTFQmw |access-date=}}
|}

====== <span id="AlgebraicStructures_To_find_out_more"></span>To find out more ======
<hr style="margin-left: 0px; width:50%;">
<ol style="list-style-type:circle;">
  <li>[[Portal:Mathematics]]</li>
  <li>And more:</li>
  <ol style="list-style-type:circle;">
    <li>[[Multiplicative group of integers modulo n]]</li>
  </ol>
</ol>
|}

==== <span style="color: rgb(0, 0, 0);">Theme 2.- Number theory</span> ====

{| class="mw-collapsible wikitable {{#ifeq: {{{plegada|no}}}|sí|mw-collapsed|}}" width="100%" style="text-align:left; padding:0px; background-color:#F9F9F9;"
| align="left"   |

===== Number theory =====
|-
| style="background-color:#F9F9F9; text-align:left; font-size:100%; padding:6px" |
{|
|[[File:Symmetrical 5-set Venn diagram LCM 2 3 4 5 7.svg|200px]]
|<div id="toc_NumberTheory" class="toc">
<div class="toctitle">
'''Contents'''
</div>
# [[#NumberTheory_Key_concepts|Key concepts]]
# [[#NumberTheory_Bibliography|Bibliography]]
# [[#NumberTheory_Software|Software]]
# [[#NumberTheory_Multimedia|Multimedia]]
# [[#NumberTheory_See_also|See also]]
# [[#NumberTheory_To_find_out_more|To find out more]]
</div>
|}

====== <span id="NumberTheory_Key_concepts"></span>Key concepts ======
<hr style="margin-left: 0px; width:50%;">
{{Main|Number theory}}
{{Main category|Number theory}}

<div id="Divisibilityandmodulararithmetic">'''— Divisibility and modular arithmetic'''</div>
{{see also|Division (mathematics)|Category:Division (mathematics)}}
* [[Arithmetic]]
* [[Divisibility]]
* [[Euclidean division]]
* [[Modular arithmetic]]

<div id="Primesandgreatestcommondivisor">'''— Primes and greatest common divisor'''</div>
* [[Prime number]]
* [[Euclid's theorem]]
* [[Primorial]]
* [[Euclid's lemma]]
* [[Fundamental theorem of arithmetic]]
* [[Integer factorization]]
* [[:Category:Conjectures about prime numbers]]
* [[Greatest common divisor]]
* [[Coprime integers]]
* [[Least common multiple]]
* [[Euclidean algorithm]]
* [[Bézout's identity]]
* [[Extended Euclidean algorithm]]

<div id="Solvingcongruences">'''— Solving congruences'''</div>
* [[Chinese remainder theorem]]
* [[Solving congruences]] <sup>([[:es:Aritmética_modular#Resolución_de_congruencias|es]])</sup>
* [[Modular multiplicative inverse]] (Inverse modulo ''m'' of an integer)
* [[Solving linear congruences]] <sup>([[:es:Teorema de congruencia lineal|es]])</sup>
* [[Method of successive substitution]]
* [[Euler's totient function]] (Euler's φ)
* [[Fermat-Euler theorem]]
* [[Fermat's little theorem]]
* [[Wilson's theorem]]
* [[Wolstenholme's theorem]]
* [[Table of congruences]]
* [[Josephus problem]] (first contact)

<div id="Applicationsofcongruences">'''— Applications of congruences'''</div>
* [[Hash function]]
* [[Pseudorandom number]]
<div style="margin-left: 1.5em; font-style: normal;">
'''- Cryptology'''
* [[Cryptology]]
* [[Cryptography]]
** [[RSA (cryptosystem)|RSA public-key cryptosystem]]
** [[RSA Factoring Challenge]]
</div>

<div id="Divisibilityrules">'''— Divisibility rules'''</div>
* [[Numeral system]]
* [[Power residue]]
* [[Pascal's tape]] <sup>([[:es:Cinta de Pascal|es]]; [[:fr:Ruban de Pascal|fr]])</sup>
* [[Divisibility rule]]

<div id="Diophantineequations">'''— Diophantine equations'''</div>
<ul>
<li> Some useful background: <div class="hlist inline">
* [[Algebraic equation]]
* [[System of equations]]
* [[Vieta's formulas]]
* [[Ruffini's rule]]
</div></li>
<li> [[Diophantine equation]]</li>
<li> [[Diophantine_equation#System_of_linear_Diophantine_equations|System of linear Diophantine equations]]</li>
</ul>

<div id="Paradoxes">'''— Paradoxes'''</div>
* [[Will Rogers phenomenon]]
<!--
{| border=0 align=center cellspacing=20
|-
| <center>{{pp-book-cover</center>
 |title=
 |subtitle=
 |cover-image=Symmetrical 5-set Venn diagram LCM 2 3 4 5 7.svg
 |cover-color=#76b3df
}} Libro wiki (itinerario en Wikipedia)<br />¡Pulse en el título, no en la imagen de portada!</center>
|}
-->

====== <span id="NumberTheory_Bibliography"></span>Bibliography: theory and proposed and solved exercises ======
<hr style="margin-left: 0px; width:50%;">
{| style="width: 100%;"
|In English: || In Spanish:
|- style="vertical-align:top;"
| style="width: 50%;" |
* —¤— Thomas Koshy. <i>Elementary number theory with applications</i>. Academic Press (an imprint of Elsevier Inc.), New York, United States, 2nd edition, 2007, ISBN: 978-0-12-372487-8
* —¤— Kenneth A. Rosen. <i>Discrete mathematics and its applications</i>. 7th edition. (Chapter 4 and related exercises). McGraw-Hill, New York, New York, United States, 2012, {{ISBN|978-0-07-338309-5}}
* Kenneth A. Rosen. <i>Elementary number theory and its applications</i>. Addison-Wesley, Reading, Massachusetts, United States, 1986, ISBN 0-201-06561
| style="width: 50%;" |
* —¤— Máximo Anzola y José Caruncho. <i>Problemas de Álgebra. Tomo 2. Anillos - Polinomios - Ecuaciones</i>. 3ª edición. Primer Ciclo, Madrid, Spain. (Capítulo 7 «Divisibilidad en <math>\mathbb{N}</math> y <math>\mathbb{Z}</math>», 94 ejercicios resueltos; Capítulo 8 «Ecuaciones diofánticas», 27 ejercicios resueltos; Capítulo 9 «Sistemas de numeración», 25 ejercicios resueltos), 1982. {{ISBN|84-300-6417-6}}.
* —¤— {{cite web
 |url=http://www2.uca.es/matematicas/Docencia/2005-2006/ESI/1710003/Apuntes/Leccion10.pdf
 |title=Apuntes de matemática discreta. 10. Divisibilidad. Algoritmo de la División
 |last=González Gutiérrez
 |first=Francisco José
 |date=2004a
 |website=
 |location=Cádiz, Andalusia (ES-AN), Spain
 |publisher=University of Cádiz (UCA)
 |access-date=
 |language=Spanish
 |format=PDF }}
* —¤— {{cite web
 |url=http://www2.uca.es/matematicas/Docencia/2005-2006/ESI/1710003/Apuntes/Leccion11.pdf
 |title=Apuntes de matemática discreta. 11. Teorema Fundamental de la Aritmética
 |last=González Gutiérrez
 |first=Francisco José
 |date=2004b
 |website=
 |location=Cádiz, Andalusia (ES-AN), Spain
 |publisher=University of Cádiz (UCA)
 |access-date=
 |language=Spanish
 |format=PDF }}
* —¤— {{cite web
 |url=http://www2.uca.es/matematicas/Docencia/2005-2006/ESI/1710003/Apuntes/Leccion12.pdf
 |title=Apuntes de matemática discreta. 12. Ecuaciones diofánticas
 |last=González Gutiérrez
 |first=Francisco José
 |date=2004c
 |website=
 |location=Cádiz, Andalusia (ES-AN), Spain
 |publisher=University of Cádiz (UCA)
 |access-date=
 |language=Spanish
 |format=PDF }}
* —¤— {{cite web
 |url=http://www2.uca.es/matematicas/Docencia/2005-2006/ESI/1710003/Apuntes/Leccion13.pdf
 |title=Apuntes de matemática discreta. 13. Clases de restos módulo m
 |last=González Gutiérrez
 |first=Francisco José
 |date=2004d
 |website=
 |location=Cádiz, Andalusia (ES-AN), Spain
 |publisher=University of Cádiz (UCA)
 |access-date=
 |language=Spanish
 |format=PDF }}
* —¤— Kenneth A. Rosen. <i>Matemática discreta y sus aplicaciones</i>. 5th edition. (Sections 2.4, 2.5, 2.6 and related exercises). McGraw-Hill/Interamericana de Spain, S.A.U., Aravaca (Madrid), Madrid, Community of Madrid (ES-MD), Spain, 2004, ISBN 84-481-4073-7
* —¤— {{cite web
 |url=http://ocw.upm.es/matematica-aplicada/matematica-discreta/contenidos/material-de-clase/tema-3
 |title=Tema 3: Ecuaciones diofánticas, congruencias y criterios de divisibilidad
 |last1=Sendra
 |first1=J.
 |last2=Martín
 |first2=E.
 |last3=Méndez
 |first3=A.
 |last4=Ortiz
 |first4=C.
 |date=2011
 |website=
 |location=Madrid, Community of Madrid (ES-MD), Spain
 |publisher=Technical University of Madrid (UPM)
 |access-date=
 |language=Spanish
 |format=PDF }}
|}

====== <span id="NumberTheory_Software"></span>Software ======
<hr style="margin-left: 0px; width:50%;">
{| style="width: 100%;"
|In English: || In Spanish:
|- style="vertical-align:top;"
| style="width: 50%;" |
* [https://www.wolframalpha.com/examples/NumberTheory.html WolframAlpha (Examples, Mathematics, Number Theory)]
* [http://www.wolframalpha.com/input/?i=rsa+encryption WolframAlpha (RSA encryption)]
| style="width: 50%;" |
|}

====== <span id="NumberTheory_Multimedia"></span>[[File:Multimedia-icon.svg|32px|link=]] Multimedia ======
<hr style="margin-left: 0px; width:50%;">
{| style="width: 100%;"
|In English: || In Spanish:
|- style="vertical-align:top;"
| style="width: 50%;" |
'''— Divisibility and modular arithmetic'''
* {{cite web
 |url=https://www.youtube.com/watch?v=_FKSIpoajbs
 |title=Section 4.1 Divisibility and Modular Arithmetic
 |last=Baugher
 |first=Greg A.
 |department=Department of Math/Science/Informatics, Penfield College, Mercer University
 |format=Video
 |publisher=
 |location=Georgia (US-GA), USA
 |date=
 |website=
 |access-date=
 |quote= }}
'''— Primes and GCD'''
* {{cite web
 |url=https://www.youtube.com/watch?v=n4T5p1-vhco
 |title=Section 4.3 Primes and Greatest Common Divisors
 |last=Baugher
 |first=Greg A.
 |department=Department of Math/Science/Informatics, Penfield College, Mercer University
 |format=Video
 |publisher=
 |location=Georgia (US-GA), USA
 |date=
 |website=
 |access-date=
 |quote= }}
* {{cite web
 |url=ftp://ftp.aduni.org/videos/02_lect_18.rm
 
 |title=Euclid's algorithm
 |last=Simonson
 |first=Shai
 |format=Video
 |publisher=ArsDigita University
 |date=
 |website=
 |access-date=
 |quote= }} ([https://www.youtube.com/watch?v=HEMny6FyVnA&index=18&list=PL__5ik6vMY50HNfWppaGxQydlSMTGQWzC Otra URL])
* {{cite web
 |url=https://www.youtube.com/watch?v=O6DNWMxG_tk&index=79&list=PLr3WmPgPWZfX1HUpeyKkP6ir2wOFhqXMO
 |title=P1--Gear Problem and LCM
 |last=Olson
 |first=Glenn R.
 |department=Maine East High School
 |location=Park Ridge, Illinois (US-IL), USA
 |format=Video
 |publisher=
 |date=
 |website=
 |access-date=
 |quote= }}
'''— Solving congruences and their applications'''
* {{cite web
 |url=https://www.youtube.com/watch?v=75rjaZeWkF4
 |title=Sections 4.4-4.5 Solving Congruences and Applications
 |last=Baugher
 |first=Greg A.
 |department=Department of Math/Science/Informatics, Penfield College, Mercer University
 |format=Video
 |publisher=
 |location=Georgia (US-GA), USA
 |date=
 |website=
 |access-date=
 |quote= }}
'''— Diophantine equations'''
* {{cite web
 |url=https://www.youtube.com/watch?v=zTXGPq0Bfb4&index=7&list=PLH83nnRMSIe4LOzMQc4jzeDbAXe9-2jrR
 |title=Number Theory 7 - Diophantine Equations
 |last=Olavsku
 |first=
 |department=
 |format=Video
 |publisher=
 |location=
 |date=
 |website=
 |access-date=
 |quote= }}
* {{cite web
 |url=https://www.youtube.com/watch?v=TIk3ujphMfk&index=63&list=PLr3WmPgPWZfX1HUpeyKkP6ir2wOFhqXMO
 |title=N1--Introduction to Linear Diophantine Equations
 |last=Olson
 |first=Glenn R.
 |department=Maine East High School
 |format=Video
 |publisher=
 |location=Park Ridge, Illinois (US-IL), USA
 |date=
 |website=
 |access-date=
 |quote= }}
* {{cite web
 |url=https://www.youtube.com/watch?v=5DcoG69NyO0&index=64&list=PLr3WmPgPWZfX1HUpeyKkP6ir2wOFhqXMO
 |title=N2--Solve Basic Linear Diophantine Equation
 |last=Olson
 |first=Glenn R.
 |department=Maine East High School
 |location=Park Ridge, Illinois (US-IL), USA
 |format=Collection of videos
 |publisher=
 |date=
 |website=
 |access-date=
 |quote= }}
* {{cite web
 |url=https://www.youtube.com/watch?v=B-0ahK4jJUM&index=65&list=PLr3WmPgPWZfX1HUpeyKkP6ir2wOFhqXMO
 |title=N3--Simplify and Solve Linear Diophantine Equation
 |last=Olson
 |first=Glenn R.
 |department=Maine East High School
 |location=Park Ridge, Illinois (US-IL), USA
 |format=Collection of videos
 |publisher=
 |date=
 |website=
 |access-date=
 |quote= }}
* {{cite web
 |url=https://www.youtube.com/watch?v=K29OtjQg35s&index=66&list=PLr3WmPgPWZfX1HUpeyKkP6ir2wOFhqXMO
 |title=N4--Solve Linear Diophantine Equation with Right Hand Side Not Equal to 1
 |last=Olson
 |first=Glenn R.
 |department=Maine East High School
 |location=Park Ridge, Illinois (US-IL), USA
 |format=Collection of videos
 |publisher=
 |date=
 |website=
 |access-date=
 |quote= }}
* {{cite web
 |url=https://www.youtube.com/watch?v=b0Z845maIuw
 |title=Linear Diophantine Equations - Ex 1
 |last=Olver
 |first=Brandon
 |department=Ballarat Grammar
 |format=Video
 |publisher=
 |location=Wendouree, Ballarat, Victoria (AU-VIC), Commonwealth of Australia
 |date=
 |website=
 |access-date=
 |quote= }}
* {{cite web
 |url=https://www.youtube.com/watch?v=Dt2-_XEj2uI
 |title=Linear Diophantine Equations - Ex 2
 |last=Olver
 |first=Brandon
 |department=Ballarat Grammar
 |format=Video
 |publisher=
 |location=Wendouree, Ballarat, Victoria (AU-VIC), Commonwealth of Australia
 |date=
 |website=
 |access-date=
 |quote= }}
'''— Cryptography'''
* {{cite web
 |url=https://www.youtube.com/watch?v=WKFRdeU52eQ
 |title=Section 4.6 Cryptography
 |last=Baugher
 |first=Greg A.
 |department=Department of Math/Science/Informatics, Penfield College, Mercer University
 |format=Video
 |publisher=
 |location=Georgia (US-GA), USA
 |date=
 |website=
 |access-date=
 |quote= }}
* {{cite web
 |url=ftp://ftp.aduni.org/videos/02_lect_19.rm
 
 |title=Cryptography
 |last=Simonson
 |first=Shai
 |format=Video
 |publisher=ArsDigita University
 |date=
 |website=
 |access-date=
 |quote= }} ([https://www.youtube.com/watch?v=ZHL4esYKhek&index=20&list=PL__5ik6vMY50HNfWppaGxQydlSMTGQWzC Otra URL])
| style="width: 50%;" |
'''— Divisibility'''
* [https://www.youtube.com/watch?v=4BmqFuebG9A Soto Espinosa, Jesús. Divisibilidad. Ejemplo 1. UCAM]
* [https://www.youtube.com/watch?v=GXvUP-dUjww Soto Espinosa, Jesús. Divisibilidad. Ejemplo 2. UCAM]
* [https://www.youtube.com/watch?v=5H8hirvHQKI Soto Espinosa, Jesús. Divisibilidad. Ejemplo 3. UCAM]
* [https://www.youtube.com/watch?v=lHA6c0d-jCc Soto Espinosa, Jesús. Divisibilidad. Ejemplo 3+. UCAM]
* [https://www.youtube.com/watch?v=w3ZmXWswCM8 Soto Espinosa, Jesús. Divisibilidad. Ejemplo 4. UCAM]
* [https://www.youtube.com/watch?v=gtpXda-tPL0 Soto Espinosa, Jesús. Divisibilidad. Ejemplo 5. UCAM]
* [https://www.youtube.com/watch?v=QZiFMtV9x_s Soto Espinosa, Jesús. Divisibilidad. Ejemplo 6 (inducción). UCAM]
* [https://www.youtube.com/watch?v=3dXEdIMX58w Soto Espinosa, Jesús. Divisibilidad. Ejemplo 4 (inducción). UCAM]
* {{cite web |last=Soto Espinosa |first=Jesús |title=Algoritmo de la división |format=Video |publisher=Escuela Politécnica Superior, Universidad Católica San Antonio de Murcia (UCAM) |location=Guadalupe, Murcia, Region of Murcia (ES-MC), Spain |url=https://www.youtube.com/watch?v=WXAcjGvQaLA&list=PL2qtqrBO8xujlH2bo2UPMQ3m9Hj9yEBur&index=117 |access-date=}}
* [https://www.youtube.com/watch?v=L70xLQzqBew Soto Espinosa, Jesús. Algoritmo de la división. Ejemplo 1. UCAM]
* [https://www.youtube.com/watch?v=lz3k1v4D16U Soto Espinosa, Jesús. Algoritmo de la división. Ejemplo 2. UCAM]
* [https://www.youtube.com/watch?v=xaOINn4p-Fs Soto Espinosa, Jesús. Algoritmo de la división. Ejercicio 3. UCAM]

'''— Primes and GCD'''
* {{cite web |last=Soto Espinosa |first=Jesús |title=Números primos, ejemplo 1 |format=Video |publisher=Escuela Politécnica Superior, Universidad Católica San Antonio de Murcia (UCAM) |location=Guadalupe, Murcia, Region of Murcia (ES-MC), Spain |url=https://www.youtube.com/watch?v=TW8-Dl4wdbw&list=PL2qtqrBO8xujlH2bo2UPMQ3m9Hj9yEBur&index=108 |access-date=}}
* {{cite web |last=Soto Espinosa |first=Jesús |title=Números primos, ejemplo 2 |format=Video |publisher=Escuela Politécnica Superior, Universidad Católica San Antonio de Murcia (UCAM) |location=Guadalupe, Murcia, Region of Murcia (ES-MC), Spain |url=https://www.youtube.com/watch?v=aWCkrzf4U-s&list=PL2qtqrBO8xujlH2bo2UPMQ3m9Hj9yEBur&index=110 |access-date=}}
* {{cite web |last=Soto Espinosa |first=Jesús |title=Números primos, ejemplo 3 |format=Video |publisher=Escuela Politécnica Superior, Universidad Católica San Antonio de Murcia (UCAM) |location=Guadalupe, Murcia, Region of Murcia (ES-MC), Spain |url=https://www.youtube.com/watch?v=4gL7Roz36go&index=21&list=PL2qtqrBO8xujlH2bo2UPMQ3m9Hj9yEBur |access-date=}}
* {{cite web |last=Soto Espinosa |first=Jesús |title=Números primos, ejemplo 5 |format=Video |publisher=Escuela Politécnica Superior, Universidad Católica San Antonio de Murcia (UCAM) |location=Guadalupe, Murcia, Region of Murcia (ES-MC), Spain |url=https://www.youtube.com/watch?v=fjcXDNs2HLA&list=PL2qtqrBO8xujlH2bo2UPMQ3m9Hj9yEBur&index=124 |access-date=}}
* {{cite web |last=Soto Espinosa |first=Jesús |title=Infinitud de los números primos |format=Video |publisher=Escuela Politécnica Superior, Universidad Católica San Antonio de Murcia (UCAM) |location=Guadalupe, Murcia, Region of Murcia (ES-MC), Spain |url=https://www.youtube.com/watch?v=cdlBaHUovAM&list=PL2qtqrBO8xujlH2bo2UPMQ3m9Hj9yEBur&index=118 |access-date=}}
* {{cite web |last=Soto Espinosa |first=Jesús |title=Teorema fundamental de la aritmética |format=Video |publisher=Escuela Politécnica Superior, Universidad Católica San Antonio de Murcia (UCAM) |location=Guadalupe, Murcia, Region of Murcia (ES-MC), Spain |url=https://www.youtube.com/watch?v=KGaM0gnttSA&list=PL2qtqrBO8xujlH2bo2UPMQ3m9Hj9yEBur&index=119 |access-date=}}
* {{cite web |last=Soto Espinosa |first=Jesús |title=Máximo común divisor |format=Video |publisher=Escuela Politécnica Superior, Universidad Católica San Antonio de Murcia (UCAM) |location=Guadalupe, Murcia, Region of Murcia (ES-MC), Spain |url=https://www.youtube.com/watch?v=fBPHE9BcCy8&list=PL2qtqrBO8xujlH2bo2UPMQ3m9Hj9yEBur&index=121 |access-date=}}
* {{cite web |last=Soto Espinosa |first=Jesús |title=Máximo común divisor, ejemplo 1 |format=Video |publisher=Escuela Politécnica Superior, Universidad Católica San Antonio de Murcia (UCAM) |location=Guadalupe, Murcia, Region of Murcia (ES-MC), Spain |url=https://www.youtube.com/watch?v=5OVyb2Pskyc&list=PL2qtqrBO8xujlH2bo2UPMQ3m9Hj9yEBur&index=107 |access-date=}}
* {{cite web |last=Soto Espinosa |first=Jesús |title=Máximo común divisor, ejemplo 2 |format=Video |publisher=Escuela Politécnica Superior, Universidad Católica San Antonio de Murcia (UCAM) |location=Guadalupe, Murcia, Region of Murcia (ES-MC), Spain |url=https://www.youtube.com/watch?v=5mIdOjTtgus&list=PL2qtqrBO8xujlH2bo2UPMQ3m9Hj9yEBur&index=106 |access-date=}}
* {{cite web |last=Soto Espinosa |first=Jesús |title=Máximo común divisor, ejemplo 3 |format=Video |publisher=Escuela Politécnica Superior, Universidad Católica San Antonio de Murcia (UCAM) |location=Guadalupe, Murcia, Region of Murcia (ES-MC), Spain |url=https://www.youtube.com/watch?v=PwuR86QogX4&list=PL2qtqrBO8xujlH2bo2UPMQ3m9Hj9yEBur&index=105 |access-date=}}
* {{cite web |last=Soto Espinosa |first=Jesús |title=Máximo Común Divisor, ejemplo 4 |format=Video |publisher=Escuela Politécnica Superior, Universidad Católica San Antonio de Murcia (UCAM) |location=Guadalupe, Murcia, Region of Murcia (ES-MC), Spain |url=https://www.youtube.com/watch?v=Dgqejj59Hw0&index=20&list=PL2qtqrBO8xujlH2bo2UPMQ3m9Hj9yEBur |access-date=}}
* {{cite web |last=Soto Espinosa |first=Jesús |title=Algoritmo de Euclides |format=Video |publisher=Universidad Católica de Murcia (UCAM) |url=https://www.youtube.com/watch?v=Yih7esYC6u8 |access-date=}}

'''— Bézout's lemma'''
* {{cite web |last=Soto Espinosa |first=Jesús |title=Identidad de Bézout |format=Video |publisher=Universidad Católica de Murcia (UCAM) |url=https://www.youtube.com/watch?v=F_Rx5zD7cUU |access-date=}}
* {{cite web |last=Soto Espinosa |first=Jesús |title=Identidad de Bézout, ejemplo 1 |format=Video |publisher=Escuela Politécnica Superior, Universidad Católica San Antonio de Murcia (UCAM) |location=Guadalupe, Murcia, Region of Murcia (ES-MC), Spain |url=https://www.youtube.com/watch?v=hPtmpQa89G8&index=18&list=PL2qtqrBO8xujlH2bo2UPMQ3m9Hj9yEBur |access-date=}}
* {{cite web |last=Soto Espinosa |first=Jesús |title=Identidad de Bézout, ejemplo 2 |format=Video |publisher=Escuela Politécnica Superior, Universidad Católica San Antonio de Murcia (UCAM) |location=Guadalupe, Murcia, Region of Murcia (ES-MC), Spain |url=https://www.youtube.com/watch?v=2i5FberlJwg&index=17&list=PL2qtqrBO8xujlH2bo2UPMQ3m9Hj9yEBur |access-date=}}

'''— Modular arithmetic. Euler's φ function (totient function)'''
* {{cite web |last=Soto Espinosa |first=Jesús |title=Función φ de Euler |format=Video |publisher=Escuela Politécnica Superior, Universidad Católica San Antonio de Murcia (UCAM) |location=Guadalupe, Murcia, Region of Murcia (ES-MC), Spain |url=https://www.youtube.com/watch?v=S6GC63Yxi-U |access-date=}}
* {{cite web |last=Soto Espinosa |first=Jesús |title=Función φ de Euler, propiedad 1 |format=Video |publisher=Escuela Politécnica Superior, Universidad Católica San Antonio de Murcia (UCAM) |location=Guadalupe, Murcia, Region of Murcia (ES-MC), Spain |url=https://www.youtube.com/watch?v=RMZJScqGbX8&index=11&list=PL2qtqrBO8xujlH2bo2UPMQ3m9Hj9yEBur |access-date=}}
* {{cite web |last=Soto Espinosa |first=Jesús |title=Función φ de Euler, propiedad 2 |format=Video |publisher=Escuela Politécnica Superior, Universidad Católica San Antonio de Murcia (UCAM) |location=Guadalupe, Murcia, Region of Murcia (ES-MC), Spain |url=https://www.youtube.com/watch?v=XqWkVFkqssw&index=10&list=PL2qtqrBO8xujlH2bo2UPMQ3m9Hj9yEBur |access-date=}}

'''— Diophantine equations'''
* {{cite web |last=Hervás Jorge |first=Antonio |title=Ecuaciones diofánticas |format=Video |publisher=Universidad Politécnica de Valencia (UPV) |url=https://www.youtube.com/watch?v=RGovvYZV7Eo |access-date=}}
* {{cite web |last=Soto Espinosa |first=Jesús |title=Ecuación diofántica |format=Video |publisher=Universidad Católica de Murcia (UCAM) |url=https://www.youtube.com/watch?v=Z8m0vfnkYQw |access-date=}}
* {{cite web |last=Soto Espinosa |first=Jesús |title=Ecuación diofántica de dos variables |format=Video |publisher=Universidad Católica de Murcia (UCAM) |url=https://www.youtube.com/watch?v=554vSoZE1dM |access-date=}}
* {{cite web |last=Soto Espinosa |first=Jesús |title=Ecuación diofántica, Ejemplo 1 |format=Video |publisher=Universidad Católica de Murcia (UCAM) |url=https://www.youtube.com/watch?v=yz4lva2EIGw |access-date=}}
* {{cite web |last=Soto Espinosa |first=Jesús |title=Ecuación diofántica, Ejemplo 2 |format=Video |publisher=Universidad Católica de Murcia (UCAM) |url=https://www.youtube.com/watch?v=Q_IZVN9SvKc |access-date=}}
* {{cite web |last=Soto Espinosa |first=Jesús |title=Ecuación diofántica, Ejemplo 3 |format=Video |publisher=Universidad Católica de Murcia (UCAM) |url=https://www.youtube.com/watch?v=0T1HrBu03BA |access-date=}}
* {{cite web |last=Soto Espinosa |first=Jesús |title=Ecuación diofántica, Ejemplo 4 |format=Video |publisher=Universidad Católica de Murcia (UCAM) |url=https://www.youtube.com/watch?v=NUkn1PkxkKQ |access-date=}}
* {{cite web |last=Soto Espinosa |first=Jesús |title=Ecuación diofántica, Ejemplo 5 |format=Video |publisher=Universidad Católica de Murcia (UCAM) |url=https://www.youtube.com/watch?v=8MAoz998T9Y |access-date=}}
* {{cite web |last=Soto Espinosa |first=Jesús |title=Ecuación diofántica de tres variables, Ejercicio 1 |format=Video |publisher=Universidad Católica de Murcia (UCAM) |url=https://www.youtube.com/watch?v=_ZgAnTUvaYI |access-date=}}
* {{cite web |last=Soto Espinosa |first=Jesús |title=Ecuación diofántica de tres variables, Ejercicio 2 |format=Video |publisher=Universidad Católica de Murcia (UCAM) |url=https://www.youtube.com/watch?v=tSl1VjVCpq4 |access-date=}}
* {{cite web |last=Soto Espinosa |first=Jesús |title=Sistemas de ecuaciones diofánticas |format=Video |publisher=Universidad Católica de Murcia (UCAM) |url=https://www.youtube.com/watch?v=eI9ZQ8l8BUQ |access-date=}}

'''— Congruences'''
* [https://www.youtube.com/watch?v=RGRrMsL0DqM Congruencias. TurboOpenFing]
* {{cite web |last=Soto Espinosa |first=Jesús |title=Congruencias |format=Video |publisher=Universidad Católica de Murcia (UCAM) |url=https://www.youtube.com/watch?v=0VYrajHNVJE |access-date=}}
* {{cite web |last=Soto Espinosa |first=Jesús |title=Congruencias: relación de equivalencia |format=Video |publisher=Universidad Católica de Murcia (UCAM) |url=https://www.youtube.com/watch?v=QMhoJgIls2A |access-date=}}
* {{cite web |last=Soto Espinosa |first=Jesús |title=Congruencias. Propiedad 1 |format=Video |publisher=Universidad Católica de Murcia (UCAM) |url=https://www.youtube.com/watch?v=EcBNusrX0pQ |access-date=}}
* {{cite web |last=Soto Espinosa |first=Jesús |title=Congruencias. Propiedad 2 |format=Video |publisher=Universidad Católica de Murcia (UCAM) |url=https://www.youtube.com/watch?v=d7diZgK8nHs |access-date=}}
* {{cite web |last=Soto Espinosa |first=Jesús |title=Congruencias. Propiedad 3 |format=Video |publisher=Universidad Católica de Murcia (UCAM) |url=https://www.youtube.com/watch?v=CIPLnJhrHUI |access-date=}}
* {{cite web |last=Soto Espinosa |first=Jesús |title=Congruencias. Ejercicio 1 |format=Video |publisher=Universidad Católica de Murcia (UCAM) |url=https://www.youtube.com/watch?v=XqfNbmOHNHQ |access-date=}}
* {{cite web |last=Soto Espinosa |first=Jesús |title=Congruencias. Ejercicio 2 |format=Video |publisher=Universidad Católica de Murcia (UCAM) |url=https://www.youtube.com/watch?v=5BiPKyLv9y8 |access-date=}}
* {{cite web |last=Soto Espinosa |first=Jesús |title=Congruencias. Ejercicio 3 |format=Video |publisher=Universidad Católica de Murcia (UCAM) |url=https://www.youtube.com/watch?v=C2eOqjzFgOs |access-date=}}
* {{cite web |last=Soto Espinosa |first=Jesús |title=Congruencias. Ejercicio 4 |format=Video |publisher=Universidad Católica de Murcia (UCAM) |url=https://www.youtube.com/watch?v=CsUcHJS824Q |access-date=}}
: Ecuaciones
* {{cite web |last=Soto Espinosa |first=Jesús |title=Ecuación de congruencias |format=Video |publisher=Universidad Católica de Murcia (UCAM) |url=https://www.youtube.com/watch?v=mKv3uqeC_Gc |access-date=}}
* {{cite web |last=Soto Espinosa |first=Jesús |title=Ecuación de congruencias. Ejemplo 1 |format=Video |publisher=Universidad Católica de Murcia (UCAM) |url=https://www.youtube.com/watch?v=aa8dRPJkzA0 |access-date=}}
* {{cite web |last=Soto Espinosa |first=Jesús |title=Ecuación de congruencias. Ejemplo 2 |format=Video |publisher=Universidad Católica de Murcia (UCAM) |url=https://www.youtube.com/watch?v=JmiJiUR9lFo |access-date=}}
* {{cite web |last=Soto Espinosa |first=Jesús |title=Ecuación de congruencias. Ejercicio 3 |format=Video |publisher=Universidad Católica de Murcia (UCAM) |url=https://www.youtube.com/watch?v=ArFzf-33lrM |access-date=}}
* {{cite web |last=Soto Espinosa |first=Jesús |title=Ecuación de congruencias. Ejercicio 4 |format=Video |publisher=Universidad Católica de Murcia (UCAM) |url=https://www.youtube.com/watch?v=Z0OqUzq46pI |access-date=}}
* {{cite web |last=Soto Espinosa |first=Jesús |title=Ecuación de congruencias. Ejercicio 7 |format=Video |publisher=Universidad Católica de Murcia (UCAM) |url=https://www.youtube.com/watch?v=1fCbvGwqAzA |access-date=}}
: Sistemas
* {{cite web |last=Soto Espinosa |first=Jesús |title=Sistemas de congruencias |format=Video |publisher=Universidad Católica de Murcia (UCAM) |url=https://www.youtube.com/watch?v=RGcCAjnwslg |access-date=}}

'''— Congruences: Casting out nines'''
* {{cite web |last=Hervás Jorge |first=Antonio |title=Congruencias de números enteros: La prueba del 9 |format=Video |publisher=Universidad Politécnica de Valencia (UPV) |url=https://www.youtube.com/watch?v=foX4YlHEht4 |access-date=}}

'''— False positives: Casting out elevens'''
* {{cite web |last=Hervás Jorge |first=Antonio |title=Los falsos positivos: La prueba del 11 |format=Video |publisher=Universidad Politécnica de Valencia (UPV) |url=https://www.youtube.com/watch?v=xPkkOCmbSI0 |access-date=}}

'''— Power residues'''
* {{cite web |last=Soto Espinosa |first=Jesús |title=Restos potenciales |format=Video |publisher=Universidad Católica de Murcia (UCAM) |url=https://www.youtube.com/watch?v=H_OPU-I7Qhk |access-date=}}
* {{cite web |last=Soto Espinosa |first=Jesús |title=Restos potenciales. Ejercicio 1 |format=Video |publisher=Universidad Católica de Murcia (UCAM) |url=https://www.youtube.com/watch?v=GjaS5Semo-8 |access-date=}}
* {{cite web |last=Soto Espinosa |first=Jesús |title=Restos potenciales. Ejercicio 2 |format=Video |publisher=Universidad Católica de Murcia (UCAM) |url=https://www.youtube.com/watch?v=154W6fvgUks |access-date=}}

'''— Divisibility rules'''
* {{cite web |last=Soto Espinosa |first=Jesús |title=Criterios de divisibilidad |format=Video |publisher=Universidad Católica de Murcia (UCAM) |url=https://www.youtube.com/watch?v=PgECRJqDpoY |access-date=}}
* {{cite web |last=Soto Espinosa |first=Jesús |title=Criterios de divisibilidad. Ejercicio 1 |format=Video |publisher=Universidad Católica de Murcia (UCAM) |url=https://www.youtube.com/watch?v=F1VdUk4M_Ak |access-date=}}
|}

====== <span id="NumberTheory_See_also"></span>See also ======
<hr style="margin-left: 0px; width:50%;">
* [[1729 (number)]]
* [[5040 (number)]]
<!-- * <del>[http://cala.unex.es/cala/epistemowikia/index.php/Ejercicios_propuestos_sobre_números_naturales_y_enteros_y_algunas_soluciones Ejercicios propuestos sobre números naturales y enteros y algunas soluciones] (Proposed exercises about natural numbers and integers and some solutions) (On Epistemowikia and in Spanish, for the time being)</del> -->

====== <span id="NumberTheory_To_find_out_more"></span>To find out more ======
<hr style="margin-left: 0px; width:50%;">
<ol style="list-style-type:circle;">
  <li>[[Portal:Mathematics]]</li>
  <li>And more:</li>
  <ol style="list-style-type:circle;">
    <li>Divisibility</li>
    <ol style="list-style-type:circle;">
      <li>[[Division algorithm]] (Algorithms for division)</li>
    </ol>
    <li>Primality</li>
    <ol style="list-style-type:circle;">
      <li>[[Quadratic residue]]</li>
      <li>[[Quadratic reciprocity]]</li>
      <li>[[Primality test]]</li>
    </ol>
    <li>Pseudo-random number generation</li>
    <ol style="list-style-type:circle;">
      <li>[[List of random number generators]]</li>
      <!-- <li>[[Linear congruential generator|Linear congruential generators (LCG)]]</li>
      <li>[[Combined Linear Congruential Generator|Combined Linear Congruential Generators]]</li>
      <li>[[Inversive congruential generator]]</li>
      <li>[[Generalized inversive congruential pseudorandom numbers|Generalised inversive congruential pseudorandom numbers]]</li> -->
    </ol>
    <li>[[Cryptography]]</li>
    <ol style="list-style-type:circle;">
      <li>[[Highly totient number]]</li>
      <li>[[Highly composite number]]</li>
      <li>[[Smooth number]]</li>
      <li>[[Rough number]]</li>
      <li>[[Semiprime]]</li>
      <li>[[Elliptic curve cryptography]]</li>
    </ol>
    <li>[[List of prime numbers]]</li>
    <li>[[List of numbers]]</li>
  </ol>
</ol>
{{Number theory|collapsed}}
{{Divisor classes|Divisibility-based sets of integers|state=collapsed}}
{{Classes of natural numbers|state=collapsed}}
{{Number theoretic algorithms|state=collapsed|width=33%}}
{{Prime number classes|collapsed}}
{{Prime number conjectures|collapsed}}
|}

==== <span style="color: rgb(0, 0, 0);">Theme 3.- Combinatorics</span> ====

{| class="mw-collapsible wikitable {{#ifeq: {{{plegada|no}}}|sí|mw-collapsed|}}" width="100%" style="text-align:left; padding:0px; background-color:#F9F9F9;"
| align="left"   |

===== Combinatorics =====
|-
| style="background-color:#F9F9F9; text-align:left; font-size:100%; padding:6px" |
{|
|[[File:Proofs-of-Fermats-Little-Theorem-bracelet1.svg|200px]]
|<div id="toc_Combinatorics" class="toc">
<div class="toctitle">
'''Contents'''
</div>
# [[#Combinatorics_Key_concepts|Key concepts]]
# [[#Combinatorics_Bibliography|Bibliography]]
# [[#Combinatorics_Software|Software]]
# [[#Combinatorics_Multimedia|Multimedia]]
# [[#Combinatorics_See_also|See also]]
# [[#Combinatorics_To_find_out_more|To find out more]]
</div>
|}

====== <span id="Combinatorics_Key_concepts"></span>Key concepts ======
<hr style="margin-left: 0px; width:50%;">
{{Main|Combinatorics}}
{{Main category|Combinatorics}}
{{see also|Combinatorial principles}}

'''<div id="Thebasicsofcounting">— The basics of counting</div>'''
<div style="margin-left: 1.5em; font-style: normal;">

'''- Rules of sum, product, substraction and division'''
* [[Rule of sum|Rule of sum]] (addition principle)
* [[Rule of product|Rule of product]] (multiplication principle)
* [[Inclusion-exclusion principle|Inclusion-exclusion principle]] (sieve principle or substraction rule)
* [[Complement rule]]
* [[Division rule]]

'''- Drawer principle and its generalisation'''
* [[Pigeonhole principle]] (Dirichlet's box/drawer/shelf principle or Dirichlet's distribution principle)

'''- Binomial coefficients and identities'''
* [[Factorial]]
* [[Binomial coefficient]]
* [[Binomial theorem]]
* [[Pascal's triangle]]
* [[Trinomial expansion]]
* [[Multinomial theorem]] (and multinomial coefficient)
* [[Pascal's rule]]

'''- (Ordinary) (i.e., without repetition) variations, permutations and combinations, and with repetition, and circular permutations'''
* [[Permutation#k-permutations_of_n|(Ordinary) variations]] (k-permutations of n, also called [[Partial permutation#Restricted partial permutations|restricted partial permutations]])
* [[Permutation#Permutations with repetition|Variations with repetition]]
* [[Permutation|(Ordinary) permutations]]
* [[Permutation#Circular permutations|Circular permutations]]
* [[Permutation#Permutations with repetition|Permutations with repetition]]
* [[Combination|(Ordinary) combinations]]
* [[Combinations#Number of combinations with repetition|Combinations with repetition]]

'''- Counting with restrictions'''
* [[Rook polynomial]]
</div>

'''<div id="Combinatorialproofs">— Combinatorial proofs: 1st, bijective proofs; 2nd, double counting proofs; 3rd, using distinguished element, and 4th, using the inclusion-exclusion principle</div>'''
{{main|Combinatorial proof}}
{{main category|Enumerative combinatorics}}
* [[Bijective proof]]
* [[Double counting (proof technique)]]
* [[Method of distinguished element]]
* [[Derangement]] (an example of proof, [[Random permutation statistics#Number of permutations that are derangements|here]])

'''<div id="Combinatorialmodeling">— Combinatorial modeling</div>'''
* [[[Mathematical model]]
* [[Combinatorial modelling]]
* [[Twelvefold way]]
<div style="margin-left: 1.5em; font-style: normal;">

'''- I: Sample selection and unit labelling with and without repetition'''
* [[Sample selection and unit labelling (combinatorics)]]

'''- II: Grouping units (distribution, storage or placement of objects into recipients)'''<br />(Occupancy problems)
* [[Distributions of objects into bins]] <sup>([[:es:Distribución de objetos en recipientes|es]])</sup>
* [[Stars and bars (combinatorics)]]

'''- III: Partition of sets'''
* [[Partition of a set|Partitions of a set]]
* [[Stirling numbers of the second kind]]
* [[Bell number|Bell numbers]]
* [[Lah number|Lah numbers]]
<div style="margin-left: 2em; font-style: normal;">
'''· Catalan and Narayana numbers. Noncrossing partitions'''
* [[Noncrossing partition]]
* [[Catalan number]]
* [[Narayana number]]
</div>

'''- IV: Additive decompositions of numbers'''
* [[Partition (number theory)|Partitions of integers (additive decompositions)]]
</div>

'''<div id="Paradoxes">— Paradoxes</div>'''
* [[Ellsberg paradox]]
* [[Birthday problem]]
* [[Base rate fallacy#False positive paradox|False positive paradox]]
* [[Monty Hall problem]]
* [[Simpson's paradox]]
<!--
{| border=0 align=center cellspacing=20
|-
| <center>{{pp-book-cover</center>
 |title=
 |subtitle=
 |cover-image=Proofs-of-Fermats-Little-Theorem-bracelet1.svg
 |cover-color=#76b3df
}} Libro wiki (itinerario en Wikipedia)<br />¡Pulse en el título, no en la imagen de portada!</center>
|}
-->

====== <span id="Combinatorics_Bibliography"></span>Bibliography: theory and proposed and solved exercises  ======
<hr style="margin-left: 0px; width:50%;">
{| style="width: 100%;"
|In English: || In Spanish:
|- style="vertical-align:top;"
| style="width: 50%;" |
* —¤— Kenneth P. Bogart. <i>Combinatorics through guided discovery</i>. 2004. https://math.dartmouth.edu/news-resources/electronic/kpbogart/
* —¤— Kenneth A. Rosen. <i>Discrete mathematics and its applications</i>. 7th edition. (Chapters 6 and 8 and related exercises). McGraw-Hill, New York, New York, United States, 2012, {{ISBN|978-0-07-338309-5}}
* <cite id="Sedgewick1977" style="font-style: normal;">[[#Sedgewick1977|—¤—]]</cite> {{cite journal |last=Sedgewick |first=Robert |title=Permutation Generation Methods |date=June 1977 |journal=Computing Surveys |volume=9 |issue=2 |pages=137-164 |url=http://homepage.math.uiowa.edu/~goodman/22m150.dir/2007/Permutation%20Generation%20Methods.pdf |isbn=}} © ARR.
| style="width: 50%;" |
* Máximo Anzola and José Caruncho. <i>Problemas de Álgebra. Tomo 1. Conjuntos-Grupos</i>. Primer Ciclo, Madrid, Spain. (Chapter 8 'Combinatoria', 31 solved problems), 1981.
* L. Barrios Calmaestra. Combinatoria. In: <i>Proyecto Descartes</i>. Ministry of Education, Government of Spain, 2007. (Open access). http://descartes.cnice.mec.es/materiales_didacticos/Combinatoria/combinatoria.htm
* M. Delgado Pineda. Material from «Curso 0: Matemáticas». Part: Combinatoria: Variaciones, Permutaciones y Combinaciones. Potencias de un binomio. OCW UNED. (Theory and exercises). 2010. (CC BY-NC-ND). http://ocw.innova.uned.es/matematicas-industriales/contenidos/pdf/tema5.pdf
* I. Espejo Miranda, F. Fernández Palacín, M. A. López Sánchez, M. Muñoz Márquez, A. M. Rodríguez Chía, A. Sánchez Navas and C. Valero Franco. <i>Estadística Descriptiva y Probabilidad</i>. Servicio de Publicaciones de la Universidad de Cádiz. (Appendix 1: Combinatoria). 2006. (GNU FDL). http://knuth.uca.es/repos/l_edyp/pdf/febrero06/lib_edyp.apendices.pdf
* <cite id="FrancoBranas2008" style="font-style: normal;">[[#FrancoBranas2008|—¤—]]</cite> {{cite book |last1=Franco Brañas |first1=José Ramón |last2=Espinel Febles |first2=María Candelaria |last3=Almeida Benítez |first3=Pedro Ramón |title=Manual de combinatoria |year=2008 |publisher=@becedario |location=Badajoz, Extremadura (ES-EX), España |isbn=978-84-96560-73-4}} © ARR.
* —¤— Kenneth A. Rosen. <i>Matemática discreta y sus aplicaciones</i>. 5th edition. (Chapters 4 and 6 and related exercises). McGraw-Hill/Interamericana de España, S.A.U., Aravaca (Madrid), Madrid, Spain, 2004, {{ISBN|84-481-4073-7}}
|}

======  <span id="Combinatorics_Software"></span>Software ======
<hr style="margin-left: 0px; width:50%;">
{| style="width: 100%;"
|In English: || In Spanish:
|- style="vertical-align:top;"
| style="width: 50%;" |
* [https://www.wolframalpha.com/examples/Combinatorics.html WolframAlpha (Examples, Mathematics, Combinatorics)]
| style="width: 50%;" |
|}

====== <span id="Combinatorics_Multimedia"></span>[[File:Multimedia-icon.svg|32px|link=]] Multimedia ======
<hr style="margin-left: 0px; width:50%;">
{| style="width: 100%;"
|In English: || In Spanish:
|- style="vertical-align:top;"
| style="width: 50%;" |
* {{cite web
 |url=https://www.youtube.com/watch?v=JI6f0Gg1Uyo
 |title=Section 6.1 The Basics of Counting
 |last=Baugher
 |first=Greg A.
 |department=Department of Math/Science/Informatics, Penfield College, Mercer University
 |location=Georgia (US-GA), USA
 |format=Video
 |publisher=
 |date=
 |website=
 |access-date=
 |quote= }}
* {{cite web
 |url=https://www.youtube.com/watch?v=oEBUQVLLSdI
 |title=Sections 6.3 & 6.5 Permutations and Combinations
 |last=Baugher
 |first=Greg A.
 |department=Department of Math/Science/Informatics, Penfield College, Mercer University
 |location=Georgia (US-GA), USA
 |format=Video
 |publisher=
 |date=
 |website=
 |access-date=
 |quote= }}
* {{cite web
 |url=https://www.youtube.com/watch?v=7SWK8lPOkEE
 |title=Section 6.6 Generating Permutations and Combinations
 |last=Baugher
 |first=Greg A.
 |department=Department of Math/Science/Informatics, Penfield College, Mercer University
 |location=Georgia (US-GA), USA
 |format=Video
 |publisher=
 |date=
 |website=
 |access-date=
 |quote= }}
* {{cite web
 |url=https://www.youtube.com/watch?v=Eick163K5Eo&list=PL0862D1A947252D20&index=27
 |title=Lecture 27 - Pigeonhole Principle 
 |last=Krithivasan
 |first=Kamala
 |format=Video
 |publisher=Indian Institute of Technology Madras
 |location=Adyar, Chennay (formerly Madras), Tamil Nadu (IN-TN), India
 |date=
 |website=
 |access-date=
 |quote= }}
* {{cite web
 |url=https://www.youtube.com/watch?v=Dsi7x-A89Mw&list=PL0862D1A947252D20&index=28
 |title=Lecture 28 - Permutations and combinations
 |last=Krithivasan
 |first=Kamala
 |format=Video
 |publisher=Indian Institute of Technology Madras
 |location=Adyar, Chennay (formerly Madras), Tamil Nadu (IN-TN), India
 |date=
 |website=
 |access-date=
 |quote= }}
* {{cite web
 |url=https://www.youtube.com/watch?v=74l6t4_4pDg&list=PL0862D1A947252D20&index=29
 |title=Lecture 29 - Permutations and Combinations (cont.) 
 |last=Krithivasan
 |first=Kamala
 |format=Video
 |publisher=Indian Institute of Technology Madras
 |location=Adyar, Chennay (formerly Madras), Tamil Nadu (IN-TN), India
 |date=
 |website=
 |access-date=
 |quote= }}
* {{cite web
 |url=https://www.youtube.com/watch?v=4d2XEn1j_q4&list=PL0862D1A947252D20&index=30
 |title=Lecture 30 - Generating Functions 
 |last=Krithivasan
 |first=Kamala
 |format=Video
 |publisher=Indian Institute of Technology Madras
 |location=Adyar, Chennay (formerly Madras), Tamil Nadu (IN-TN), India
 |date=
 |website=
 |access-date=
 |quote= }}
* {{cite web
 |url=https://www.youtube.com/watch?v=FDKXPDK8ZhQ&list=PL0862D1A947252D20&index=31
 |title=Lecture 31 - Generating Functions (cont.)
 |last=Krithivasan
 |first=Kamala
 |format=Video
 |publisher=Indian Institute of Technology Madras
 |location=Adyar, Chennay (formerly Madras), Tamil Nadu (IN-TN), India
 |date=
 |website=
 |access-date=
 |quote= }}
* {{cite web
 |url=ftp://ftp.aduni.org/videos/02_lect_11.rm
 |title=Lecture 11 - Combinations and permutations
 |last=Simonson
 |first=Shai
 |format=Video
 |publisher=ArsDigita University
 |date=
 |website=
 |access-date=
 |quote= }} ([https://www.youtube.com/watch?v=GODDuoQr3wI&index=11&list=PL__5ik6vMY50HNfWppaGxQydlSMTGQWzC Another URL])
* {{cite web
 |url=ftp://ftp.aduni.org/videos/02_lect_12.rm
 
 |title=Lecture 12 - Counting
 |last=Simonson
 |first=Shai
 |format=Video
 |publisher=ArsDigita University
 |date=
 |website=
 |access-date=
 |quote= }} ([https://www.youtube.com/watch?v=lSuu0AOh6Ig&index=12&list=PL__5ik6vMY50HNfWppaGxQydlSMTGQWzC Another URL])
* {{cite web
 |url=ftp://ftp.aduni.org/videos/02_lect_13.rm
 
 |title=Lecture 13 - Counting (cont.)
 |last=Simonson
 |first=Shai
 |format=Video
 |publisher=ArsDigita University
 |date=
 |website=
 |access-date=
 |quote= }} ([https://www.youtube.com/watch?v=vDW8rs8bVjU&index=13&list=PL__5ik6vMY50HNfWppaGxQydlSMTGQWzC Another URL])
* {{cite web
 |url=ftp://ftp.aduni.org/videos/02_lect_14.rm
 
 |title=Lecture 14 - Counting problems using combinations, distributions
 |last=Simonson
 |first=Shai
 |format=Video
 |publisher=ArsDigita University
 |date=
 |website=
 |access-date=
 |quote= }} ([https://www.youtube.com/watch?v=RdHdBRm1TCY&index=14&list=PL__5ik6vMY50HNfWppaGxQydlSMTGQWzC Another URL])
* {{cite web
 |url=ftp://ftp.aduni.org/videos/02_lect_15.rm
 
 |title=Lecture 15 - Counting problems using combinations, distributions (cont.)
 |last=Simonson
 |first=Shai
 |format=Video
 |publisher=ArsDigita University
 |date=
 |website=
 |access-date=
 |quote= }} ([https://www.youtube.com/watch?v=q8nVXCqDeZU&index=15&list=PL__5ik6vMY50HNfWppaGxQydlSMTGQWzC Another URL])
* {{cite web
 |url=ftp://ftp.aduni.org/videos/02_lect_16.rm
 
 |title=Lecture 16 - The pigeonhole principle and examples. The inclusion/exclusion theorem and advanced examples. A combinatorial card trick
 |last=Simonson
 |first=Shai
 |format=Video
 |publisher=ArsDigita University
 |date=
 |website=
 |access-date=
 |quote= }} ([https://www.youtube.com/watch?v=owVusG8FXsg&index=16&list=PL__5ik6vMY50HNfWppaGxQydlSMTGQWzC Another URL])
* {{cite web
 |url=ftp://ftp.aduni.org/videos/02_reci_01.rm
 
 |title=Recitation 1 - A combinatorial card trick
 |last=Simonson
 |first=Shai
 |format=Video
 |publisher=ArsDigita University
 |date=
 |website=
 |access-date=
 |quote= }} ([https://www.youtube.com/watch?v=P2PmyHoG89w&index=19&list=PL__5ik6vMY50HNfWppaGxQydlSMTGQWzC Another URL])
| style="width: 50%;" |
'''— Basic principles'''
* {{cite web |last=Soto Espinosa |first=Jesús |title=Principios básicos de conteo |format=Video |publisher=Escuela Politécnica Superior, Universidad Católica San Antonio de Murcia (UCAM) |location=Guadalupe, Murcia, Region of Murcia (ES-MC), Spain |url=https://www.youtube.com/watch?v=Kl62fUnnzT8&index=62&list=PL2qtqrBO8xujlH2bo2UPMQ3m9Hj9yEBur |access-date=}}
* {{cite web |last=Soto Espinosa |first=Jesús |title=Ejercicios |format=Video |publisher=Escuela Politécnica Superior, Universidad Católica San Antonio de Murcia (UCAM) |location=Guadalupe, Murcia, Region of Murcia (ES-MC), Spain |url=https://www.youtube.com/watch?v=GnF0-N--ckI&index=67&list=PL2qtqrBO8xujlH2bo2UPMQ3m9Hj9yEBur |access-date=}} (Ejercicio 3).
'''— Variations, permutations and combinations'''
* {{cite web |last=Soto Espinosa |first=Jesús |title=Variaciones |format=Video |publisher=Escuela Politécnica Superior, Universidad Católica San Antonio de Murcia (UCAM) |location=Guadalupe, Murcia, Region of Murcia (ES-MC), Spain |url=https://www.youtube.com/watch?v=SWaYLSpGnXc&index=57&list=PL2qtqrBO8xujlH2bo2UPMQ3m9Hj9yEBur |access-date=}}
* {{cite web |last=Soto Espinosa |first=Jesús |title=Variaciones con repetición |format=Video |publisher=Escuela Politécnica Superior, Universidad Católica San Antonio de Murcia (UCAM) |location=Guadalupe, Murcia, Region of Murcia (ES-MC), Spain |url=https://www.youtube.com/watch?v=kp5sZ8SF9pQ&index=56&list=PL2qtqrBO8xujlH2bo2UPMQ3m9Hj9yEBur |access-date=}}
* {{cite web |last=Soto Espinosa |first=Jesús |title=Permutaciones |format=Video |publisher=Escuela Politécnica Superior, Universidad Católica San Antonio de Murcia (UCAM) |location=Guadalupe, Murcia, Region of Murcia (ES-MC), Spain |url=https://www.youtube.com/watch?v=7UhetaiTlXs&index=55&list=PL2qtqrBO8xujlH2bo2UPMQ3m9Hj9yEBur |access-date=}}
* {{cite web |last=Soto Espinosa |first=Jesús |title=Permutaciones, ejemplo 1 |format=Video |publisher=Escuela Politécnica Superior, Universidad Católica San Antonio de Murcia (UCAM) |location=Guadalupe, Murcia, Region of Murcia (ES-MC), Spain |url=https://www.youtube.com/watch?v=xqL4VY0GV0Q&index=64&list=PL2qtqrBO8xujlH2bo2UPMQ3m9Hj9yEBur |access-date=}}
* {{cite web |last=Soto Espinosa |first=Jesús |title=Permutaciones circulares |format=Video |publisher=Escuela Politécnica Superior, Universidad Católica San Antonio de Murcia (UCAM) |location=Guadalupe, Murcia, Region of Murcia (ES-MC), Spain |url=https://www.youtube.com/watch?v=-Mp0I3LtGHI&index=54&list=PL2qtqrBO8xujlH2bo2UPMQ3m9Hj9yEBur |access-date=}}
* {{cite web |last=Soto Espinosa |first=Jesús |title=Permutaciones con repetición |format=Video |publisher=Escuela Politécnica Superior, Universidad Católica San Antonio de Murcia (UCAM) |location=Guadalupe, Murcia, Region of Murcia (ES-MC), Spain |url=https://www.youtube.com/watch?v=sqIp8xPYWio&index=53&list=PL2qtqrBO8xujlH2bo2UPMQ3m9Hj9yEBur |access-date=}}
* {{cite web |last=Soto Espinosa |first=Jesús |title=Combinaciones |format=Video |publisher=Escuela Politécnica Superior, Universidad Católica San Antonio de Murcia (UCAM) |location=Guadalupe, Murcia, Region of Murcia (ES-MC), Spain |url=https://www.youtube.com/watch?v=8eV8DR3TKvE&index=52&list=PL2qtqrBO8xujlH2bo2UPMQ3m9Hj9yEBur |access-date=}}
* {{cite web |last=Soto Espinosa |first=Jesús |title=Combinaciones con repetición |format=Video |publisher=Escuela Politécnica Superior, Universidad Católica San Antonio de Murcia (UCAM) |location=Guadalupe, Murcia, Region of Murcia (ES-MC), Spain |url=https://www.youtube.com/watch?v=nAiblNllCOI&index=51&list=PL2qtqrBO8xujlH2bo2UPMQ3m9Hj9yEBur |access-date=}}
* {{cite web |last=Soto Espinosa |first=Jesús |title=Ejercicios |format=Video |publisher=Escuela Politécnica Superior, Universidad Católica San Antonio de Murcia (UCAM) |location=Guadalupe, Murcia, Region of Murcia (ES-MC), Spain |url=https://www.youtube.com/watch?v=ggG26wurw3U&index=60&list=PL2qtqrBO8xujlH2bo2UPMQ3m9Hj9yEBur |access-date=}}
'''— Binomial numbers'''
* {{cite web |last=Soto Espinosa |first=Jesús |title=Número binomial |format=Video |publisher=Escuela Politécnica Superior, Universidad Católica San Antonio de Murcia (UCAM) |location=Guadalupe, Murcia, Region of Murcia (ES-MC), Spain |url=https://www.youtube.com/watch?v=KL19bVBoz1U&index=61&list=PL2qtqrBO8xujlH2bo2UPMQ3m9Hj9yEBur |access-date=}}
* {{cite web |last=Soto Espinosa |first=Jesús |title=Número binomial, ejercicio 2 |format=Video |publisher=Escuela Politécnica Superior, Universidad Católica San Antonio de Murcia (UCAM) |location=Guadalupe, Murcia, Region of Murcia (ES-MC), Spain |url=https://www.youtube.com/watch?v=Tjj7IWXj4Vw&index=28&list=PL2qtqrBO8xujlH2bo2UPMQ3m9Hj9yEBur |access-date=}}
* {{cite web |last=Soto Espinosa |first=Jesús |title=Número binomial, ejercicio 3 |format=Video |publisher=Escuela Politécnica Superior, Universidad Católica San Antonio de Murcia (UCAM) |location=Guadalupe, Murcia, Region of Murcia (ES-MC), Spain |url=https://www.youtube.com/watch?v=SW_SXZZDffc&index=29&list=PL2qtqrBO8xujlH2bo2UPMQ3m9Hj9yEBur |access-date=}}
* {{cite web |last=Soto Espinosa |first=Jesús |title=Número binomial, ejercicio 5 |format=Video |publisher=Escuela Politécnica Superior, Universidad Católica San Antonio de Murcia (UCAM) |location=Guadalupe, Murcia, Region of Murcia (ES-MC), Spain |url=https://www.youtube.com/watch?v=ez6fJnnD2e4&index=5&list=PL2qtqrBO8xujlH2bo2UPMQ3m9Hj9yEBur |access-date=}}
* {{cite web |last=Soto Espinosa |first=Jesús |title=Número binomial, fórmula de Stifel |format=Video |publisher=Escuela Politécnica Superior, Universidad Católica San Antonio de Murcia (UCAM) |location=Guadalupe, Murcia, Region of Murcia (ES-MC), Spain |url=https://www.youtube.com/watch?v=FSlJdLTXeAU&index=43&list=PL2qtqrBO8xujlH2bo2UPMQ3m9Hj9yEBur |access-date=}}
* {{cite web |last=Soto Espinosa |first=Jesús |title=Coeficiente Multinomial, ejercicio 1 |format=Video |publisher=Escuela Politécnica Superior, Universidad Católica San Antonio de Murcia (UCAM) |location=Guadalupe, Murcia, Region of Murcia (ES-MC), Spain |url=https://www.youtube.com/watch?v=RfVTNR0oFY4&index=2&list=PL2qtqrBO8xujlH2bo2UPMQ3m9Hj9yEBur |access-date=}}
* {{cite web |last=Soto Espinosa |first=Jesús |title=Combinatoria, ejemplo 6 |format=Video |publisher=Escuela Politécnica Superior, Universidad Católica San Antonio de Murcia (UCAM) |location=Guadalupe, Murcia, Region of Murcia (ES-MC), Spain |url=https://www.youtube.com/watch?v=5t_EDHqmG-g&index=66&list=PL2qtqrBO8xujlH2bo2UPMQ3m9Hj9yEBur |access-date=}}
* {{cite web |last=Soto Espinosa |first=Jesús |title=Combinatoria, ejemplo 7 |format=Video |publisher=Escuela Politécnica Superior, Universidad Católica San Antonio de Murcia (UCAM) |location=Guadalupe, Murcia, Region of Murcia (ES-MC), Spain |url=https://www.youtube.com/watch?v=UxIFACqHi9k&index=65&list=PL2qtqrBO8xujlH2bo2UPMQ3m9Hj9yEBur |access-date=}}
* {{cite web |last=Soto Espinosa |first=Jesús |title=Combinatoria, ejemplo 8 |format=Video |publisher=Escuela Politécnica Superior, Universidad Católica San Antonio de Murcia (UCAM) |location=Guadalupe, Murcia, Region of Murcia (ES-MC), Spain |url=https://www.youtube.com/watch?v=BmN4ID8JhxQ&index=9&list=PL2qtqrBO8xujlH2bo2UPMQ3m9Hj9yEBur |access-date=}}
* {{cite web |last=Soto Espinosa |first=Jesús |title=Combinatoria, ejemplo 9 |format=Video |publisher=Escuela Politécnica Superior, Universidad Católica San Antonio de Murcia (UCAM) |location=Guadalupe, Murcia, Region of Murcia (ES-MC), Spain |url=https://www.youtube.com/watch?v=BSyBoX-Kyv8&index=8&list=PL2qtqrBO8xujlH2bo2UPMQ3m9Hj9yEBur |access-date=}}
* {{cite web |last=Soto Espinosa |first=Jesús |title=Combinatoria, ejemplo 10 |format=Video |publisher=Escuela Politécnica Superior, Universidad Católica San Antonio de Murcia (UCAM) |location=Guadalupe, Murcia, Region of Murcia (ES-MC), Spain |url=https://www.youtube.com/watch?v=Da9fiqcB6LI&index=7&list=PL2qtqrBO8xujlH2bo2UPMQ3m9Hj9yEBur |access-date=}}
* {{cite web |last=Soto Espinosa |first=Jesús |title=Combinatoria, ejemplo 11 |format=Video |publisher=Escuela Politécnica Superior, Universidad Católica San Antonio de Murcia (UCAM) |location=Guadalupe, Murcia, Region of Murcia (ES-MC), Spain |url=https://www.youtube.com/watch?v=k5kgrTeRVjA&index=6&list=PL2qtqrBO8xujlH2bo2UPMQ3m9Hj9yEBur |access-date=}}
* {{cite web |last=Soto Espinosa |first=Jesús |title=Teorema del binomio |format=Video |publisher=Escuela Politécnica Superior, Universidad Católica San Antonio de Murcia (UCAM) |location=Guadalupe, Murcia, Region of Murcia (ES-MC), Spain |url=https://www.youtube.com/watch?v=FmOWznaXJtM&index=42&list=PL2qtqrBO8xujlH2bo2UPMQ3m9Hj9yEBur |access-date=}}
* {{cite web |last=Soto Espinosa |first=Jesús |title=Fórmula de Leibniz |format=Video |publisher=Escuela Politécnica Superior, Universidad Católica San Antonio de Murcia (UCAM) |location=Guadalupe, Murcia, Region of Murcia (ES-MC), Spain |url=https://www.youtube.com/watch?v=4yTx8T-1yVY&index=41&list=PL2qtqrBO8xujlH2bo2UPMQ3m9Hj9yEBur |access-date=}}
* {{cite web |last=Soto Espinosa |first=Jesús |title=Ejercicios |format=Video |publisher=Escuela Politécnica Superior, Universidad Católica San Antonio de Murcia (UCAM) |location=Guadalupe, Murcia, Region of Murcia (ES-MC), Spain |url=https://www.youtube.com/watch?v=5B5nrFu0704&index=69&list=PL2qtqrBO8xujlH2bo2UPMQ3m9Hj9yEBur |access-date=}}
'''— Inclusion-exclusion principle'''
* {{cite web |last=Soto Espinosa |first=Jesús |title=Principio de inclusión-exclusión |format=Video |publisher=Escuela Politécnica Superior, Universidad Católica San Antonio de Murcia (UCAM) |location=Guadalupe, Murcia, Region of Murcia (ES-MC), Spain |url=https://www.youtube.com/watch?v=ulWUfA1v1Yg&index=35&list=PL2qtqrBO8xujlH2bo2UPMQ3m9Hj9yEBur |access-date=}}
* {{cite web |last=Soto Espinosa |first=Jesús |title=Generalización del principio de inclusión-exclusión |format=Video |publisher=Escuela Politécnica Superior, Universidad Católica San Antonio de Murcia (UCAM) |location=Guadalupe, Murcia, Region of Murcia (ES-MC), Spain |url=https://www.youtube.com/watch?v=nugR2mChJYU&index=34&list=PL2qtqrBO8xujlH2bo2UPMQ3m9Hj9yEBur |access-date=}}
* {{cite web |last=Soto Espinosa |first=Jesús |title=Principio de inclusión-exclusión - Ejemplo 1 |format=Video |publisher=Escuela Politécnica Superior, Universidad Católica San Antonio de Murcia (UCAM) |location=Guadalupe, Murcia, Region of Murcia (ES-MC), Spain |url=https://www.youtube.com/watch?v=cx-zcOhmBcY&index=36&list=PL2qtqrBO8xujlH2bo2UPMQ3m9Hj9yEBur |access-date=}}
* {{cite web |last=Soto Espinosa |first=Jesús |title=Desarreglos |format=Video |publisher=Escuela Politécnica Superior, Universidad Católica San Antonio de Murcia (UCAM) |location=Guadalupe, Murcia, Region of Murcia (ES-MC), Spain |url=https://www.youtube.com/watch?v=lCMuk0aVzUI&index=40&list=PL2qtqrBO8xujlH2bo2UPMQ3m9Hj9yEBur |access-date=}}
* {{cite web |last=Soto Espinosa |first=Jesús |title=Contando desarreglos |format=Video |publisher=Escuela Politécnica Superior, Universidad Católica San Antonio de Murcia (UCAM) |location=Guadalupe, Murcia, Region of Murcia (ES-MC), Spain |url=https://www.youtube.com/watch?v=jsODf-yfTEc&index=39&list=PL2qtqrBO8xujlH2bo2UPMQ3m9Hj9yEBur |access-date=}}
* {{cite web |last=Soto Espinosa |first=Jesús |title=Ejercicios |format=Video |publisher=Escuela Politécnica Superior, Universidad Católica San Antonio de Murcia (UCAM) |location=Guadalupe, Murcia, Region of Murcia (ES-MC), Spain |url=https://www.youtube.com/watch?v=92LBS423qhQ&index=68&list=PL2qtqrBO8xujlH2bo2UPMQ3m9Hj9yEBur |access-date=}}
'''— Partitions'''
* {{cite web |last=Soto Espinosa |first=Jesús |title=Particiones. Número de Bell |format=Video |publisher=Escuela Politécnica Superior, Universidad Católica San Antonio de Murcia (UCAM) |location=Guadalupe, Murcia, Region of Murcia (ES-MC), Spain |url=https://www.youtube.com/watch?v=Sx6xpCcyECE&index=37&list=PL2qtqrBO8xujlH2bo2UPMQ3m9Hj9yEBur |access-date=}}
* {{cite web |last=Soto Espinosa |first=Jesús |title=Número de Stirling de segunda clase |format=Video |publisher=Escuela Politécnica Superior, Universidad Católica San Antonio de Murcia (UCAM) |location=Guadalupe, Murcia, Region of Murcia (ES-MC), Spain |url=https://www.youtube.com/watch?v=UKQLyrM6dZg&index=38&list=PL2qtqrBO8xujlH2bo2UPMQ3m9Hj9yEBur |access-date=}}
* {{cite web |last=Soto Espinosa |first=Jesús |title=Ejercicios |format=Video |publisher=Escuela Politécnica Superior, Universidad Católica San Antonio de Murcia (UCAM) |location=Guadalupe, Murcia, Region of Murcia (ES-MC), Spain |url=https://www.youtube.com/watch?v=WizhEPApP7I&index=63&list=PL2qtqrBO8xujlH2bo2UPMQ3m9Hj9yEBur |access-date=}}
|}

====== <span id="Combinatorics_See_also"></span>See also ======
<hr style="margin-left: 0px; width:50%;">
* [[Kirkman's schoolgirl problem]]
* [[Magic square]]
* [[Latin square]]
* [[Trinomial triangle]]
* [[Permutation generation methods]] <sup>([[Wikipedia:School_and_university_projects/Discrete_and_numerical_mathematics/Learning_plan/Sandbox#Permutation generation methods|ref]])</sup>
** [[Heap's algorithm]]
** [[Steinhaus–Johnson–Trotter algorithm]] (and its relation to [[change ringing]]).
<!--* <del>[http://cala.unex.es/cala/epistemowikia/index.php/Ejercicios_propuestos_sobre_combinatoria_y_algunas_soluciones Ejercicios propuestos sobre combinatoria y algunas soluciones] (Proposed exercises about combinatorics and some solutions) (On Epistemowikia and in Spanish, for the time being)</del>-->

====== <span id="Combinatorics_To_find_out_more"></span>To find out more ======
<hr style="margin-left: 0px; width:50%;">
<ol style="list-style-type:circle;">
  <li>[[Portal:Mathematics]]</li>
  <li>And more:</li>
  <ol style="list-style-type:circle;">
    <li>[[Generating function|Generating functions]]</li>
    <li>[[Examples of generating functions]]</li>
  </ol>
</ol>
|}

==== <span style="color: rgb(0, 0, 0);">Theme 4.- Difference equations</span> ====

{| class="mw-collapsible wikitable {{#ifeq: {{{plegada|no}}}|sí|mw-collapsed|}}" width="100%" style="text-align:left; padding:0px; background-color:#F9F9F9;"
| align="left"   |

===== [[Recurrence relation|Finite difference equations (recurrence relations)]] =====
|-
| style="background-color:#F9F9F9; text-align:left; font-size:100%; padding:6px" |
{|
|[[File:ADI-stencil.svg|200px]]
|<div id="toc_RecurrenceRelations" class="toc">
<div class="toctitle">
'''Contents'''
</div>
# [[#RecurrenceRelations_Key_concepts|Key concepts]]
# [[#RecurrenceRelations_Bibliography|Bibliography]]
# [[#RecurrenceRelations_Software|Software]]
# [[#RecurrenceRelations_Multimedia|Multimedia]]
# [[#RecurrenceRelations_To_find_out_more|To find out more]]
</div>
|}

====== <span id="RecurrenceRelations_Key_concepts"></span>Key concepts ======
<hr style="margin-left: 0px; width:50%;">
{{Main|Recurrence relation}}
{{Main category|Recurrence relations}}
<div role="note" class="hatnote">Some useful previous concepts: [[Recursive definition]], [[Recursion]] and [[Recursion (computer science)]]</div>
'''<div id="Lineardifferenceequations">— Linear difference equations</div>'''
{{Main|Linear difference equation}}
* [[Recurrence relation|Recurrence relation (difference equation)]]
* [[Recurrence relation#Solving homogeneous linear recurrence relations with constant coefficients|Solving homogeneous linear recurrence relations with constant coefficients]]
* [[Recurrence relation#Solving non-homogeneous linear recurrence relations with constant coefficients|Solving non-homogeneous linear recurrence relations with constant coefficients]]
'''<div id="Lineardiscretedynamicalsystems">— Linear discrete dynamical systems</div>'''
<div style="margin-left: 1.5em; font-style: normal;">
'''- Population dynamics'''
* [[Population dynamics]]
'''- Linear discrete dynamical models'''
* [[Dynamical system]]
'''- BIDE models'''
* [[Matrix population models]]
'''- Markov chains'''
* [[Markov chain]]
</div>
'''<div id="Solvingequationsnumerically">— Solving equations numerically</div>'''
{{main|Root-finding algorithm}}
{{Main category|Root-finding algorithms}}
{{see also|Numerical analysis}}
* [[Fixed-point iteration]]
* [[Secant method]]
====== <span id="RecurrenceRelations_Connections"></span>Connections ======
<hr style="margin-left: 0px; width:50%;">
'''— Computational complexity'''
{{Main|Computational complexity theory}}
<div class="dablink" style="font-style:normal;">
'''- Analysis of algorithms'''
{{main|Analysis of algorithms}}
</div>
<!--
{| border=0 align=center cellspacing=20
|-
| <center>{{pp-book-cover</center>
 |title=[[Usuario:Jmleonrojas/Libros/Ecuaciones_en_diferencias_finitas|Ecuaciones en diferencias finitas]]
 |subtitle=
 |cover-image=ADI-stencil.svg
 |cover-color=#76b3df
}} Libro wiki (itinerario en Wikipedia)<br />¡Pulse en el título, no en la imagen de portada!</center>
|}
-->

====== <span id="RecurrenceRelations_Bibliography"></span>Bibliography: theory and proposed and solved exercises  ======
<hr style="margin-left: 0px; width:50%;">
{| style="width: 100%;"
|En español: || En inglés:
|- style="vertical-align:top;"
| style="width: 50%;" |
* Richard Johnsonbaugh. <i>Discrete mathematics</i>. 6th edition. (Chapter 7 and corresponding exercises). Prentice Hall Inc., New York, New York, United States, 2005, {{ISBN|0-13-117686-2}}
* —¤— Eric Lehman, F. Thomson Leighton, Albert R. Meyer. Mathematics for Computer Science. (Chapter V). 2017 (2 May). CC BY-SA. https://courses.csail.mit.edu/6.042/spring17/mcs.pdf
* —¤— Kenneth A. Rosen. <i>Discrete mathematics and its applications</i>. 7th edition. (Chapters 5 and 8 and corresponding exercises). McGraw-Hill, Nueva York, Nueva York, Estados Unidos, 2012, {{ISBN|978-0-07-338309-5}}
* Wikibooks contributors. Discrete Mathematics/Recursion. <i>Wikibooks</i>. https://en.wikibooks.org/wiki/Discrete_Mathematics/Recursion
| style="width: 50%;" |
* Javier Cobos Gavala. <i>Apuntes de introducción a la matemática discreta</i>. (E.T.S. de ingeniería informática, University of Seville). http://ma1.eii.us.es/material/IMD_ii_Ap.pdf
* Colaboradores de Wikilibros. Matemática Discreta/Relaciones de Recurrencia. <i>Wikilibros</i>. https://es.wikibooks.org/wiki/Matem%C3%A1tica_Discreta/Relaciones_de_Recurrencia
* —¤— Richard Johnsonbaugh. <i>Matemáticas discretas</i>. 6th edition. (Chapter 7 and corresponding exercises). Pearson Educación de México, S.A. de C.V., Naucalpan de Juárez, Edo. de México, México, 2005, {{ISBN|970-26-0637-3}}
* Alfredo Méndez Alonso, Eduardo Martín Novo, Carmen Ortiz Martínez and Juana Sendra Pons. <i>Tema 4: Relaciones de recurrencia</i>. OCW-UPM, 2011. CC BY-NC-SA. http://ocw.upm.es/matematica-aplicada/matematica-discreta/contenidos/material-de-clase/tema-4
* <cite id="NavasEstebanQuesada2009a" style="font-style: normal;">[[#NavasEstebanQuesada2009a|—¤—]]</cite> {{cite book |last1=Navas Ureña |first1=Juan |last2=Esteban Ruiz |first2=Francisco José |last3=Quesada Teruel |first3=José María |title=Modelos matemáticos en biología. Teoría |year=2009a |chapter=Tema 9. Ecuaciones y sistemas en diferencias |publisher=Universidad de Jaén. Departamento de Matemáticas |location=Jaén, Andalucía (ES-AN), España |url=http://matema.ujaen.es/jnavas/web_modelos/pdf_mmb08_09/texto%20completo.pdf |chapter-url=http://matema.ujaen.es/jnavas/web_modelos/pdf_mmb08_09/sistemas%20dinamicos.pdf |isbn=}} © TDR.
* <cite id="NavasEstebanQuesada2009b" style="font-style: normal;">[[#NavasEstebanQuesada2009b|—¤—]]</cite> {{cite book |last1=Navas Ureña |first1=Juan |last2=Esteban Ruiz |first2=Francisco José |last3=Quesada Teruel |first3=José María |title=Modelos matemáticos en biología. Ejercicios resueltos y propuestos |year=2009b |chapter=Tema 2. Modelos discretos II |publisher=Universidad de Jaén. Departamento de Matemáticas |location=Jaén, Andalucía (ES-AN), España |url=http://matema.ujaen.es/jnavas/web_modelos/problemas/problemas%20completo.pdf |chapter-url=http://matema.ujaen.es/jnavas/web_modelos/problemas/ptema2.pdf |isbn=}} © TDR.
* —¤— Kenneth A. Rosen. <i>Matemática discreta y sus aplicaciones</i>. 5th edition. (Chapters 4 and 6 and corresponding exercises). McGraw-Hill/Interamericana de España, S.A.U., Aravaca (Madrid), Madrid, Spain, 2004, {{ISBN|84-481-4073-7}}
* Enrique Vílchez Quesada. Resolución de relaciones de recurrencia lineales no homogéneas con coeficientes constantes a través de valores y vectores propios. <i>Uniciencia</i>, <i>24</i>, 2010, pp. 121-132. https://dialnet.unirioja.es/servlet/articulo?codigo=5381351
* Álex Zeballos. Solución a Ecuaciones de Recurrencia Online - Usando Wolfram Alpha. http://wolframalpha0.blogspot.com.es/2012/01/solucion-ecuaciones-de-recurrencia.html
|}

====== <span id="RecurrenceRelations_Software"></span>Software ======
<hr style="margin-left: 0px; width:50%;">
{| style="width: 100%;"
|In English: || In Spanish:
|- style="vertical-align:top;"
| style="width: 50%;" |
* [https://www.wolframalpha.com/examples/Recurrences.html WolframAlpha (Examples, Mathematics, Recurrences)]
| style="width: 50%;" |
|}

====== <span id="RecurrenceRelations_Multimedia"></span>[[File:Multimedia-icon.svg|32px|link=]] Multimedia ======
<hr style="margin-left: 0px; width:50%;">
{| style="width: 100%;"
|In English: || In Spanish:
|- style="vertical-align:top;"
| style="width: 50%;" |
* {{cite web
 |url=https://www.youtube.com/watch?v=qvw1GX93JSY
 |title=Lecture 32 - Recurrence relations
 |last=Krithivasan
 |first=Kamala
 |format=Video
 |publisher=Indian Institute of Technology Madras
 |location=Adyar, Chennay (formerly Madras), Tamil Nadu (IN-TN), India
 |date=
 |website=
 |access-date=
 |quote= }}
* {{cite web
 |url=https://www.youtube.com/watch?v=ImLM1Vsr35c
 |title=Lecture 33 - Recurrence relations (cont.)
 |last=Krithivasan
 |first=Kamala
 |format=Video
 |publisher=Indian Institute of Technology Madras
 |location=Adyar, Chennay (formerly Madras), Tamil Nadu (IN-TN), India
 |date=
 |website=
 |access-date=
 |quote= }}
* {{cite web
 |url=https://www.youtube.com/watch?v=vNf_XY7G0uo
 |title=Lecture 34 - Recurrence relations (cont.)
 |last=Krithivasan
 |first=Kamala
 |format=Video
 |publisher=Indian Institute of Technology Madras
 |location=Adyar, Chennay (formerly Madras), Tamil Nadu (IN-TN), India
 |date=
 |website=
 |access-date=
 |quote= }}
* {{cite web
 |url=ftp://ftp.aduni.org/videos/02_lect_06.rm
 
 |title=Lecture 6 - Basic arithmetic and geometric sums, closed forms.
 |last=Simonson
 |first=Shai
 |format=Video
 |publisher=ArsDigita University
 |date=
 |website=
 |access-date=
 |quote= }} ([https://www.youtube.com/watch?v=492OmzjIsw8&index=6&list=PL__5ik6vMY50HNfWppaGxQydlSMTGQWzC Another URL])
* {{cite web
 |url=ftp://ftp.aduni.org/videos/02_lect_07.rm
 
 |title=Lecture 7 - Chinese rings puzzle
 |last=Simonson
 |first=Shai
 |format=Video
 |publisher=ArsDigita University
 |date=
 |website=
 |access-date=
 |quote= }} ([https://www.youtube.com/watch?v=hupn2J8Wvys&index=7&list=PL__5ik6vMY50HNfWppaGxQydlSMTGQWzC Another URL])
* {{cite web
 |url=ftp://ftp.aduni.org/videos/02_lect_08.rm
 
 |title=Lecture 8 - Solving recurrence equations
 |last=Simonson
 |first=Shai
 |format=Video
 |publisher=ArsDigita University
 |date=
 |website=
 |access-date=
 |quote= }} ([https://www.youtube.com/watch?v=s2f15X4-GDU&index=8&list=PL__5ik6vMY50HNfWppaGxQydlSMTGQWzC Another URL])
* {{cite web
 |url=ftp://ftp.aduni.org/videos/02_lect_09.rm
 
 |title=Lecture 9 - Solving recurrence equations (cont.)
 |last=Simonson
 |first=Shai
 |format=Video
 |publisher=ArsDigita University
 |date=
 |website=
 |access-date=
 |quote= }} ([https://www.youtube.com/watch?v=TPb_gnCczpY&index=9&list=PL__5ik6vMY50HNfWppaGxQydlSMTGQWzC Another URL])
| style="width: 50%;" |
* {{cite web |last=Rodríguez Álvarez |first=María José |department=Departamento de Matemática Aplicada, Escuela Técnica Superior de Ingeniería Informática |title=Método de la ecuación característica para recurrencias homogéneas lineales de segundo orden con coeficientes constantes (raíces reales) |format=Video |publisher=Universidad Politécnica de Valencia (UPV) |location=Valencia, Valencian Community (ES-VC), Spain |url=https://media.upv.es/#/portal/video/25060ec1-3934-b74a-bb81-4c146afc0e75  |access-date=}} ([https://www.youtube.com/watch?v=JHeVpfXyaB4 Otra URL])
* {{cite web |last=Rodríguez Álvarez |first=María José |department=Departamento de Matemática Aplicada, Escuela Técnica Superior de Ingeniería Informática |title=Recurrencias lineales de segundo orden con coeficientes constantes no homogénea (constante) |format=Video |publisher=Universidad Politécnica de Valencia (UPV) |location=Valencia, Valencian Community (ES-VC), Spain |url=https://media.upv.es/#/portal/video/a30514ee-d1fc-4c47-aa04-082249a47eaa  |access-date=}} ([https://www.youtube.com/watch?v=GNUzf3EAGuY Otra URL])
* {{cite web |last=Rodríguez Álvarez |first=María José |department=Departamento de Matemática Aplicada, Escuela Técnica Superior de Ingeniería Informática |title=Recurrencias no homogéneas lineales de segundo orden con coeficientes constantes |format=Video |publisher=Universidad Politécnica de Valencia (UPV) |location=Valencia, Valencian Community (ES-VC), Spain |url=https://media.upv.es/#/portal/video/c617011e-c399-be43-b316-33c6b53d9385  |access-date=}} ([https://www.youtube.com/watch?v=muRVDIhumBQ Otra URL])
* {{cite web |last=Rodríguez Álvarez |first=María José |department=Departamento de Matemática Aplicada, Escuela Técnica Superior de Ingeniería Informática |title=Resolución de recurrencias con Mathematica |format=Video |publisher=Universidad Politécnica de Valencia (UPV) |location=Valencia, Valencian Community (ES-VC), Spain |url=https://media.upv.es/#/portal/video/b40afff0-30f5-11e7-9247-516e3f451f9f  |access-date=}} ([https://www.youtube.com/watch?v=WVDXJ_QvVK4 Otra URL])
|}

====== <span id="RecurrenceRelations_To_find_out_more"></span>To find out more ======
<hr style="margin-left: 0px; width:50%;">
<ol style="list-style-type:circle;">
  <li>[[:Category:Integer sequences|Integer sequences]]</li>
  <li>[[List of OEIS sequences|List of integer sequences in the OEIS that have their own English Wikipedia entries]]</li>
  <li>[http://oeis.org/wiki/Index_to_OEIS:_Section_Rec Index to OEIS: Section Recurrent Sequencies]</li>
  <li>[[Recursion (computer science)]]</li>
  <li>[[Exponential factorial]]</li>
  <li>[[Ackermann function]]</li>
  <li>[[McCarthy 91 function]]</li>
  <li>[[Tower of Hanoi]]</li>
  <li>[[Josephus problem]]</li>
</ol>
|}

==== <span style="color: rgb(0, 0, 0);">Appendices</span> ====

{| class="mw-collapsible wikitable {{#ifeq: {{{plegada|no}}}|sí|mw-collapsed|}}" width="100%" style="text-align:left; padding:0px; background-color:#F9F9F9;"
| align="left"   |

===== Graphs =====
|-
| style="background-color:#F9F9F9; text-align:left; font-size:100%; padding:6px" |
{|
|[[File:Graph isomorphism.svg|172px]]
|<div id="toc_Graphs" class="toc">
<div class="toctitle">
'''Contents'''
</div>
# [[#Graphs_Key_concepts|Key concepts]]
# [[#Graphs_Bibliography|Bibliography]]
# [[#Graphs_Software|Software]]
# [[#Graphs_Multimedia|Multimedia]]
# [[#Graphs_See_also|See also]]
# [[#Graphs_To_find_out_more|To find out more]]
</div>
|}

====== <span id="Graphs_Key_concepts"></span>Key concepts ======
<hr style="margin-left: 0px; width:50%;">
* [[Graph theory]]
* [[Graph (discrete mathematics)]]
* [[Seven Bridges of Königsberg]]
* [[Five room puzzle]]
* [[Eulerian path]]
* [[Hamiltonian path]]
* [[Hamiltonian path problem]]
* [[Graph isomorphism]]
* [[Graph isomorphism problem]]
* [[Graph connectivity]]
* [[Matching (graph theory)]]
* [[Assignment problem]]
* [[:Category:Graph algorithms|Graph algorithms]]
<!--
{| border=0 align=center cellspacing=20
|-
| <center>{{pp-book-cover</center>
 |title=[[Usuario:Jmleonrojas/Libros/Ecuaciones_en_diferencias_finitas|Ecuaciones en diferencias finitas]]
 |subtitle=
 |cover-image=Graph isomorphism.svg
 |cover-color=#76b3df
}} Libro wiki (itinerario en Wikipedia)<br />¡Pulse en el título, no en la imagen de portada!</center>
|}
-->

====== <span id="Graphs_Bibliography"></span>Bibliography: theory and proposed and solved exercises  ======
<hr style="margin-left: 0px; width:50%;">
{| style="width: 100%;"
|In English: || In Spanish:
|- style="vertical-align:top;"
| style="width: 50%;" |
* —¤— Kenneth A. Rosen. <i>Discrete mathematics and its applications</i>. 7th edition. (Chapters 10 and 11 and corresponding exercises). McGraw-Hill, New York, New York, United States, 2012, {{ISBN|978-0-07-338309-5}}
| style="width: 50%;" |
* —¤— Kenneth A. Rosen. <i>Matemática discreta y sus aplicaciones</i>. 5th edition. (Chapters 8 and 9 and corresponding exercises). McGraw-Hill/Interamericana de España, S.A.U., Aravaca (Madrid), Madrid, Spain, 2004, {{ISBN|84-481-4073-7}}
|}

====== <span id="Graphs_Software"></span>Software ======
<hr style="margin-left: 0px; width:50%;">
{| style="width: 100%;"
|In English: || In Spanish:
|- style="vertical-align:top;"
| style="width: 50%;" |
* [https://www.wolframalpha.com/examples/GraphTheory.html WolframAlpha (Examples, Mathematics, Graph Theory)]
| style="width: 50%;" |
|}

====== <span id="Graphs_Multimedia"></span>[[File:Multimedia-icon.svg|32px|link=]] Multimedia ======
<hr style="margin-left: 0px; width:50%;">
{| style="width: 100%;"
|In English: || In Spanish:
|- style="vertical-align:top;"
| style="width: 50%;" |
* {{cite web
 |url=https://www.youtube.com/user/DrSaradaHerke
 |title=Sarada Herke's Youtube channel focused on graph theory and discrete mathematics
 |last=Herke
 |first=Sarada
 |format=Collection of videos
 |publisher=
 |location=Brisbane, Queensland (AU-QLD), Commonwealth of Australia
 |date=
 |department=School of Mathematics and Physics, The University of Queensland
 |website=
 |access-date=
 |quote= }}
* {{cite web
 |url=https://www.youtube.com/watch?v=RMLR2JHHeWo
 |title=Lecture 14 - Graphs
 |last=Krithivasan
 |first=Kamala
 |format=Video
 |publisher=Indian Institute of Technology Madras
 |location=Adyar, Chennay (formerly Madras), Tamil Nadu (IN-TN), India
 |date=
 |website=
 |access-date=
 |quote= }}
* {{cite web
 |url=https://www.youtube.com/watch?v=nf9e0_ylGdc
 |title=Lecture 15 - Graphs (Contd.)
 |last=Krithivasan
 |first=Kamala
 |format=Video
 |publisher=Indian Institute of Technology Madras
 |location=Adyar, Chennay (formerly Madras), Tamil Nadu (IN-TN), India
 |date=
 |website=
 |access-date=
 |quote= }}
| style="width: 50%;" |
* {{cite web |last=UPV |first= |title=Teoría básica de grafos y análisis de 4 conocidos problemas |format=Course |publisher=Escuela Técnica Superior de Ingeniería Informática, Universidad Politécnica de Valencia (UPV) |location=Valencia, Valencian Community (ES-VC), Spain |url=https://www.youtube.com/playlist?list=PL5098BF5A01819B3B |access-date=}} © gratis OA.
* {{cite web |last=UPV |first= |title=Aplicaciones de la Teoría de Grafos a la vida real I |format=MOOC |publisher=Escuela Técnica Superior de Ingeniería Informática, Universidad Politécnica de Valencia (UPV) |location=Valencia, Valencian Community (ES-VC), Spain |url=https://www.youtube.com/playlist?list=PL6kQim6ljTJu44dsVeZifHHiuDC1MEZ7q |access-date=}} © gratis OA.
* {{cite web |last=UPV |first= |title=Aplicaciones de la Teoría de Grafos a la vida real II |format=MOOC |publisher=Escuela Técnica Superior de Ingeniería Informática, Universidad Politécnica de Valencia (UPV) |location=Valencia, Valencian Community (ES-VC), Spain |url=https://www.youtube.com/playlist?list=PL6kQim6ljTJs2p7W1xMPhelLR-xPuhtJj |access-date=}} © gratis OA.
* {{cite web |last=Soto Espinosa |first=Jesús |title=Matemática discreta |format=Collection of videos |publisher=Escuela Politécnica Superior, Universidad Católica San Antonio de Murcia (UCAM) |location=Guadalupe, Murcia, Region of Murcia (ES-MC), Spain |url=https://www.youtube.com/playlist?list=PL2qtqrBO8xujlH2bo2UPMQ3m9Hj9yEBur |access-date=}}
|}

====== <span id="Graphs_See_also"></span>See also ======
<hr style="margin-left: 0px; width:50%;">
* [[Handshaking lemma]]
* [[Route inspection problem]]

====== <span id="Graphs_To_find_out_more"></span>To find out more ======
<hr style="margin-left: 0px; width:50%;">
<ol style="list-style-type:circle;">
  <li>[[Gallery of named graphs]]</li>
  <li>[[Portal:Mathematics]]</li>
  <li>[[Mesh networking]]</li>
</ol>
|}

{| class="mw-collapsible wikitable {{#ifeq: {{{plegada|no}}}|sí|mw-collapsed|}}" width="100%" style="text-align:left; padding:0px; background-color:#F9F9F9;"
| align="left"   |

===== Numerical calculus =====
|-
| style="background-color:#F9F9F9; text-align:left; font-size:100%; padding:6px" |
{|
|[[File:BilinearInterpolExample2.png|200px]]
|<div id="toc_NumericalCalculus" class="toc">
<div class="toctitle">
'''Contents'''
</div>
# [[#NumericalCalculus_Key_concepts|Key concepts]]
# [[#NumericalCalculus_Bibliography|Bibliography]]
# [[#NumericalCalculus_Software|Software]]
# [[#NumericalCalculus_Multimedia|Multimedia]]
# [[#NumericalCalculus_To_find_out_more|To find out more]]
</div>
|}

====== <span id="NumericalCalculus_Key_concepts"></span>Key concepts ======
<hr style="margin-left: 0px; width:50%;">
<div role="note" class="hatnote">Main articles: [[Numerical analysis]] and [[Numerical linear algebra]]</div>
<div role="note" class="hatnote">Main categories: [[:Category:Numerical analysis|Numerical analysis]] and [[:Category:Numerical linear algebra|Numerical linear algebra]]</div>
'''— Interpolation '''
* [[Polynomial interpolation]]
** [[Newton polynomial|Newton's divided differences interpolation polynomial]]
** [[Lagrange polynomial]]
<!--
** [[Bernstein polynomial]]
* [[Root-finding_algorithm|Root-finding_algorithms]]
** [[Bisection method]]
** [[False position method]]
** [[Newton's method|Newton-Raphson method]]
** [[Secant method]]
* [[Numerical linear algebra]]
* [[Continued fraction|Continued fractions]]
* [[Mathematical constants by continued fraction representation]]
* [[Pi]]
* [[:Category:Pi algorithms|Pi algorithms]]
-->
<!--
{| border=0 align=center cellspacing=20
|-
| <center>{{pp-book-cover</center>
 |title=[[Usuario:Jmleonrojas/Libros/Ecuaciones_en_diferencias_finitas|Ecuaciones en diferencias finitas]]
 |subtitle=
 |cover-image=BilinearInterpolExample2.png
 |cover-color=#76b3df
}} Libro wiki (itinerario en Wikipedia)<br />¡Pulse en el título, no en la imagen de portada!</center>
|}
-->

====== <span id="NumericalCalculus_Bibliography"></span>Bibliography: theory and proposed and solved exercises  ======
<hr style="margin-left: 0px; width:50%;">
{| style="width: 100%;"
|In English: || In Spanish:
|- style="vertical-align:top;"
| style="width: 50%;" |
* —¤— Leon Q. Brin. <i>Tea Time Numerical Analysis. Experiences in Mathematics</i>. 2nd edition, 2016, CC BY-SA. http://lqbrin.github.io/tea-time-numerical/
* —¤— Steven C. Chapra and Raymond P. Canale. <i>Numerical methods for Engineers</i>. McGraw-Hill Science/Engineering/Math, 6th edition, 2010, ISBN: 9780072918731. http://openlibrary.org/books/OL7305712M/Numerical_Methods_for_Engineers
| style="width: 50%;" |
* —¤— Steven C. Chapra and Raymond P. Canale. <i>Métodos numéricos para ingenieros</i>. 5th edition, 2007.
* María José Ibáñez Pérez. <i>Notas sobre Interpolación polinómica</i>. http://www.ugr.es/~mibanez/ejemplos/interpolacion.pdf
* —¤— Walter Mora. <i>Introducción a los métodos numéricos</i>. 2016. https://tecdigital.tec.ac.cr/revistamatematica/Libros/WMora_MetodosNumericos/WMora-ITCR-MetodosNumericos.pdf
|}

====== <span id="NumericalCalculus_Software"></span>Software  ======
<hr style="margin-left: 0px; width:50%;">
{| style="width: 100%;"
|In English: || In Spanish:
|- style="vertical-align:top;"
| style="width: 50%;" |
* [https://www.wolframalpha.com/examples/Polynomials.html WolframAlpha (Examples, Mathematics, Polynomials)]
| style="width: 50%;" |
|}

====== <span id="NumericalCalculus_Multimedia"></span>[[File:Multimedia-icon.svg|32px|link=]] Multimedia ======
<hr style="margin-left: 0px; width:50%;">
{| style="width: 100%;"
|In English: || In Spanish:
|- style="vertical-align:top;"
| style="width: 50%;" |
'''— Interpolation'''
* {{cite web |last=Kaw |first=Autar |title=Primer on interpolation |format=Video |publisher=University of South Florida (USF) |location=Tampa, Florida (US-FL), USA |url=http://nm.mathforcollege.com/topics/primer_interpolation.html |access-date=}}
'''— Newton's divided differences interpolation polynomial'''
* {{cite web |last=Kaw |first=Autar |title=Newton's divided difference polynomial interpolation. Linear interpolation: Theory |format=Video |publisher=University of South Florida (USF) |location=Tampa, Florida (US-FL), USA |url=http://mathforcollege.com/nm/videos/youtube/05inp/newton/newton_05inp_lineartheory.html |access-date=}}
* {{cite web |last=Kaw |first=Autar |title=Newton's divided difference polynomial interpolation. Linear interpolation: Example |format=Video |publisher=University of South Florida (USF) |location=Tampa, Florida (US-FL), USA |url=http://mathforcollege.com/nm/videos/youtube/05inp/newton/newton_05inp_linearexample.html |access-date=}}
'''— Lagrange polynomial'''
* {{cite web |last=Kaw |first=Autar |title=Lagrangian Interpolation: Theory |format=Video |publisher=University of South Florida (USF) |location=Tampa, Florida (US-FL), USA |url=http://mathforcollege.com/nm/videos/youtube/05inp/lagrangian/lagrangian_05inp_theory.html |access-date=}}
* {{cite web |last=Kaw |first=Autar |title=Lagrangian Interpolation: Linear Interpolation: Example |format=Video |publisher=University of South Florida (USF) |location=Tampa, Florida (US-FL), USA |url=http://mathforcollege.com/nm/videos/youtube/05inp/lagrangian/lagrangian_05inp_linear.html |access-date=}}
* {{cite web |last=Kaw |first=Autar |title=Lagrangian Interpolation: Theory |format=Video |publisher=University of South Florida (USF) |location=Tampa, Florida (US-FL), USA |url=http://mathforcollege.com/nm/videos/youtube/05inp/lagrangian/lagrangian_05inp_theory.html |access-date=}}
| style="width: 50%;" |
'''— Interpolation'''
<!--* {{cite web |last=Soto Espinosa |first=Jesús |title=Polinomio Interpolador |format=Video |publisher=Escuela Politécnica Superior, Universidad Católica San Antonio de Murcia (UCAM) |location=Guadalupe, Murcia, Región de Murcia (ES-MC), España |url=https://www.youtube.com/watch?v=GnD3JqYDSWk |access-date=}}-->
* {{cite web |last=Martín Barreiro |first=Carlos |title=El problema de la interpolación lineal |format=Video |publisher=Facultad de Ciencias Naturales y Matemáticas, Escuela Superior Politécnica del Litoral (ESPOL) |location=Santiago de Guayaquil, Provincia del Guayas (EC-G), Ecuador |url=https://www.youtube.com/watch?v=Er8DRO1mea8&t=11s |access-date=}}
'''— Newton's divided differences interpolation polynomial'''
* {{cite web |last=Martín Barreiro |first=Carlos |title=Polinomio de Lagrange |format=Video |publisher=Facultad de Ciencias Naturales y Matemáticas, Escuela Superior Politécnica del Litoral (ESPOL) |location=Santiago de Guayaquil, Provincia del Guayas (EC-G), Ecuador |url=https://www.youtube.com/watch?v=Er8DRO1mea8&t=1160s |access-date=}}
* {{cite web |last=Martín Barreiro |first=Carlos |title=Polinomio de Lagrange. Ejemplo |format=Video |publisher=Facultad de Ciencias Naturales y Matemáticas, Escuela Superior Politécnica del Litoral (ESPOL) |location=Santiago de Guayaquil, Provincia del Guayas (EC-G), Ecuador |url=https://www.youtube.com/watch?v=tMV3HfxKQKc&t=1706s |access-date=}}
'''— Lagrange polynomial'''
* {{cite web |last=Martín Barreiro |first=Carlos |title=Polinomio de Newton |format=Video |publisher=Facultad de Ciencias Naturales y Matemáticas, Escuela Superior Politécnica del Litoral (ESPOL) |location=Santiago de Guayaquil, Provincia del Guayas (EC-G), Ecuador |url=https://www.youtube.com/watch?v=uJzzqnbnKRg&t=305s |access-date=}}
* {{cite web |last=Martín Barreiro |first=Carlos |title=Polinomio de Newton. Ejemplo |format=Video |publisher=Facultad de Ciencias Naturales y Matemáticas, Escuela Superior Politécnica del Litoral (ESPOL) |location=Santiago de Guayaquil, Provincia del Guayas (EC-G), Ecuador |url=https://www.youtube.com/watch?v=uJzzqnbnKRg&t=1810s |access-date=}}
|}

====== <span id="NumericalCalculus_To_find_out_more"></span>To find out more ======
<hr style="margin-left: 0px; width:50%;">
{| style="width: 100%;"
|In English: || In Spanish:
|- style="vertical-align:top;"
| style="width: 50%;" |
<ol style="list-style-type:circle;">
  <li>[http://mathworld.wolfram.com/topics/Archer.html Archer, Branden] and [http://mathworld.wolfram.com/about/author.html Weisstein, Eric W.] &quot;Lagrange Interpolating Polynomial.&quot; From [http://mathworld.wolfram.com/ ''MathWorld'']--A Wolfram Web Resource. http://mathworld.wolfram.com/LagrangeInterpolatingPolynomial.html</li>
  <li>[http://mathworld.wolfram.com/about/author.html Weisstein, Eric W.] &quot;Newton's Divided Difference Interpolation Formula.&quot; From [http://mathworld.wolfram.com/ ''MathWorld'']--A Wolfram Web Resource. http://mathworld.wolfram.com/NewtonsDividedDifferenceInterpolationFormula.html</li>
  <li>{{cite web |last=Kaw |first=Autar |title=Holistic Numerical Methods |format=Vídeo |publisher=University of South Florida (USF) |location=Tampa, Florida (US-FL), EUA |url=http://mathforcollege.com/nm |access-date=}}</li>
  <li>[[Portal:Mathematics]]</li>
</ol>
| style="width: 50%;" |
<ol style="list-style-type:circle;">
  <li>{{cite web |last=Martín Barreiro |first=Carlos |title=Análisis numérico |format=Vídeo |publisher=Facultad de Ciencias Naturales y Matemáticas, Escuela Superior Politécnica del Litoral (ESPOL) |location=Santiago de Guayaquil, Provincia del Guayas (EC-G), Ecuador |url=https://www.youtube.com/playlist?list=PLKoPE99M_ulXtxS00hFRpnpOth1lXk0uL |access-date=}}</li>
</ol>
|}
|}

{| class="mw-collapsible wikitable {{#ifeq: {{{plegada|no}}}|sí|mw-collapsed|}}" width="100%" style="text-align:left; padding:0px; background-color:#F9F9F9;"
| align="left"   |

===== More complimentary knowledge pills =====
|-
| style="background-color:#F9F9F9; text-align:left; font-size:100%; padding:6px" |<!--
====== Big data ======
* [[Computational X]]
-->
* [[Red pill and blue pill]]
====== Conjectures ======
<hr style="margin-left: 0px; width:50%;">
{{Main|Conjecture}}
{{Main category|Conjectures}}
* [[List of conjectures]] (list of mathematical conjectures)
* [[Undecidable conjectures]]

====== Open problems ======
<hr style="margin-left: 0px; width:50%;">
{{Main|Open problem|l1=Open problem (unsolved problem)}}
{{Main category|Open problems}}
* [[List of unsolved problems in mathematics]]
* [[:Category:Unsolved problems in mathematics]]
* [[List of unsolved problems in computer science]]
* [[:Category:Unsolved problems in computer science]]
* [[Lists of unsolved problems]]
* [[:Category:Lists of unsolved problems]]

====== Paradoxes ======
<hr style="margin-left: 0px; width:50%;">
* [[List of paradoxes]]

====== Some more problems, either not solved or solved ======
<hr style="margin-left: 0px; width:50%;">
* [[:Category:Hypotheses]]
* [[:Category:Philosophical problems]]
* [[:Category:Mathematical problems]]
* [[:Category:Computational problems]]
* [[:Category:Recreational mathematics]]
* [[:Category:Mathematics-related lists]], for instance:
** [[List of proved mathematical statements]] <sup>([[:es:Anexo:Enunciados matemáticos|es]])</sup>
** [[List of theorems]]
** [[List of lemmas]]
** [[List of mathematical proofs]]
** [[List of inequalities]]
** [[List of disproved mathematical ideas]]
** [[List of incomplete proofs]]
** [[List of undecidable problems]]
* [[:Category:Lists of unsolved problems]]
* [[:Category:Statistics-related lists]]
* [[:Category:Computing-related lists]]
* [[:Category:Philosophy-related lists]]

====== Philosophy ======
<hr style="margin-left: 0px; width:50%;">
* [[Philosophy of logic]]
* [[Philosophy of language]]
* [[Philosophy of Arithmetic]]
* [[Philosophy of statistics]]
* [[Philosophy of mathematics]]
* [[Philosophy of computer science]]
* [[Philosophy of science]]
* [[Philosophy of technology]]
* [[Philosophy of technique]] <sup>([[:es:Filosofía de la técnica|es]]; [[:de:Technikphilosophie|de]])</sup>
* [[Philosophy of mind]]
* [[Philosophy of artificial intelligence]]
* [[Philosophy of information]]
* [[Philosophy of social science]]
* [[Science and technology studies]]
* [[Technoscience]]
{{see also|:es:Wikipedia:Proyecto educativo/Matemática discreta y numérica/Plan de aprendizaje#Ex post I: Aula de Humanidades Juanelo Turriano (Segunda edición)|l1=Aula de Humanidades Juanelo Turriano (Segunda edición) [Arts & Humanities Classroom 'Juanelo Turriano' (2nd edition)] (for the time being, in Spanish)}}

====== History ======
<hr style="margin-left: 0px; width:50%;">
* [[History of logic]]
* [[Historical linguistics]]
* [[History of algebra]]
* [[History of arithmetic]]
* [[Number#History|History of numbers]]
* [[Number theory#History|History of number theory]]
* [[History of the Theory of Numbers|History of the Theory of Numbers]] (three-volume work by [[Leonard Eugene Dickson]] summarizing work in number theory up to about 1920)
* [[History of combinatorics]]
* [[History of statistics]]
* [[History of probability]]
* [[History of mathematics]]
* [[History of computer science]]
* [[History of science]]
* [[History of technology]]
* [[History of science and technology]]
* [[History of computing]]
* [[History of artificial intelligence]]
* [[History of philosophy]]
* [[History of sociology]]
* [[History of ideas]]

====== Imagination ======
<hr style="margin-left: 0px; width:50%;">
* [[:Category:Imagination]]
* [[Analog computer]]
* [[Future of mathematics]]
* [[:Category:Future]]

====== Languages ======
<hr style="margin-left: 0px; width:50%;">
* [[Language of mathematics]]
* [[Mathematical notation]]
* [[:Category:Glossaries]], for instance:
** [[:Category:Glossaries of mathematics]]
*** Of particular relevance: [[Closed-form expression]] • ...
* [[:Category:Terminology]], for instance:
** [[:Category:Mathematical terminology]]
** [[:Category:Computing terminology]]
** [[:Category:Philosophical terminology]]
** [[:Category:Social sciences terminology]]

====== [[File:Multimedia-icon.svg|32px|link=]] Multimedia ======
<hr style="margin-left: 0px; width:50%;">
{| style="width: 100%;"
|In English: || In Spanish:
|- style="vertical-align:top;"
| style="width: 50%;" |
* ...
| style="width: 50%;" |
* {{cite web |last=Soto Espinosa |first=Jesús |title=Momentos de ciencia |format=Collection of videos |publisher=Escuela Politécnica Superior, Universidad Católica San Antonio de Murcia (UCAM) |location=Guadalupe, Murcia, Region of Murcia (ES-MC), Spain |url=https://www.youtube.com/playlist?list=PL2qtqrBO8xugdaQMPvm7AmV78kgKnNE91 |access-date=}}
|}

====== To know more ======
<hr style="margin-left: 0px; width:50%;">
* [http://mathematics-in-europe.eu/?p=334 Richard Elwes (2016) The top 10 mathematical achievements of the last 5ish years, maybe. En: www.mathematics-in-europe.eu, European Mathematical Society (EMS)]
|}

{| class="mw-collapsible wikitable {{#ifeq: {{{plegada|no}}}|sí|mw-collapsed|}}" width="100%" style="text-align:left; padding:0px; background-color:#F9F9F9;"
| align="left"   |

===== Editathons (intensive collaborative learning meetings) =====
|-
| style="background-color:#F9F9F9; text-align:left; font-size:100%; padding:6px" |
* [[Wikipedia:School_and_university_projects/Discrete_and_numerical_mathematics/Learning_plan/Sandbox#Editathons|Here]].
<!--
* Sobre lógica / About logic:
** [http://cala.unex.es/cala/epistemowikia/index.php/Árbol_semántico Tablas semánticas] / [http://cala.unex.es/cala/epistemowikia/index.php/Semantic_tableau Semantic tableaux]
* Sobre conjuntos / About sets:
** [http://cala.unex.es/cala/epistemowikia/index.php/Algoritmos_de_unión_y_búsqueda_en_conjuntos‎‎ Algoritmos de unión y búsqueda en conjuntos‎‎] / [http://cala.unex.es/cala/epistemowikia/index.php/Union-find_algorithms_for_sets Union-find algorithms for sets]
* Sobre funciones / About functions:
** [http://cala.unex.es/cala/epistemowikia/index.php/Inversión_de_programas‎‎ Inversión de programas‎‎] / [http://cala.unex.es/cala/epistemowikia/index.php/Inverting_programs Inverting programs]
* Sobre relaciones / About relations:
** [http://cala.unex.es/cala/epistemowikia/index.php/Cierre_transitivo_de_una_relación_binaria‎‎ Cierre transitivo de una relación binaria‎‎] / [http://cala.unex.es/cala/epistemowikia/index.php/Transitive_closure_of_a_binary_relation Transitive closure of a binary relation]
* Sobre interpolación / About interpolation:
** [[Interpolación polinomial de Newton en diferencias divididas]] / [[Newton's divided-difference interpolating polynomials]]
* Sobre teoría de números y conjeturas / About number theory and conjectures:
** [[Conjetura de Collatz]] / [[Collatz conjecture]]
* Sobre criptografía / About cryptography:
** [[RSA (sistema criptográfico)]] / [[RSA (cryptosystem)]]
* Sobre combinatoria / About combinatorics:
** [[Variaciones, permutaciones y combinaciones]] / [[Variations, permutations and combinations]]
* Métodos de generación de permutaciones / Permutation generation methods:
** [[Métodos de generación de permutaciones]] / [[Permutation generation methods]]
* Estructuras algebraicas / Algebraic structures
** [[Estructura algebraica|Estructuras algebraicas]] / [[Algebraic structure|Algebraic structures]]
-->
|}

{| class="mw-collapsible wikitable {{#ifeq: {{{plegada|no}}}|sí|mw-collapsed|}}" width="100%" style="text-align:left; padding:0px; background-color:#F9F9F9;"
| align="left"   |

===== [[File:Multimedia-icon.svg|32px|link=]] More multimedia by the mentioned authors and by others =====
|-
| style="background-color:#F9F9F9; text-align:left; font-size:100%; padding:6px" |
{| style="width: 100%;"
|In English: || In Spanish:
|- style="vertical-align:top;"
| style="width: 50%;" |
* {{cite web
 |url=https://www.youtube.com/channel/UCZrFqXWKZZIXiBE4A8oQToA/videos
 |title=Greg A. Baugher YouTube channel
 |last=Baugher
 |first=Greg A.
 |department=Department of Math/Science/Informatics, Penfield College, Mercer University
 |location=Georgia (US-GA), USA
 |format= Collection of videos
 |publisher=
 |date=
 |website=
 |access-date=
 |quote= }}
* {{cite web
 |url=https://www.youtube.com/user/DrSaradaHerke
 |title=Sarada Herke's Youtube channel focused on graph theory and discrete mathematics
 |last=Herke
 |first=Sarada
 |format=Collection of videos
 |publisher=
 |location=Brisbane, Queensland (AU-QLD), Commonwealth of Australia
 |date=
 |department=School of Mathematics and Physics, The University of Queensland
 |website=
 |access-date=
 |quote= }}
* {{cite web
 |url=https://www.youtube.com/channel/UCxx5xjd5O36gSnt-GEAVAJQ
 |title=Spoonful of Maths
 |last=Herke
 |first=Sarada
 |format=Collection of videos
 |publisher=
 |location=Brisbane, Queensland (AU-QLD), Commonwealth of Australia
 |date=
 |department=School of Mathematics and Physics, The University of Queensland
 |website=
 |access-date=
 |quote= }}
* {{cite web |last=Kaw |first=Autar |title=The Numerical Methods Guy |format= |publisher= |location= |url=https://autarkaw.org/author/autarkaw/ |access-date=}}
* {{cite web
 |url=https://www.youtube.com/playlist?list=PL0862D1A947252D20
 |title=Computer Sc - Discrete Mathematical Structures
 |last=Krithivasan
 |first=Kamala
 |format=Course
 |publisher=Indian Institute of Technology Madras
 |location=Adyar, Chennay (ant. Madrás), Tamil Nadu (IN-TN), India
 |date=
 |website=
 |access-date=
 |quote= }}
* {{cite web
 |url=https://www.youtube.com/playlist?list=PLH83nnRMSIe4LOzMQc4jzeDbAXe9-2jrR
 |title=Number Theory
 |last=Olavsku
 |first=
 |department=
 |format=Collection of videos
 |publisher=
 |location=
 |date=
 |website=
 |access-date=
 |quote= }}
* {{cite web
 |url=https://www.youtube.com/playlist?list=PLr3WmPgPWZfX1HUpeyKkP6ir2wOFhqXMO
 |title=Number Theory
 |last=Olson
 |first=Glenn R.
 |department=Maine East High School
 |location=Park Ridge, Illinois (US-IL), USA
 |format=Collection of videos
 |publisher=
 |date=
 |website=
 |access-date=
 |quote= }}
* {{cite web
 |url=https://www.youtube.com/channel/UCTx0ZsgYgrXJ-i-Nhx0WX-g/videos
 |title=Brandon Olver Youtube channel
 |last=Olver
 |first=Brandon
 |department=Ballarat Grammar
 |location=Wendouree, Ballarat, Victoria (AU-VIC), Commonwealth of Australia
 |format=Collection of videos
 |publisher=
 |date=
 |website=
 |access-date=
 |quote= }}
* {{cite web
 |url=https://www.youtube.com/channel/UC1_uAIS3r8Vu6JjXWvastJg
 |title=Mathologer
 |last=Polster
 |first=Burkard
 |department=Monash University
 |location=Melbourne, Victoria (AU-VIC), Commonwealth of Australia
 |format=Collection of videos
 |publisher=
 |date=
 |website=
 |access-date=
 |quote= }}
* {{cite web
 |url=http://www.aduni.org/courses/discrete/
 |title=Discrete Mathematics
 |last=Simonson
 |first=Shai
 |format=Course
 |publisher=ArsDigita University
 |date=
 |website=
 |access-date=
 |quote= }} ([https://www.youtube.com/playlist?list=PL__5ik6vMY50HNfWppaGxQydlSMTGQWzC Another URL] [only the videos])
* https://www.youtube.com/playlist?list=PLbMVogVj5nJQ-tIzyygzzLLxomzlbIiE4
* https://www.professorserna.com/videos/#discrete-math
* https://www.youtube.com/playlist?list=PLHjOMouVJ7UXITXJPhN6lzKyYaLapsAxK (https://www.maths.unsw.edu.au/sites/default/files/math1081-2017s1.pdf)
* https://www.youtube.com/user/CosmoLearning/search?query=LSU+Number+Theory+Lecture (https://www.youtube.com/watch?v=tUNjx2kgj_M)
* https://www.youtube.com/playlist?list=PL018X5Hlr4RnB78091B1SghJNb98T-XUs
* https://www.youtube.com/playlist?list=PL71FE85723FD414D7
* https://www.youtube.com/playlist?list=PLbMVogVj5nJTA6ZeVkqHEiUup_3ccdg_C
* https://www.youtube.com/playlist?list=PLbMVogVj5nJSxFihV-ec4A3z_FOGPRCo-
* https://www.youtube.com/watch?v=2esURv_dtBk
| style="width: 50%;" |
* {{cite web |last=Hervás Jorge |first=Antonio |department=Departamento de Matemática Aplicada, Escuela Técnica Superior de Ingeniería Informática |title=Colección de vídeos |format= |publisher=Universidad Politécnica de Valencia (UPV) |location=Valencia, Comunidad Valenciana (ES-VC), España |url=https://media.upv.es/#/portal/user/d68676eb-232c-4527-9365-a14b74a594ed/videos |access-date=}}
* {{cite web |last=Jordán Lluch |first=Cristina |department=Departamento de Matemática Aplicada, Escuela Técnica Superior de Ingeniería Informática |title=Colección de vídeos |format= |publisher=Universidad Politécnica de Valencia (UPV) |location=Valencia, Comunidad Valenciana (ES-VC), España |url=https://media.upv.es/#/portal/user/58cc4c2a-6459-4311-b124-23658c85b778/videos |access-date=}}
* {{cite web |last=Martín Barreiro |first=Carlos |title=Análisis numérico |format=Course |publisher=Facultad de Ciencias Naturales y Matemáticas, Escuela Superior Politécnica del Litoral (ESPOL) |location=Santiago de Guayaquil, Provincia del Guayas (EC-G), Ecuador |url=https://www.youtube.com/playlist?list=PLKoPE99M_ulXtxS00hFRpnpOth1lXk0uL |access-date=}}
* {{cite web |last=Rodríguez Álvarez |first=María José |department=Departamento de Matemática Aplicada, Escuela Técnica Superior de Ingeniería Informática |title=Colección de vídeos |format= |publisher=Universidad Politécnica de Valencia (UPV) |location=Valencia, Comunidad Valenciana (ES-VC), España |url=https://media.upv.es/#/portal/user/ce4531fd-7c85-4292-b7a8-4b69b3761354/videos |access-date=}}
* {{cite web |last=Soto Espinosa |first=Jesús |department=Unidad Central de Informática / Escuela Politécnica Superior |title=Colección de vídeos |format= |publisher=Universidad Católica San Antonio de Murcia (UCAM) |location=Guadalupe, Murcia, Región de Murcia (ES-MC), España |url=https://www.youtube.com/channel/UC2SGcUXrGzJwKO_sMNzZXMA |access-date=}}
|}
|}
</div>
|-
<!--          EJEMPLOS DE EXAMEN / EXAMPLES OF EXAM          -->
| style="padding:2px;" | <h2 id="pamdan-ede-h2" style="margin:3px; background:#deecc8; font-family:inherit; font-size:110%; font-weight:bold; border:1px solid #deecc8; text-align:left; color:rgb(0, 0, 0); padding:0.2em 0.4em;">Sample exam questions, instrumental and relational{{#tag:ref|Although [[relational understanding|relational questions]] are of paramount importance for training [[critical thinking]] and [[creativity]], [[instrumental understanding|instrumental exercises]] cannot be ruled out since they are essential to build the foundations and gain the knowledge necessary to set the scene of [[mathematical thinking]].<ref>{{Cite journal |last=Skemp |first=Richard R. |year=1976 |title=Relational understanding and instrumental understanding |journal=Mathematics Teaching |volume=77 |pages=20-26 |format=PDF |url=https://www.atm.org.uk/write/MediaUploads/Resources/Richard_Skemp.pdf}}</ref>|group="note"}}, and some answers<hr style="width: 100%; height: 4px; background-color: rgb(0, 102, 0);" /></h2>
<span style="color: rgb(0, 0, 0); font-style: italic;">(Illustrative examples, cases, exercises, problems).</span>
<div style="margin-left: 48%; text-align: right; color: rgb(0, 102, 0); background-color: rgb(255, 255, 223); background-image: none; padding: 0px; border-top: 4px solid rgb(0, 102, 0); border-bottom: 4px solid rgb(0, 102, 0); vertical-align: middle;">
'It is axiomatic that the greater the student's individual effort, the more thorough will be his (sic) learning.'<br />
<small>[[Timothy J. Fitikides]]: ''Common mistakes in English'', Longmans, 6ª edición / 6th edition, 2000, p. vii.</small>
</div>
|-
| style="color:#000; padding:2px 5px 5px;" | <div id="pamdan-ede">

==== Very important warning ====

{{ambox
|nocat=true
| type  = notice
| text  =
<ul>
  <li>The following '''exercises with solution''' are <strong>sample exam questions and answers</strong> for the contents of the course. 
    <ol style="list-style-type:circle;">
      <li>You should be able to respond any of them in a <strong>maximum of 30 minutes</strong>.</li>
      <li>It is highly important that you keep in mind that when you sit an exam, both theoretical (including proofs of theorems) and practical questions about all topics worked out in class may be asked.</li>
      <li>The concreteness of the questions within the context of these examples <strong>implies in no case a course content cut</strong>.</li>
    </ol>
  </li>
  <li>Rigor when it comes to carrying out the solutions must be the central focus.
    <ol style="list-style-type:circle;">
      <li><strong>Show what you do and how you do it</strong>.</li>
      <li>Do not simplify your answers, write legibly and explain your work, ''showing all intermediate steps'', seeking to ensure that your arguments are clear and logically sound.</li>
      <li>Label, or define, any variables and notation you choose to work with; also explain how you are solving each question as you go along, for example, ''stating clearly any theorem or result you use''; neatness, cleanliness and organization count.</li>
    </ol>
  </li>
  <li>Once you consider an exercise solved, you can '''check your work''' against our resolution (however, keep in mind the possibility of correct alternative paths to the solution).</li>
  <li>'''Important if you have joined MATDIN ([[Wikipedia:School and university projects/Discrete and numerical mathematics|the learning project on Wikipedia]]):''' You might like to see these exercises as seeds to help you in finding examples to <strong>illustrate your contributions to the project</strong>.</li>
</ul>
}}

==== Theme 1.- Fundamentals ====

===== Logic =====

====== Propositional logic ======

<div style="margin-left: 5%; margin-right: 5%; background: #FAFBFC; border: 1px solid rgb(153, 153, 153); padding: 1em; text-align:left;">
<strong>Question L1.</strong> (2.5 points).<br/>
On the island of truthfuls and deceitfuls — another nomenclature in the literature for the
couple have been [[knights and knaves|(knights, knaves)]] and (truth-tellers/'truthers', liars) — there are two types of inhabitants, 'truthfuls' who always tell the truth and 'deceitfuls' who always lie. It is assumed that every inhabitant is either a truthful or a deceitful person. There were two inhabitants, <math>A</math> and <math>B</math>, standing together in the front yard of a house. You passed by and asked them, 'Are you truthful or deceitful persons?' 
<ul style = "list-style: none; margin-left: 5; padding-left: 1.35em; text-indent: -1.35em;">
<li>a) <math>A</math> answered, 'If <math>B</math> is a truthful person then I am a deceitful person.' Can it be determined whether <math>A</math> and <math>B</math> were truthfuls or deceitfuls? (1.25 p.)</li>
<li>b) Afterward, <math>B</math> said, 'Don't believe <math>A</math>; he's lying.' With this new information, can it be determined whether <math>A</math> and <math>B</math> were truthfuls or deceitfuls? (1.25 p.)</li>
</ul>
</div>
{| class="mw-collapsible wikitable {{#ifeq: {{{plegada|sí}}}|no||mw-collapsed}}" style="margin-left: 5%; margin-right: 5%; text-align:left; padding:0px; background-color:white;"
| align="left"   | <span style="text-align:left; font-weight:bold;">Solution:</span>
|-
| style="text-align:left; font-size:100%; padding:6px" | Let us use <math>X</math> for '<math>X</math> is a truthful person' --- therefore, <math>\neg X</math> is an abbreviation for '<math>X</math> is a deceitful person' ---.
	<ul style = "list-style: none; margin-left: 5; padding-left: 1.35em; text-indent: -1.35em;">
        <li>a) <math>A</math>'s statement, 'If <math>B</math> is a truthful person then I am a deceitful person', can be expressed as <math>B \rightarrow \neg A</math> and the fact that <math>A</math> says it, as <math>A \leftrightarrow (B \rightarrow \neg A)</math>. In view of the [[truth table]]:
		<math display=block>
		\begin{array}{cc|cccccc}
			A & B & A & \leftrightarrow & (B & \rightarrow & \neg & A) \\
			\hline
			1 & 1 & 1 & \mathbf{0} & 1 & 0 & 0 & 1 \\
			1 & 0 & 1 & \mathbf{1} & 0 & 1 & 0 & 1 \\
			0 & 1 & 0 & \mathbf{0} & 1 & 1 & 1 & 0 \\
			0 & 0 & 0 & \mathbf{0} & 0 & 1 & 1 & 0
		\end{array}
		</math>
		the only model for <math>A \leftrightarrow (B \rightarrow A)</math> is the 2nd [[Interpretation (logic)|interpretation]], so it can be determined that <math>A</math> is a truthful person and <math>B</math> a deceitful person.</li>
		<li>b) <math>B</math>'s statement, 'Don't believe <math>A</math>; he's lying', is equivalent to '<math>A</math> is a deceitful person', which can be expressed as <math>\neg A</math> and the fact that <math>B</math> says it, as <math>B \leftrightarrow \neg A</math>, which simply says that neither both <math>A</math> and <math>B</math> can be truthfuls nor deceitfuls at the same time, and this adds nothing new, as expected because it was already determined. Indeed, in view of the truth table:
		<math display=block>
		\begin{array}{cc|ccccccccccc}
			A & B & (A & \leftrightarrow & (B & \rightarrow & \neg & A)) & \wedge & (B & \leftrightarrow & \neg & A) \\
			\hline
			1 & 1 & 1 & 0 & 1 & 0 & 0 & 1 & \mathbf{0} & 1 & 0 & 0 & 1 \\
			1 & 0 & 1 & 1 & 0 & 1 & 0 & 1 & \mathbf{1} & 0 & 1 & 0 & 1 \\
			0 & 1 & 0 & 0 & 1 & 1 & 1 & 0 & \mathbf{0} & 1 & 1 & 1 & 0 \\
			0 & 0 & 0 & 0 & 0 & 1 & 1 & 0 & \mathbf{0} & 0 & 0 & 1 & 0
		\end{array}
		</math>
		we notice that everything remains the way it was, the 2nd interpretation is again a model, but now for <math>(A \leftrightarrow (B \rightarrow \neg A)) \wedge (B \leftrightarrow \neg A)</math>.</li>
	</ul>
|}<br/>

<div style="margin-left: 5%; margin-right: 5%; background: #FAFBFC; border: 1px solid rgb(153, 153, 153); padding: 1em; text-align:left;">
<strong>Question L2.</strong> (2.5 points)<br/>
With the help of [[Propositional calculus|propositional logic]], prove that the following [[Argument|argument]] is [[Validity|valid]] or not. 'This program will compile whenever we have declared the variables. However, in truth, we will declare the variables precisely if we do not forget to do so. It turns out that the program has not compiled. Then it follows that we have forgotten to declare the variables.'<br/>Important: Do not solve it using [[Truth table|truth tables]]. 
</div>
{| class="mw-collapsible wikitable {{#ifeq: {{{plegada|sí}}}|no||mw-collapsed}}" style="margin-left: 5%; margin-right: 5%; text-align:left; padding:0px; background-color:white;"
| align="left"   | <span style="text-align:left; font-weight:bold;">Solution:</span>
|-
| style="text-align:left; font-size:100%; padding:6px" | You can check the complete solution through [[Method of analytic tableaux|semantic tableaux]] of several examples in [https://zurbaran.unex.es/cloud/index.php/s/FS2Jh9svKdY0Oup this document] (in Spanish, for the time being); (in particular, see exercise 7).<!-- http://cala.unex.es/cala/epistemowikia/index.php/Archivo:Ejemplo.de.tabla.semantica.pdf -->
|}<br/>

<div style="margin-left: 5%; margin-right: 5%; background: #FAFBFC; border: 1px solid rgb(153, 153, 153); padding: 1em; text-align:left;">
<strong>Question L3.</strong> (2.5 points).<br/>
<ul style = "list-style: none; margin-left: 5; padding-left: 1.35em; text-indent: -1.35em;">
<li>a) Define adequate set of connectives (asc), also called completely expressive or functionally complete set of connectives.</li>
<li>b) Provide two examples of two-element asc, explaining why they are so and assuming that we know the asc which elements are the most usual connectives <math>\left\lbrace \neg, \wedge, \vee, \rightarrow, \leftrightarrow \right\rbrace</math>.</li>
</ul>
</div>
{| class="mw-collapsible wikitable {{#ifeq: {{{plegada|sí}}}|no||mw-collapsed}}" style="margin-left: 5%; margin-right: 5%; text-align:left; padding:0px; background-color:white;"
| align="left"   | <span style="text-align:left; font-weight:bold;">Solution:</span>
|-
| style="text-align:left; font-size:100%; padding:6px" | <ul style = "list-style: none; margin-left: 5; padding-left: 1.35em; text-indent: -1.35em;">
<li>a) In Propositional Logic, an adequate set of connectives (asc) is any set of connectives such that every logical connective can be represented as an expression involving only those belonging to the asc.</li>
<li>b) As it is said in the wording, we assume that we know that the set of the most usual connectives, <math> \lbrace \neg, \wedge, \vee, \rightarrow, \leftrightarrow \rbrace </math> is an asc. Two two-element asc are the sets <math> \lbrace \neg, \wedge \rbrace </math> and <math> \lbrace \neg, \vee \rbrace </math>. In effect, we only have to check, for each two-element set, that the missing most usual connectives may be represented only with the ones in the set:
<math display="block">
\begin{align}
\lbrace \neg, \wedge \rbrace: p \vee q & \equiv \neg (\neg p \wedge \neg q)\\
p \rightarrow q & \equiv \neg (p \wedge \neg q)\\
p \leftrightarrow q & \equiv \neg (p \wedge \neg q) \wedge \neg (\neg p \wedge q)\\
\lbrace \neg, \vee \rbrace: p \wedge q & \equiv \neg (\neg p \vee \neg q)\\
p \rightarrow q & \equiv \neg p \vee q\\
p \leftrightarrow q & \equiv \neg (\neg (\neg p \vee q) \vee \neg (p \vee \neg q))
\end{align}
</math>
</li>
</ul>
|}<br/>

====== Predicate logic ======

<div style="margin-left: 5%; margin-right: 5%; background: #FAFBFC; border: 1px solid rgb(0, 153, 153); padding: 1em; text-align:left;">
<strong>Question L4.</strong> (2.5 points).<br/>
There are <math display="inline">77</math> animals on a hill, they are two-legged or four-legged. A villager says: 'At least one of the animals has two legs and given any pair of animals, at least one of them has four legs.'
<ul style = "list-style: none; margin-left: 5; padding-left: 1.75em; text-indent: -1.75em;">
        <li>(a) Formalise in predicate logic what the local said.</li>
        <li>(b) How many of them are two-legged and how many are four-legged?</li>
</ul>
</div>
{| class="mw-collapsible wikitable {{#ifeq: {{{plegada|sí}}}|no||mw-collapsed}}" style="margin-left: 5%; margin-right: 5%; text-align:left; padding:0px; background-color:white;"
| align="left"   | <span style="text-align:left; font-weight:bold;">Solution:</span>
|-
| style="text-align:left; font-size:100%; padding:6px" | Considering the set of animals on the hill as the universe of discourse, let be:
<math display="block">\begin{aligned}
Dx &\leftarrow \mbox{'}x \text{ is two-legged'}, \\
Cx &\leftarrow \mbox{'}x \text{ is four-legged'}.
\end{aligned}</math>
<ul style = "list-style: none; margin-left: 5; padding-left: 1.75em; text-indent: -1.75em;">
        <li>(a) <math display="inline">\exists x Dx \wedge \forall xy \left( Cx \vee Cy \right)</math>;</li>
        <li>(b) <br />
<math>
\begin{array}{ll}
\mathbf{Step} & \mathbf{Reason} \\
1. \; \exists x Dx & \text{Premise} \\
2. \; \forall xy \left( Cx \vee Cy \right) & \text{Premise} \\
3. \; Ca \vee Cb & \text{UI 2 }\text{(universal instantiation)} \\
4. \; \forall x \left( Cx \leftrightarrow \neg Dx \right) & \text{Premise} \\
5. \; Ca \leftrightarrow \neg Da & \text{UI 4 }\text{(universal instantiation)} \\
6. \; Ca \rightarrow \neg Da & \mathrm{BIE_{1}\text{ 5 }}\text{(biconditional elimination)} \\
7. \; Cb \leftrightarrow \neg Db & \text{UI 4 }\text{(universal instantiation)} \\
8. \; Cb \rightarrow \neg Db & \mathrm{BIE_{1}\text{ 7 }}\text{(biconditional elimination)} \\
9. \; \neg Da \vee \neg Db & \mathrm{Dil_{3}\text{ 3, 6, 8 }}\text{(complex constructive dilemma: }\{A \vee B, A \rightarrow C, B \rightarrow D\} \vdash C \vee D\text{ )} \\
10. \; \neg \left( Da \wedge Db \right) & \mathrm{DM_{1}\text{ 9 }}\text{(De Morgan's law, negation of the conjunction)} \\
11. \; \forall x \forall y \neg \left( Dx \wedge Dy \right) & \mathrm{UG^{2} 10 }\text{ (multiple universal generalisation)} \\
12. \; \forall y \neg \left( Da \wedge Dy \right) & \text{UE 11 }\text{(universal instantiation)} \\
13. \; \neg \exists y \left( Da \wedge Dy \right) & \text{EN 12 }\text{(existential negation)} \\
14. \; \forall x \neg \exists y \left( Dx \wedge Dy \right) & \text{UG 13 }\text{(universal generalisation)} \\
15. \; \neg \exists x \exists y \left( Dx \wedge Dy \right) & \text{EN 14 }\text{(existential negation)} \\
\end{array}
</math>
<br />
Translating (1) and (15) into English: (1) there is a two-legged animal and (15) there is no pair of animals in which both are two-legged, so there is only one two-legged animal and therefore, 76 four-legged animals.
</li>
</ul>
|}<br/>

<div style="margin-left: 5%; margin-right: 5%; background: #FAFBFC; border: 1px solid rgb(0, 153, 153); padding: 1em; text-align:left;">
<strong>Question L5.</strong> (2.5 points).<br/>
Source: Lewis Carroll, Symbolic Logic: Part I. Elementary (Macmillan, 1896), pg. 118. Public Domain.<br />
40.<br />
(1) No kitten, that loves fish, is unteachable;<br />
(2) No kitten without a tail will play with a gorilla;<br />
(3) Kittens with whiskers always love fish;<br />
(4) No teachable kitten has green eyes;<br />
(5) No kittens have tails unless they have whiskers.<br />
Universe = 'kittens'; A = loving fish; B = green-eyed; C = tailed; D = teachable; E = whiskered; H = will play with a gorilla.<br />

You are required to:
<ul style = "list-style: none; margin-left: 5; padding-left: 1.75em; text-indent: -1.75em;">
        <li>(a) Formalise all these statements into Predicate Logic.</li>
        <li>(b) In the universe of kittens and using Predicate Logic, deduce the one conclusion that follows from these statements and makes the argument valid.</li>
        <li>(c) Translate your symbolic answer into English.</li>
</ul>
</div>
{| class="mw-collapsible wikitable {{#ifeq: {{{plegada|sí}}}|no||mw-collapsed}}" style="margin-left: 5%; margin-right: 5%; text-align:left; padding:0px; background-color:white;"
| align="left"   | <span style="text-align:left; font-weight:bold;">Solution:</span>
|-
| style="text-align:left; font-size:100%; padding:6px" | Considering the set of kittens as the universe of discourse, let be: <math display="block">\begin{aligned}
Dx &\leftarrow \text{«} x \text{ is teachable»}, \\
Ax &\leftarrow \text{«} x \text{ loves fish»}, \\
Cx &\leftarrow \text{«} x \text{ has tail»}, \\
Hx &\leftarrow \text{«} x \text{ plays with a gorilla»}, \\
Ex &\leftarrow \text{«} x \text{ has whiskers»}, \\
Bx &\leftarrow \text{«} x \text{ has green eyes»}.\end{aligned}</math>
<ul style = "list-style: none; margin-left: 5; padding-left: 1.75em; text-indent: -1.75em;">
        <li>(a) (1) <math>\forall x (\neg Dx \rightarrow \neg Ax)</math>;<br />
(2) <math>\forall x (\neg Cx \rightarrow \neg Hx)</math>;<br />
(3) <math>\forall x (Ex \rightarrow Ax)</math>;<br />
(4) <math>\forall x (Dx \rightarrow \neg Bx)</math>;<br />
(5) <math>\forall x (\neg Ex \rightarrow \neg Cx)</math>.<br /></li>
        <li>(b) <br />
<math>
\begin{array}{ll}
\mathbf{Step} & \mathbf{Reason} \\
1. \; \forall x (\neg Dx \rightarrow \neg Ax) & \text{Premise} \\
2. \; \forall x (\neg Cx \rightarrow \neg Hx) & \text{Premise} \\
3. \; \forall x (Ex \rightarrow Ax) & \text{Premise} \\
4. \; \forall x (Dx \rightarrow \neg Bx) & \text{Premise} \\
5. \; \forall x (\neg Ex \rightarrow \neg Cx) & \text{Premise} \\
6. \; \neg Da \rightarrow \neg Aa & \text{UI 1 }\text{(universal instantiation)} \\
7. \; \neg Ca \rightarrow \neg Ha & \text{UI 2 }\text{(universal instantiation)} \\
8. \; Ea \rightarrow Aa & \text{UI 3 }\text{(universal instantiation)} \\
9. \; Da \rightarrow \neg Ba & \text{UI 4 }\text{(universal instantiation)} \\
10. \; \neg Ea \rightarrow \neg Ca & \text{UI 5 }\text{(universal instantiation)} \\
11. \; \neg Aa \rightarrow \neg Ea & \mathrm{Cp_{1}\text{ 8 }}\text{(contraposition [also called transposition] law)} \\
12. \; Ba \rightarrow \neg Da & \mathrm{Cp_{2}\text{ 9 }}\text{(contraposition [also called transposition] law)} \\
13. \; (Ba \rightarrow \neg Da) \wedge (\neg Da \rightarrow \neg Aa) & \text{CI 12, 6 }\text{(conjunction introduction)} \\
14. \; (Ba \rightarrow \neg Aa) & \text{Syl 13 }\text{(hypothetical syllogism)} \\
15. \; (Ba \rightarrow \neg Aa) \wedge (\neg Aa \rightarrow \neg Ea) & \text{CI 14, 11 }\text{(conjunction introduction)} \\
16. \; (Ba \rightarrow \neg Ea) & \text{Syl 15 }\text{(hypothetical syllogism)} \\
17. \; (Ba \rightarrow \neg Ea) \wedge (\neg Ea \rightarrow \neg Ca) & \text{CI 16, 10 }\text{(conjunction introduction)} \\
18. \; (Ba \rightarrow \neg Ca) & \text{Syl 17 }\text{(hypothetical syllogism)} \\
19. \; (Ba \rightarrow \neg Ca) \wedge (\neg Ca \rightarrow \neg Ha) & \text{CI 18, 7 }\text{(conjunction introduction)} \\
20. \; (Ba \rightarrow \neg Ha) & \text{Syl 19 }\text{(hypothetical syllogism)} \\
21 \; \forall x (Bx \rightarrow \neg Hx) & \text{UG 20 }\text{(universal generalisation)}
\end{array}
</math></li>
        <li>(c) No green-eyed kitten will play with a gorilla.</li>
</ul>
|}<br/>

<div style="margin-left: 5%; margin-right: 5%; background: #FAFBFC; border: 1px solid rgb(0, 153, 153); padding: 1em; text-align:left;">
<strong>Question L6.</strong> (2.5 points).<br/>
Formalise into Predicate Logic:
<ul style = "list-style: none; margin-left: 5; padding-left: 1.75em; text-indent: -1.75em;">
        <li>(a) 'All <math>A</math> that is <math>B</math>, it is also <math>C</math>.'</li>
        <li>(b) 'If all <math>A</math> is <math>B</math>, then it is also <math>C</math>.'</li>
        <li>(c) 'There is none which is <math>A</math> or <math>B</math> and is not <math>C</math>.'</li>
</ul>
</div>
{| class="mw-collapsible wikitable {{#ifeq: {{{plegada|sí}}}|no||mw-collapsed}}" style="margin-left: 5%; margin-right: 5%; text-align:left; padding:0px; background-color:white;"
| align="left"   | <span style="text-align:left; font-weight:bold;">Solution:</span>
|-
| style="text-align:left; font-size:100%; padding:6px" | <ul style = "list-style: none; margin-left: 5; padding-left: 1.75em; text-indent: -1.75em;">
        <li>(a) <math>(\forall x)((Ax \wedge Bx) \rightarrow Cx)</math>.</li>
        <li>(b) It can be rewritten as 'if all <math>A</math> is <math>B</math>, then all <math>A</math> is <math>C</math> too,' therefore, <math>(\forall x (Ax \rightarrow Bx)) \rightarrow (\forall x (Ax \rightarrow Cx))</math>. Warning!, <math>(\forall x(Px \rightarrow Qx)) \Rightarrow ((\forall x Px) \rightarrow (\forall x Qx))</math> but <math>((\forall x Px) \rightarrow (\forall x Qx)) \nRightarrow (\forall x(Px \rightarrow Qx))</math>.</li>
        <li>(c) <math>(\neg \exists x)((Ax \vee Bx) \wedge \neg Cx)</math>. Or put it in other words, 'if something is <math>A</math> or <math>B</math> then it is <math>C</math>,' which in the language of Predicate Logic is written as follows: <math>(\forall x)((Ax \vee Bx) \rightarrow Cx)</math>.</li>
</ul>
|}<br/>

===== Sets, relations and functions =====

<div style="margin-left: 5%; margin-right: 5%; background: #FAFBFC; border: 1px solid rgb(153, 153, 153); padding: 1em; text-align:left;">
<strong>Question SRF1.</strong> (2.5 points)<br/>
<ul style = "list-style: none; margin-left: 5; padding-left: 1.75em; text-indent: -1.75em;">
<li>(a) Propose three [[Set (mathematics)|sets]] <math>A</math>, <math>B</math> and <math>C</math>, such that <math>A \in B</math>, <math>B \in C</math> and <math>A \notin C</math>. (0,5 points).</li>
<li>(b) According to a [[Survey methodology|survey]] of a certain group of students, they said that, if they had to decide between two courses, equally interesting because of their contents, they [[Preference|prefer]] that one for which the time they dedicate to study it is the lowest and for which they foresee the best results in exams. In case of equality of study times and of exam results forecasts, they are indifferent to them. Study the properties of this [[Binary relation|binary relation]]. (2 points).</li>
</ul>
</div><br/>

<!--
<div style="margin-left: 5%; margin-right: 5%; background: #FAFBFC; border: 1px solid rgb(0, 153, 153); padding: 1em; text-align:left;">
<strong>Question SRF2.</strong> (2.5 points).<br/>
Formalise the following statements into the language of Set Theory:
<ul style = "list-style: none; margin-left: 5; padding-left: 1.75em; text-indent: -1.75em;">
        <li>(a) 'All <math>A</math> that is <math>B</math>, it is also <math>C</math>'.</li>
        <li>(b) 'If any <math>A</math> is <math>B</math>, then it is also <math>C</math>'.</li>
        <li>(c) 'There is none which is <math>A</math> or <math>B</math> and is not <math>C</math>'.</li>
</ul>
</div>
{| class="mw-collapsible wikitable {{#ifeq: {{{plegada|sí}}}|no||mw-collapsed}}" style="margin-left: 5%; margin-right: 5%; text-align:left; padding:0px; background-color:white;"
| align="left"   | <span style="text-align:left; font-weight:bold;">Solution:</span>
|-
| style="text-align:left; font-size:100%; padding:6px" | <ul style = "list-style: none; margin-left: 5; padding-left: 1.75em; text-indent: -1.75em;">
        <li>(a) <math>(\forall x) (x \in A \cap B \Rightarrow x \in C)</math>, that is to say, <math>A \cap B \subseteq C</math>.</li>
        <li>(b) As shown in L6, this expression may be rewritten as 'if any <math>A</math> is <math>B</math>, then any <math>A</math> is <math>C</math> too', which in the language of set theory is <math>(\forall x) (x \in A \Rightarrow x \in B) \Rightarrow (\forall x) (x \in A \Rightarrow x \in C)</math>, or equivalently, <math>A \subseteq B \Rightarrow A \subseteq C</math>.</li>
        <li>(c) As shown in L6, this expression may be rewritten as 'if something is <math>A</math> or <math>B</math> then it is <math>C</math>', which in the language of set theory is <math>(\forall x) (x \in A \cup B \Rightarrow x \in C)</math>, or equivalently, <math>A \cup B \subseteq C</math>.</li>
</ul>
|}<br/>
-->

===== Algebraic structures =====

<div style="margin-left: 5%; margin-right: 5%; background: #FAFBFC; border: 1px solid rgb(153, 153, 153); padding: 1em; text-align:left;">
<strong>Question AS1.</strong> (2.5 points)<br/>
Let
<math display=block>
x \ast y = u
</math>
be a binary relation defined on the set <math>A = \left\lbrace 1,3,5,7,9 \right\rbrace</math>, for every two elements <math>x</math> and <math>y</math> in <math>A</math>, where <math>u</math> is the figure of the units of the usual product <math>x \cdot y</math> between two natural numbers (for example, <math>3 \ast 9 = 7</math>).
<ul style = "list-style: none; margin-left: 5; padding-left: 1.35em; text-indent: -1.35em;">
<li>a) Find out theCayley table for the operation <math>\ast</math> on <math>A</math>.</li>
<li>b) Is <math>(A;\ast)</math> an abelian group? (You can reason using the Cayley table).</li>
</ul>
</div>
{| class="mw-collapsible wikitable {{#ifeq: {{{plegada|sí}}}|no||mw-collapsed}}" style="margin-left: 5%; margin-right: 5%; text-align:left; padding:0px; background-color:white;"
| align="left"   | <span style="text-align:left; font-weight:bold;">Solution:</span>
|-
| style="text-align:left; font-size:100%; padding:6px" | <ul style = "list-style: none; margin-left: 5; padding-left: 1.35em; text-indent: -1.35em;">
<li>a) Here is the <strong>[[Cayley table]]</strong> of the binary operation <math>\ast</math> defined on the set <math>A</math>:
<math display=block>
\begin{array}{c|ccccc}
\ast & 1 & 3 & 5 & 7 & 9 \\
\hline
1 & 1 & 3 & 5 & 7 & 9 \\
3 & 3 & 9 & 5 & 1 & 7 \\
5 & 5 & 5 & 5 & 5 & 5 \\
7 & 7 & 1 & 5 & 9 & 3 \\
9 & 9 & 7 & 5 & 3 & 1 \\
\end{array}
</math>
</li>
<li>b) Let us check if <math>(A, \ast)</math> satisfies the five requirements (axioms) to be qualified as an [[abelian group]]:
<ul style = "list-style: none; margin-left: 5; padding-left: 1.35em; text-indent: -1.35em;">
<li>a) <math>A</math> is <strong>closed</strong> under <math>\ast</math> (it is also said that <math>\ast</math> is a <i>closed operation</i> or an <i>[[internal composition law]]</i> on <math>A</math>) since for all <math>a</math> and <math>b</math> in <math>A</math>, <math>a \ast b \in A</math>. It is easy to reason using the table: every number in the table is an element of <math>A</math>.</li>
<li>b) <math>\ast</math> is <strong>associative</strong> on <math>A</math> — we could check every triad, <math>1 \ast (3 \ast 5) = 1 \ast 5 = 5 = 3 \ast 5 = (1 \ast 3) \ast 5)</math>, and so on, however it is easier to reason using the fact that the product (<math>\cdot</math>) of natural numbers is associative: simply, <math>\forall x,y,z \in \mathbb{N}</math>, <math>x \cdot (y \cdot z) = (x \cdot y) \cdot z \Rightarrow x \ast (y \ast z) = (x \ast y) \ast z</math> is true since when we multiply the unit digits among natural numbers we have to carry no number —;</li>
<li>c) <math>\ast</math> is <strong>conmutative</strong> (the table is symmetric with respect to the main diagonal);</li>
<li>d) the <strong>identity</strong> for <math>\ast</math> in <math>A</math> is <math>1</math> as can be seen from the Cayley table since the first row and the first column are the same than the heading row and column, respectively;</li>
<li>e) <strong>not every element is invertible</strong> — we can use the table for checking this: given a certain number, heading a row, we only have to find out which other number gives the identity when operated with the former one, and this can be done by searching for the identity on that row (for example, <math>7</math> is the inverse of <math>3</math> because <math>3 \ast 7 = 7 \ast 3 = 1</math> [we search for <math>1</math> on the second row (headed by <math>3</math>), finding it on the fourth column (headed by <math>7</math>])—: the inverse of <math>1</math> is <math>1</math>, of <math>3</math> is <math>7</math>, of <math>7</math> is <math>3</math> and of <math>9</math> is <math>9</math>, but <math>5</math> has no inverse — when operating any other number with <math>5</math> the result is always <math>5</math> (such a number, as <math>5</math> in this case, is called an <i>[[absorbing element]]</i> for <math>\ast</math> in <math>A</math> [as zero for the product of integers]), so it is impossible to obtain the identity as a result —.</li>
</ul>
In short, <math>(A;\ast)</math> <strong>has not abelian group structure</strong> (it has an [[abelian monoid]] structure).
</li>
</ul>
|}<br />

===== Cardinality, induction and recursion =====

<div style="margin-left: 5%; margin-right: 5%; background: #FAFBFC; border: 1px solid rgb(153, 153, 153); padding: 1em; text-align:left;">
<strong>Question C1.</strong> (2.5 points).<br/>
Proof by definition that <math>\mathbb{N}</math> is an infinite set.
</div>
{| class="mw-collapsible wikitable {{#ifeq: {{{plegada|sí}}}|no||mw-collapsed}}" style="margin-left: 5%; margin-right: 5%; text-align:left; padding:0px; background-color:white;"
| align="left"   | <span style="text-align:left; font-weight:bold;">Solution:</span>
|-
| style="text-align:left; font-size:100%; padding:6px" | A set is infinite precisely if there exists a bijection between it and one of its proper subsets (definition by [[Richard Dedekind|Dedekind]]). As an example, consider <math>f:\mathbb{N} \rightarrow \mathbb{N}\setminus\{0\}</math>, defined by <math>n \mapsto f(n) = n+1</math>. Let us prove that it is a bijective mapping. In effect:
* <math>f</math> is a mapping <math>\Leftrightarrow \left( \forall x \in \mathbb{N} \right) \left( \exists y \in \mathbb{N}\setminus\{0\} \right) \left( f(x) = y \right) \wedge \left( \forall x,x' \in \mathbb{N} \right) \left( x = x' \rightarrow f(x) = f(x') \right)</math>, which is trivial, as if <math>x \in \mathbb{N}</math> is given and because of the definition of <math>f</math>, there exists <math>y_x = x+1 \in \mathbb{N}\setminus\{0\}</math>, this <math>y_x</math> being unique for each <math>x</math>, that is, that if <math>x = x'</math>, then, because of the definition of <math>f</math>, <math>f(x) = x+1 = y_x = y_{x'} = x'+1 = f(x')</math>;
* <math>f</math> is injective <math>\Leftrightarrow \left( \forall x,x' \in \mathbb{N} \right) \left( f(x) = f(x') \rightarrow x = x' \right)</math>, which is trivial because of the definition of <math>f</math>, as if <math>f(x) = f(x')</math>, that is, if <math>x+1 = x'+1</math>, then, <math>x = x'</math>;
* <math>f</math> is surjective <math>\Leftrightarrow \left( \forall y \in \mathbb{N}\setminus\{0\} \right) \left( \exists x \in \mathbb{N} \right) \left( f(x) = y \right)</math>, which is also trivial due to the definition of <math>f</math>, as if <math>y</math> is given, then <math>x = y-1</math> satisfies <math>f(x) = f(y-1) = (y-1) + 1 = y</math>.
|}<br/>

<div style="margin-left: 5%; margin-right: 5%; background: #FAFBFC; border: 1px solid rgb(153, 153, 153); padding: 1em; text-align:left;">
<strong>Question C2.</strong> (2.5 points).<br/>
Knowing that <math>\mathbb{Z}</math> (integers) is a [[denumerable set]] and that the denumerable union of denumerable sets is a denumerable set, prove that <math>\mathbb{Q}</math> (rationals) is a denumerable set.
</div>
{| class="mw-collapsible wikitable {{#ifeq: {{{plegada|sí}}}|no||mw-collapsed}}" style="margin-left: 5%; margin-right: 5%; text-align:left; padding:0px; background-color:white;"
| align="left"   | <span style="text-align:left; font-weight:bold;">Solution:</span>
|-
| style="text-align:left; font-size:100%; padding:6px" | <math>\mathbb{Q}</math> is a denumerable set as it can be expressed by the denumerable union <math>\mathbb{Q} = A_1 \cup A_2 \cup \ldots \cup A_n \cup \ldots</math>, where every <math>A_i = \left\lbrace 0, \frac{-1}{i}, \frac{1}{i}, \ldots, \frac{-k}{i}, \frac{k}{i}, \ldots \right\rbrace</math> is a denumerable set, since <math>f: \mathbb{Z} \rightarrow A_i</math>, defined by <math>f(0) = 0</math> and <math>f(\pm n) = \frac{\pm n}{i}</math> is a bijection. Note that the set <math>A_i</math> is the set of all the rational numbers that have the same denominator <math>i</math>.
|}<br/>

<div style="margin-left: 5%; margin-right: 5%; background: #FAFBFC; border: 1px solid rgb(0, 153, 153); padding: 1em; text-align:left;">
<strong>Cuestión C3.</strong> (2.5 points).<br/>
Let <math>A</math> be a denumerable set and let <math>x \notin A</math>. Prove that <math>A \cup \{x\}</math> is a denumerable set.
</div>
{| class="mw-collapsible wikitable {{#ifeq: {{{plegada|sí}}}|no||mw-collapsed}}" style="margin-left: 5%; margin-right: 5%; text-align:left; padding:0px; background-color:white;"
| align="left"   | <span style="text-align:left; font-weight:bold;">Solution:</span>
|-
| style="text-align:left; font-size:100%; padding:6px" | 
As <math>A</math> is a denumerable set then --- by definition of denumerable set --- there is a bijection <math>f: \mathbb{N} \rightarrow A</math>. Let <math>g: \mathbb{N} \rightarrow A \cup \{x\}</math>, defined by <math>g(n) = x</math>, if <math>n = 0</math> and by <math>g(n) = f(n - 1)</math>, if <math>n \neq 0</math>, that is, the correspondence <math>g</math> is defined on two subdomains, on <math>\{0\}</math> as the constant correspondence <math>x</math>, a bijection, and on <math>\mathbb{N}^+</math> as <math>f</math>, also a bijection, and as such subdomains are disjoint and their images, <math>A</math> and <math>\{x\}</math> are disjoint too, then <math>g</math> is a bijection. 
|}<br/>

==== Theme 2.- Number theory ====

===== Congruences =====

<div style="margin-left: 5%; margin-right: 5%; background: #FAFBFC; border: 1px solid rgb(153, 153, 153); padding: 1em; text-align:left;">
<strong>Question NT1.</strong> (2.5 points).<br/>
Use [[congruence relation]] theory to respond.
<ul style = "list-style: none; margin-left: 5; padding-left: 1.35em; text-indent: -1.35em;">
<li>a) Prove that, for any <math>n \in \mathbb{N}</math>, <math>21 \cdot 2^{5n+1} - 4 \cdot 3^{3n+1}</math> is divisible by <math>5</math>. (1.25 p.)</li>
<li>b) Calculate the remainder of <math>3^{6n+1} + 3^{2n+1} \cdot 19^{2n} - 3</math> (for any <math>n \in \mathbb{N}</math>), when it is divided by <math>28</math>. (1.25 p.)</li>
</ul>
</div>
{| class="mw-collapsible wikitable {{#ifeq: {{{plegada|sí}}}|no||mw-collapsed}}" style="margin-left: 5%; margin-right: 5%; text-align:left; padding:0px; background-color:white;"
| align="left"   | <span style="text-align:left; font-weight:bold;">Solution:</span>
|-
| style="text-align:left; font-size:100%; padding:6px" | We use congruence relation theory.
	<ul style = "list-style: none; margin-left: 5; padding-left: 1.35em; text-indent: -1.35em;">
        <li>a) On the one hand: <br /><math display=block>\left.
            \begin{align}
                2^5 = 32 &\equiv 2 \pmod{5} \\
                3^3 = 27 &\equiv 2 \pmod{5}
            \end{align}
            \right\rbrace</math>
<br /><math display=block>\begin{align}
                &\stackrel{(i)}{\Rightarrow} &2^5 &\equiv 3^3 \pmod{5} \\
                &\stackrel{(ii)}{\Rightarrow} &2^{5n} &\equiv 3^{3n} \pmod{5}   \;\;\;\;\; {\scriptstyle(1)}
            \end{align}</math>
<br />
On the other: <br /><math display=block>\begin{align}
                &            &7 &\equiv 2 \pmod{5} \\
                &\stackrel{(iii)}{\Rightarrow} &7 \cdot 2 &\equiv 2 \cdot 2 \pmod{5} \\
                &\stackrel{(iv)}{\Rightarrow} &7 \cdot 2 \cdot 3 &\equiv 2 \cdot 2 \cdot 3 \pmod{5}   \;\;\;\;\; {\scriptstyle(2)}
            \end{align}</math>
<br />
Substituting <math>(2)</math> in <math>(1)</math>:
<br /><math display=block>\begin{align}
                &            &7 \cdot 3 \cdot 2 \cdot 2^{5n} &\equiv 2 \cdot 2 \cdot 3 \cdot 3^{3n} \pmod{5} \\
                &            &21 \cdot 2^{5n+1} &\equiv 4 \cdot 3^{3n+1} \pmod{5}
            \end{align}</math>
<br /> which, by definition of congruence relations, means that <math>21 \cdot 2^{5n+1} - 4 \cdot 3^{3n+1}</math> is divisible by <math>5</math>.</li>
		<li>b) On the one hand: <br /><math display=block>\begin{align}
                &            &3^3 = 27 &\equiv -1 \pmod{28} \\
                &\stackrel{(v)}{\Rightarrow} &3^{6n} = \left( 3^3 \right)^{2n} &\equiv (-1)^{2n} = 1 \pmod{28} \\
                &\stackrel{(iv)}{\Rightarrow} &3^{6n+1} &\equiv 3 \pmod{28}   \;\;\;\;\; {\scriptstyle(3)}
            \end{align}</math>
<br />
On the other: <br /><math display=block>\begin{align}
                &            &57 = 3 \cdot 19 &\equiv 1 \pmod{28} \\
                &\stackrel{(v)}{\Rightarrow} &3^{2n} \cdot 19^{2n} &\equiv 1 \pmod{28} \\
                &\stackrel{(iv)}{\Rightarrow} &3^{2n+1} \cdot 19^{2n} &\equiv 3 \pmod{28}   \;\;\;\;\; {\scriptstyle(4)}
            \end{align}</math>
<br />
If we add side by side <math>(3)</math> and <math>(4)</math>:<br /><math display=block>
			\begin{align}
				&            &3^{6n+1} + 3^{2n+1} \cdot 19^{2n} &\equiv 6 \pmod{28} \\
				&\Rightarrow &3^{6n+1} + 3^{2n+1} \cdot 19^{2n} - 3 &\equiv 3 \pmod{28}
			\end{align}
		</math><br />
		In other words, the requested remainder is <math>3</math>.</li>
	</ul>
<br /><hr width=25%/>
<small>(i) Because congruence relations are symmetric and transitive.<br />(ii) Rising each side of the congruence relation to the power <math>n</math>.<br />(iii) Multiplying each side of the congruence relation by <math>2</math>.<br />(iv) Multiplying each side of the congruence relation by <math>3</math>.<br />(v) Rising each side of the congruence relation to the power <math>2n</math>.</small>
|}<br/>

===== Power residues and divisibility rules =====

<div style="margin-left: 5%; margin-right: 5%; background: #FAFBFC; border: 1px solid rgb(153, 153, 153); padding: 1em; text-align:left;">
<strong>Question NT2.</strong> (2.5 points)<br/>
In [[Decimal|base-ten]] (decimal numeral system), find the [[Numerical digit|digits]] <math>x,y</math> such that the [[Number|number]] <math>12xy567</math> be [[Divisor|divisible]] by <math>33</math>.
</div>
{| class="mw-collapsible wikitable {{#ifeq: {{{plegada|sí}}}|no||mw-collapsed}}" style="margin-left: 5%; margin-right: 5%; text-align:left; padding:0px; background-color:white;"
| align="left"   | <span style="text-align:left; font-weight:bold;">Solution:</span>
|-
| style="text-align:left; font-size:100%; padding:6px" | <math>33 = 3 \times 11</math>.<br />
A number is divisible by <math>3</math> precisely if the sum of all its digits is divisible by <math>3</math>:
<math display="block">
3 \,\vert\, 12xy567 \Leftrightarrow 3 \,\vert\, (7 + 6 + 5 + y + x + 2 + 1)
</math>
this is:
<math display="block">
\begin{align}
3 \,\vert\, 12xy567 &\Leftrightarrow 3 \,\vert\, (21 + x + y) \\
                    &\Leftrightarrow 21 + x + y = \dot{3} \\
                    &\Leftrightarrow x + y = \dot{3} - 21
\end{align}
</math>
Moreover:
<math display="block">
x,y \mbox{ are digits in base-ten } \Rightarrow 0 \leqslant x,y \leqslant 9
</math>
then:
<math display="block">
0 \leqslant x + y \leqslant 18
</math>
We have to find out what differences <math>\dot{3} - 21</math> satisfy the fact of belonging to <math>\left[ 0, 18 \right]</math>:
<math display="block">\dot{3} - 21 = \left\lbrace \ldots, 0(=21-21), 3(=24-21), 6(=27-21), 9(=30-21), 12(=33-21), 15(=36-21), 18(=39-21), \ldots \right\rbrace</math>
so there are <math>7</math> possible cases:
<math display="block">x + y = 0 \veebar x + y = 3 \veebar x + y = 6 \veebar x + y = 9 \veebar x + y = 12 \veebar x + y = 15 \veebar x + y = 18</math>

A number is divisible by <math>11</math> precisely if the sum of its digits at even places minus the sum of its digits at odd places is divisible by <math>11</math>:
<math display="block">
11 \,\vert\, 12xy567 \Leftrightarrow 11 \,\vert\, ((7 + 5 + x + 1) - (6 + y + 2))
</math>
this is:
<math display="block">
\begin{align}
11 \,\vert\, 12xy567 &\Leftrightarrow 11 \,\vert\, (5 + x - y) \\
                    &\Leftrightarrow 5 + x - y = \dot{11} \\
                    &\Leftrightarrow x - y = \dot{11} - 5
\end{align}
</math>
Moreover:
<math display="block">
x,y \mbox{ are digits in base-ten } \Rightarrow 0 \leqslant x,y \leqslant 9
</math>
then:
<math display="block">
-9 \leqslant x - y \leqslant 9
</math>
We have to find out what differences <math>\dot{11} - 5</math> satisfy the fact of belonging to <math>\left[ -9, 9 \right]</math>:
<math display="block">\dot{11} - 5 = \left\lbrace \ldots, -5(=0-5), 6(=11-5), \ldots \right\rbrace</math>
so there are <math>2</math> possible cases:
<math display="block">x - y = -5 \veebar x - y = 6</math>

Therefore, there are <math>7 \times 2 = 14</math> possible cases:

{| class="wikitable" style="text-align:center"
|+Table of possible cases
|-
! &Lambda; !! <math>x + y = 0</math> !! <math>x + y = 3</math> !! <math>x + y = 6</math> !! <math>x + y = 9</math> !! <math>x + y = 12</math> !! <math>x + y = 15</math> !! <math>x + y = 18</math>
|-
! rowspan="2" | <math>x - y = -5</math>
| <math>2x = -5</math><br/><math>x = \frac{-5}{2}</math> || <math>2x = -2</math><br/><math>x = -1</math> || <math>2x = 1</math><br/><math>x = \frac{1}{2}</math> || <math>2x = 4</math><br/><math>x = 2</math><br/><math>y = 7</math> || <math>2x = 7</math><br/><math>x = \frac{7}{2}</math> || <math>2x = 10</math><br/><math>x = 5</math><br/><math>y = 10</math> || <math>2x = 13</math><br/><math>x = \frac{13}{2}</math>
|-
| <span style="color: Red;">No:</span>{{pad|0.2em}}<math>x</math> is not a digit<br/>in base-ten || <span style="color: Red;">No:</span>{{pad|0.2em}}<math>x</math> is not a digit<br/>in base-ten || <span style="color: Red;">No:</span>{{pad|0.2em}}<math>x</math> is not a digit<br/>in base-ten || <span style="color: Green;">Yes:</span>{{pad|0.2em}}<math>x,y</math> are digits<br/>in base-ten || <span style="color: Red;">No:</span>{{pad|0.2em}}<math>x</math> is not a digit<br/>in base-ten || <span style="color: Red;">No:</span>{{pad|0.2em}}<math>y</math> is not a digit<br/>in base-ten || <span style="color: Red;">No:</span>{{pad|0.2em}}<math>x</math> is not a digit<br/>in base-ten
|-
! rowspan="2" | <math>x - y = 6</math>
| <math>2x = 6</math><br/><math>x = 3</math><br/><math>y = -3</math> || <math>2x = 9</math><br/><math>x = \frac{9}{2}</math> || <math>2x = 12</math><br/><math>x = 6</math><br/><math>y = 0</math> || <math>2x = 15</math><br/><math>x = \frac{15}{2}</math> || <math>2x = 18</math><br/><math>x = 9</math><br/><math>y = 3</math> || <math>2x = 21</math><br/><math>x = \frac{21}{2}</math> || <math>2x = 24</math><br/><math>x = 12</math>
|-
| <span style="color: Red;">No:</span>{{pad|0.2em}}<math>y</math> is not a digit<br/>in base-ten || <span style="color: Red;">No:</span>{{pad|0.2em}}<math>x</math> is not a digit<br/>in base-ten || <span style="color: Green;">Yes:</span>{{pad|0.2em}}<math>x,y</math> are digits<br/>in base-ten || <span style="color: Red;">No:</span>{{pad|0.2em}}<math>x</math> is not a digit<br/>in base-ten || <span style="color: Green;">Yes:</span>{{pad|0.2em}}<math>x,y</math> are digits<br/>in base-ten || <span style="color: Red;">No:</span>{{pad|0.2em}}<math>x</math> is not a digit<br/>in base-ten || <span style="color: Red;">No:</span>{{pad|0.2em}}<math>x</math> is not a digit<br/>in base-ten
|}

So there are three possible solutions:
<math>\binom{x}{y} \in \left\lbrace \binom{2}{7}, \binom{6}{0}, \binom{9}{3} \right\rbrace</math>.

Thus, the possible numbers are:
<math>1227567, 1260567, 1293567</math>,

which are divisibles by <math>33</math>. Their quotients are:
<math>37199, 38199, 39199</math>.
|}<br/>

===== Diophantine equations =====

<div style="margin-left: 5%; margin-right: 5%; background: #FAFBFC; border: 1px solid rgb(153, 153, 153); padding: 1em; text-align:left;">
<strong>Question NT3.</strong> (2.5 points).<br/>
One company spent <math>100000</math> euros in buying <math>100</math> electronic devices, some of them ground breaking and providing maximum performance. Smartphones were <math>50</math> euros each, tablets were <math>1000</math> euros each and laptops were <math>5000</math> euros each. How many of each device did they buy? Solve this question using the theory of:
<ul style = "list-style: none; margin-left: 5; padding-left: 1.35em; text-indent: -1.35em;">
<li>a) diophantine equations;</li>
<li>b) congruence equations.</li>
</ul>
</div>
{| class="mw-collapsible wikitable {{#ifeq: {{{plegada|sí}}}|no||mw-collapsed}}" style="margin-left: 5%; margin-right: 5%; text-align:left; padding:0px; background-color:white;"
| align="left"   | <span style="text-align:left; font-weight:bold;">Solution:</span>
|-
| style="text-align:left; font-size:100%; padding:6px" | Once translated the information from the wording into a system of linear equations and simplifying the latter:
<math display=block>
\begin{align}
\left\lbrace
\begin{align}
x &+ y &+ z &= 100 \\
50x &+ 1000y &+ 5000z &= 100000
\end{align}
\right.
&\Rightarrow
\left\lbrace
\begin{align}
x &+ y &+ z &= 100 \\
x &+ 20y &+ 100z &= 2000
\end{align}
\right. \\
&\Rightarrow
\left\lbrace
\begin{align}
x + y + z &= 100 \\
x + y + z + 19y + 99z &= 2000
\end{align}
\right. \\
&\Rightarrow
100 + 19y + 99z = 2000 \\
&\Rightarrow
99z + 19y = 1900
\end{align}
</math>
<ul style = "list-style: none; margin-left: 5; padding-left: 1.35em; text-indent: -1.35em;">
<li>a) A diophantine equation <math>ax + by = c</math> has solution precisely if <math>\operatorname{mcd}{(a,b)} \mid c</math>. In such a case,
<math display=block>
\left( x_0,y_0 \right) = \left( \frac{cp}{d}, \frac{cq}{d} \right)
</math>
is a particular solution of the equation, where <math>d = \operatorname{mcd}{(a,b)}</math> and <math>p</math> and <math>q</math> are a pair of [[Bezout coefficient|Bézout coefficients]] (the coefficients of <math>a</math> and <math>b</math> in one linear combination that is equal to <math>d</math>) (Bézout's identity). The general solution of that equation is:
<math display=block>
\binom{x}{y} = \binom{x_0}{y_0} + \frac{k}{d} \binom{b}{-a}
</math>
where <math>\binom{x_0}{y_0}</math> is a particular solution and <math>k \in \mathbb{Z}</math>.
<br \>
The following table shows the use of the [[extended Euclidean algorithm]] for this case. The computation stops when the remainder is zero (in red color). The previous remainder, <math>2</math> (also in red color), is the greatest common divisor. Bézout coefficients are <math>5</math> and <math>-26</math> (in magenta color). Numbers in cyan color, <math>-19</math> and <math>99</math>, are, up to the sign, the quotients of the original numbers by the greatest common divisor.
{| class="wikitable" style="margin:auto; text-align:right;"
! index ''i''!! {{blue|quotient ''q''<sub>''i''−1</sub> }}!! {{olive|remainder ''r''<sub>''i''</sub>}}!! {{brown|''s''<sub>''i''</sub> }}!! ''t''<sub>''i''</sub>
|-
| 0 || ||{{green|99}}||{{brown|1}} || 0
|-
| 1 || ||{{green|19}} || {{brown|0}} || 1
|-
| 2 ||{{green|99}} &divide; {{green|19}} = {{blue|5}}
||{{green|99}} − {{blue|5}} × {{green|19}} = {{olive|4}}
||{{brown|1}} − {{blue|5}} × {{brown|0}} = {{brown|1}}
|| 0 − {{blue|5}} × 1 = −5
|-
| 3 ||{{green|19}} &divide; {{olive|4}} = {{blue|4}}
||{{green|19}} − {{blue|4}} × {{olive|4}} = {{olive|3}}
||{{brown|0}} − {{blue|4}} × {{brown|1}} = {{brown|−4}}
|| 1 − {{blue|4}} × (−5) = 21
|-
| 4 ||{{olive|4}} &divide; {{olive|3}} = {{blue|1}}
||{{olive|4}} − {{blue|1}} × {{olive|3}} = {{red|1}}
||{{brown|1}} − {{blue|1}} × ({{brown|−4}}) = {{magenta|5}}
|| −5 − {{blue|1}} × 21 = {{magenta|−26}}
|-
| 5 ||{{olive|3}} &divide; {{olive|1}} = {{blue|3}}
||{{olive|3}} − {{blue|3}} × {{olive|1}} = {{red|0}}
||{{brown|−4}} − {{blue|3}} × {{brown|5}} = {{cyan|−19}}
|| 21 − {{blue|3}} × (−26) = {{cyan|99}}
|} Thus,
<math display=block>
\left( z_0,y_0 \right) = \left( \frac{1900 \cdot 5}{1}, \frac{1900 \cdot (-26)}{1} \right) = \left( 9500, -49400 \right)
</math>
is a particular solution and:
<math display=block>
\binom{z}{y} = \binom{9500}{-49400} + \frac{k}{1} \binom{19}{-99}
</math>
is the general solution, where <math>k \in \mathbb{Z}</math>.
<br \>
So, how many of each device did they buy, knowing that they bought at least one device of each type? Let us see. We know that <math>z > 0</math> and that <math>y > 0</math>. Thus, <math>9500 + 19k > 0</math> and <math>-49400 - 99k > 0</math>, and consequently, <math>-500 < k</math> and <math>k < -498,9</math>. As <math>k \in \mathbb{Z}</math>, <math>k = -499</math>. Substituting in the equations, we obtain: <math>z = 19</math>, <math>y = 1</math> and <math>x = 100 - 1 - 19 = 80</math>.
<br \>
<i>Sol.</i>: <math>80</math> smartphones, <math>1</math> tablets and <math>19</math> laptops.
</li>
<li>b) As a congruence equation, it could be:
<math display=block>
99z \equiv 1900 \pmod{19}
</math>
Or, equivalently:
<math display=block>
99z \equiv 0 \pmod{19}
</math>
As <math>\operatorname{mcd}{(99,19)} = 1</math>, this way prompts us to consider all positive multiples of <math>19</math>, lesser than <math>100</math>, as possible solutions: <math>z \in \left\lbrace 19, 38, 57, 76, 95 \right\rbrace</math>.
<br \>
Let us try the other way. The diophantine equation <math>99z + 19y = 1900</math> could also be seen as the following congruence equation:
<math display=block>
19y \equiv 1900 \pmod{99}
</math>
Or, equivalently:
<math display=block>
19y \equiv 19 \pmod{99}
</math>
As <math>\operatorname{mcd}{(99,19)} = 1</math>, the equation has a unique solution <math>\pmod{99}</math>, which is:
<math display=block>
y \equiv 19 \cdot 19^{\phi(99)-1} \pmod{99}
</math>
that is to say,
<math display=block>
y \equiv 19^{60} \pmod{99}
</math>
Exploring the potential residues, we find out that:
<math display=block>
19^{10} \equiv 1 \pmod{99}
</math>
and thus, multiplying this congruence six times by itself, side by side:
<math display=block>
19^{60} \equiv 1 \pmod{99}
</math>
and, as a congruence relation is transitive:
<math display=block>
y \equiv 1 \pmod{99}
</math>
In other words, <math>y = 1</math> (since <math>y < 100</math>).
<br \>
As <math>99z + 19y = 1900</math>, we obtain that <math>z = 19</math>.
<br \>
Finally, from <math>x + y + z = 100</math>, we can assess the value of <math>x</math>, <math>x = 80</math>.
<br \>
<i>Sol.</i>: <math>80</math> smartphones, <math>1</math> tablets and <math>19</math> laptops.
</li>
</ul>
|}<br/>

<div style="margin-left: 5%; margin-right: 5%; background: #FAFBFC; border: 1px solid rgb(153, 153, 153); padding: 1em; text-align:left;">
<strong>Question NT4.</strong> (2.5 points).<br/>
How could we distribute <math>100</math> litres of water in a total of <math>40</math> different containers of <math>1</math>, <math>4</math> and <math>12</math> litres? Solve this question using the theory of Diophantine equations.
</div>
{| class="mw-collapsible wikitable {{#ifeq: {{{plegada|sí}}}|no||mw-collapsed}}" style="margin-left: 5%; margin-right: 5%; text-align:left; padding:0px; background-color:white;"
| align="left"   | <span style="text-align:left; font-weight:bold;">Solution:</span>
|-
| style="text-align:left; font-size:100%; padding:6px" | Let <math>x</math>, <math>y</math>, and <math>z</math> denote the number of containers of <math>1</math>, <math>4</math> and <math>12</math> liters, respectively, used in the solution. The conditions set forth in the statement of the question are taken up by the equations: <math>x + y + z = 40</math> and <math>x + 4y + 12z = 100</math>. By subtracting the first equation from the second equation: <math>3y + 11z = 60</math>, that is a Diophantine equation. As <math>\gcd (3,11) = 1 \mid 60</math>, such Diophantine equation has integer solutions. As <math>1 = 4 \cdot 3 + (-1) \cdot 11</math>, the Bézout coefficients are <math>p = 4</math> and <math>q = -1</math>. A particular solution is: <math>y_0 = 4 \cdot \dfrac{60}{1} = 240</math>, <math>z_0 = (-1) \cdot \dfrac{60}{1} = -60</math>. The general solution is: <math>y = 240 + 11k</math>, <math>z = -60-3k</math>, where <math>k \in \mathbb{Z}</math>. Assuming that at least one container of each type is involved in the solution, then <math>0 < y,z \leq 40-(1+1) = 38</math>. By replacing <math>y</math> and <math>z</math> with the general solution found, then, on the one side: <math>0 < 240+11k \leq 38</math>, hence <math>\dfrac{-240}{11} < k \leq \dfrac{-202}{11}</math>, thus <math>-21,82 < k \leq -18,36</math> and therefore, <math>k \in \left\lbrace -21, -20, -19 \right\rbrace</math>, and on the other: <math>0 < -60-3k \leq 38</math>, hence <math>\dfrac{60}{3} < -k \leq \dfrac{98}{3}</math>, thus <math>\dfrac{-98}{3} \leq k < \dfrac{-60}{3}</math>, that is, <math>-32,67 \leq k < -20</math> and therefore, <math>k \in \left\lbrace -32, \ldots, -22, -21 \right\rbrace</math>. Thus, <math>k = -21</math>. If <math>k</math> is replaced in the general solution with its value, <math>y = 240 + 11 \cdot (-21) = 9</math> and <math>z = -60 - 3 \cdot (-21) = 3</math>, then it is obtained that <math>x = 40 - (9 + 3) = 28</math>.<br />
'''Sol.:''' Assuming that at least one container of each type must be filled, then the <math>100</math> liters of water can be distributed using <math>28</math> containers of <math>1</math> liter, <math>9</math> of <math>4</math> liters and <math>3</math> of <math>12</math> liters. (If such a thing is not assumed, then another solution would be possible: <math>20</math> containers of <math>1</math> liter, <math>20</math> of <math>4</math> liters and <math>0</math> of <math>12</math> liters. [Verify this]).
|}<br/>

===== Cryptography =====

<div style="margin-left: 5%; margin-right: 5%; background: #FAFBFC; border: 1px solid rgb(153, 153, 153); padding: 1em; text-align:left;">
<strong>Question NT5.</strong> (2.5 points).<br/>
Abigail wants to send Balbina the most simple call message: <code>eh</code>. They can only send numbers. Abigail and Balbina use the letters' position in the alphabet to code them (thus, Abigail codes <code>e</code> as <code>06</code> and <code>h</code> as <code>08</code>). They use RSA to encrypt their messages. If Abigail choose <math>p = 3</math> and <math>q = 7</math> as the ground primes for RSA:
<ul style = "list-style: none; margin-left: 5; padding-left: 1.35em; text-indent: -1.35em;">
<li>a) imagine you are Abigail and obtain the encrypted message that you have to send to Balbina;</li>
<li>b) imagine you are Balbina and decrypt the encrypted message that Abigail has sent to you.</li>
</ul>
</div>
{| class="mw-collapsible wikitable {{#ifeq: {{{plegada|sí}}}|no||mw-collapsed}}" style="margin-left: 5%; margin-right: 5%; text-align:left; padding:0px; background-color:white;"
| align="left"   | <span style="text-align:left; font-weight:bold;">Solution:</span>
|-
| style="text-align:left; font-size:100%; padding:6px" | Following the steps of RSA algorithm:
<ul style = "list-style: none; margin-left: 5; padding-left: 1.35em; text-indent: -1.35em;">
<li>1) <math>p = 3</math>, <math>q = 7</math>.</li>
<li>2) <math>r = p \cdot q = 21</math>.</li>
<li>3) <math>\phi (21) = 12</math> (Euler phi of <math>21</math>).</li>
<li>4) We have to choose as the secret key (<math>SK</math>) a relatively prime with <math>12</math> and less than <math>12</math>, at the same time; we choose <math>SK = 5</math>.</li>
<li>5) The secret key and the public key (<math>PK</math>) are linked by the equation <math>SK \cdot PK \equiv 1 \pmod{\phi(r)}</math>, in this case: <math>5 \cdot PK \equiv 1 \pmod{12}</math>, so <math>PK = 5</math>.</li>
<li>6) If we code the original message, (<code>eh</code>), we have <code>0608</code>. It can be proven that that if the coded message <math>X</math> satisfies <math>0 \leq X \leq r-1</math>, then the ciphered message <math>Y</math>, also satisfies <math>0 \leq Y \leq r-1</math>. As we are interested in code, then cipher, then decipher and finally decode, we have to group into blocks so that each one of their individual coding is less than <math>r-1 = 20</math>. Let <math>X_1 = 06</math> and <math>X_2 = 08</math> be the coded blocks and let <math>Y_1</math> and <math>Y_2</math> be the ciphered blocks. As RSA method establishes, the <math>X_i</math> encryption is performed by solving the congruence equation <math>X_i^{PK} \equiv Y_i \pmod{r}</math> and the <math>Y_j</math> decryption by solving the congruence equation <math>Y_j^{SK} \equiv X_j \pmod{r}</math>.</li>
</ul>
Let us answer now the two sections of the question.
<ul style = "list-style: none; margin-left: 5; padding-left: 1.35em; text-indent: -1.35em;">
<li>a)  Putting ourselves in the shoes of Abigail, let us assess the ciphered message that we have to send to Balbina. Let us cipher <math>X_1 = 06</math>: the solution of <math>6^5 \equiv Y_1 \pmod{21}</math> is <math>Y_1 = 6</math>. Let us cipher <math>X_2 = 08</math>: the solution of <math>8^5 \equiv Y_2 \pmod{21}</math> is <math>Y_2 = 8</math>. Thus, the message that we have to send is: <code>0608</code>.</li>
<li>b) Putting ourselves in the shoes of Balbina, let us decipher the ciphered message that Abigail has sent us. Let us decipher <math>Y_1 = 06</math>: the solution of <math>6^5 \equiv X_1 \pmod{21}</math> is <math>X_1 = 6</math>. Let us decipher <math>Y_2 = 08</math>: the solution of <math>8^5 \equiv X_2 \pmod{21}</math> is <math>X_2 = 8</math>. Thus, the message we have just deciphered is: <code>0608</code>.</li>
</ul>
|}<br/>

==== Theme 3.- Combinatorics ====

===== Combinatorics =====

<div style="margin-left: 5%; margin-right: 5%; background: #FAFBFC; border: 1px solid rgb(153, 153, 153); padding: 1em; text-align:left;">
<strong>Question CT1.</strong> (2.5 points)<br/>
Let <math>D</math> be the set of decimal digits, that is, <math>D = \left\lbrace 0,1,2,3,4,5,6,7,8,9 \right\rbrace</math>. Using combinatorial reasoning, calculate:
<ul style = "list-style: none; margin-left: 5; padding-left: 1.35em; text-indent: -1.35em;">
<li>a) The number of subsets of <math>D</math> which elements are all primes.</li>
<li>b) The number of subsets of <math>D</math> having a prime number of elements.</li>
</ul>
</div>
{| class="mw-collapsible wikitable {{#ifeq: {{{plegada|sí}}}|no||mw-collapsed}}" style="margin-left: 5%; margin-right: 5%; text-align:left; padding:0px; background-color:white;"
| align="left"   | <span style="text-align:left; font-weight:bold;">Solution:</span>
|-
| style="text-align:left; font-size:100%; padding:6px" | <ul style = "list-style: none; margin-left: 5; padding-left: 1.35em; text-indent: -1.35em;">
<li>a) Let <math>P = \left\lbrace 2,3,5,7 \right\rbrace</math> be the set consisting of the the prime numbers in <math>D</math>. What is actually requested is the number of nonempty (nonvoid) subsets of <math>P</math>, in other words, subtracting one (the empty set) from the total number of subsets of <math>P</math>:
<math display="block">
\begin{align}
\left\vert{P}\right\vert - 1 &= 2^4 - 1 \\
          &= 15
\end{align}
</math>
<i>Sol.</i>: <math>15</math> subsets.
</li>
<li>b) The total number of subsets of <math>k</math> elements of a set of <math>n</math> elements is given by <math>C_{n,k} = \binom{n}{k}</math>. Thus, running over the prime numbers in <math>D</math>:
<math display="block">
\begin{align}
\binom{10}{2} + \binom{10}{3} + \binom{10}{5} + \binom{10}{7} &= \frac{10!}{2! \cdot 8!} \cdot \frac{10!}{3! \cdot 7!} \cdot \frac{10!}{5! \cdot 5!} \cdot \frac{10!}{7! \cdot 3!} \\
          &= 537
\end{align}
</math>
<i>Sol.</i>: <math>537</math> subsets.
</li>
</ul>
|}<br/>

<div style="margin-left: 5%; margin-right: 5%; background: #FAFBFC; border: 1px solid rgb(153, 153, 153); padding: 1em; text-align:left;">
<strong>Question CT2.</strong> (2.5 points)<br/>
A group of twelve people visit a museum. Everybody is wearing a woolen overcoat. Upon entering, they leave their coats in the attended cloakroom. On leaving, the cloakroom attendant puts the twelve coats on the counter. Each person in the group picks out one at random, completely absent-minded because of a very interesting discussion. Using combinatorial reasoning, calculate in how many ways can the coats be chosen by them so that none of them get their own coat back.
</div>
{| class="mw-collapsible wikitable {{#ifeq: {{{plegada|sí}}}|no||mw-collapsed}}" style="margin-left: 5%; margin-right: 5%; text-align:left; padding:0px; background-color:white;"
| align="left"   | <span style="text-align:left; font-weight:bold;">Solution:</span>
|-
| style="text-align:left; font-size:100%; padding:6px" | This involves finding the number of derangements of <math>12</math> objects. Instead of calculating for <math>12</math>, we are going to do for the general case of having <math>n</math> objects. Let <math> 1, 2, \ldots, n </math> denote the objects themselves. Let <math>P</math> be the set of all permutations of the objects and let <math>P_k</math> be the set of all derangements that have <math>k</math> fixed elements. Then, the set of all derangements is:
<math display="block">
D = P - \left( \bigcup\limits_{i=1}^{n} P_{i} \right)
</math>
Let us see it:
* How many permutations fix one specific number? The answer is the number of permutations of the other <math>n-1</math> (non fixed) numbers, that is to say, <math>(n-1)!</math>, and as there are <math>n</math> numbers, then, the number of permutations that fix any of these <math>n</math> numbers is <math>(n-1)! \cdot n</math>.
* How many permutations fix two specific numbers? The answer is the number of permutations of the other <math>n-2</math> (non fixed) numbers, that is to say, <math>(n-2)!</math>, and as there are <math>\binom{n}{2}</math> ways of choosing two different numbers out of <math>n</math> numbers, then, the number of permutations that fix any two of these <math>n</math> numbers is <math>\binom{n}{2} (n-2)!</math>.
Let us note that in the case of <math>n = 2</math>, that is, <math>\left\lbrace 1,2 \right\rbrace</math>, when we subtract those which fix the <math>1</math>, then we are subtracting once those which fix both the <math>1</math> and the <math>2</math> and when we subtract those that fix the <math>2</math> we are subtracting again those which fix both the <math>1</math> and the <math>2</math>. Thus, we have to add them one time. If we follow this reasoning, the number of permutations (derangements) for which no number is in its original place is:
<math display="block">
\begin{align}
\left\vert{D}\right\vert &= \left\vert{P}\right\vert - \left\vert{\left( \bigcup\limits_{i=1}^{n} P_{i} \right)}\right\vert \\
          &= \binom{n}{0} n! - \binom{n}{1} (n-1)! + \binom{n}{2} (n-2)! - \cdots + (-1)^{n+1} \binom{n}{n} 0!
\end{align}
</math>
Thus, if <math>n = 12</math>, there are:
<math display="block">
\binom{12}{0} 12! - \binom{12}{1} (12-1)! + \binom{12}{2} (12-2)! - \cdots + (-1)^{12+1} \binom{12}{12} 0! = 176 \ 214 \ 841
</math>
derangements.<br/>
<i>Sol.</i>: In <math>176 \ 214 \ 841</math> ways.
|}<br/>

<div style="margin-left: 5%; margin-right: 5%; background: #FAFBFC; border: 1px solid rgb(153, 153, 153); padding: 1em; text-align:left;">
<strong>Question CT3.</strong> (2.5 points)<br/>
An urn contains seven balls numbered one through seven. They are randomly chosen, one by one and without reposition until the urn is empty. As they are removed from the urn, we write their figures down from left to right on a first out, first writen basis. Using combinatorial reasoning calculate how many numbers thus formed start and end with an even digit.
</div>
{| class="mw-collapsible wikitable {{#ifeq: {{{plegada|sí}}}|no||mw-collapsed}}" style="margin-left: 5%; margin-right: 5%; text-align:left; padding:0px; background-color:white;"
| align="left"   | <span style="text-align:left; font-weight:bold;">Solution:</span>
|-
| style="text-align:left; font-size:100%; padding:6px" | There are <math>7</math> positions for the figures. At both ends, hypotheses imply even figure. There are three even figures between <math>1</math> and <math>7</math>: <math>2</math>, <math>4</math> and <math>6</math>. Being guided by the [[distribution of objects into recipients]] models, consider these <math>3</math> even figures (distinguishable boxes) and the two ends (distinguishable objects) of the seven-digit number, on an underlying injective mapping (at most, one end by each figure, as there are not two balls with the same figure). For each one of these cases at the ends (each one of the variations) we have to take into account all the possibilities for the <math>5</math> intermediate positions. The number of these possibilities is given by the permutations of <math>5</math> elements (one new abstraction as <math>5</math> distinguishable objects [the intermediate positions] being distributed into <math>5</math> distinguishable recipients [the figures <math>1</math>, <math>3</math> and <math>5</math> and the even figure that is at none of the ends of the seven-digit number thus formed], this time on an underlying bijective mapping). Applying the [[rule of product]]:
<math display="block">
\begin{align}
V_{3,2} \cdot P_5 &= 3^{\underline{2}} \cdot 5! \\
          &= (3 \cdot 2) \cdot (5!) \\
          &= 6 \cdot 120 \\
          &= 720
\end{align}
</math>
<i>Sol.</i>: <math>720</math> numbers.
|}<br/>

<div style="margin-left: 5%; margin-right: 5%; background: #FAFBFC; border: 1px solid rgb(153, 153, 153); padding: 1em; text-align:left;">
<strong>Question CT4.</strong> (2.5 points)<br/>
A secret ballot is made in a meeting of seventeen people. Two people have cast invalid ballots, three have cast blank ballots, five have cast dissenting votes and seven have cast assenting votes. Using combinatorial reasoning calculate in how many ways this could have occured.
</div>
{| class="mw-collapsible wikitable {{#ifeq: {{{plegada|sí}}}|no||mw-collapsed}}" style="margin-left: 5%; margin-right: 5%; text-align:left; padding:0px; background-color:white;"
| align="left"   | <span style="text-align:left; font-weight:bold;">Solution:</span>
|-
| style="text-align:left; font-size:100%; padding:6px" | Let us use the [[distribution of objects into recipients|occupancy model for the non ordered distribution of balls into boxes]]; balls and boxes representing ballots and people, respectively. Consider the <math>17</math> persons (distinguishable boxes) and the <math>7</math> assenting ballots (indistinguishable balls), on an underlying injective mapping (at most, one ballot by each person) — alternatively, we could consider the number of subsets of <math>7</math> elements from a set of <math>17</math> elements —. In any case, there are <math>C_{17,7}</math> ways of distributing the assenting ballots into the boxes. For each one of these cases (each one of the combinations), there are <math>10</math> empty boxes left. Now, using a similar reasoning, there are <math>C_{10,5}</math> ways of distributing the dissenting ballots into the boxes, with <math>5</math> empty boxes remaining for each one of these cases. Similarly, there are <math>C_{5,3}</math> ways of distributing the blank ballots into the boxes, with <math>2</math> empty boxes remaining for each one of the cases. So, lastly, there are <math>C_{2,2}</math> ways in which invalid ballots can be placed into the boxes. Applying the [[rule of product]]:
<math display="block">
\begin{align}
C_{17,7} \cdot C_{10,5} \cdot C_{5,3} \cdot C_{2,2} &= \binom{17}{7} \cdot \binom{10}{5} \cdot \binom{5}{3} \cdot \binom{2}{2} \\
          &= \frac{17!}{7! \cdot 10!} \cdot \frac{10!}{5! \cdot 5!} \cdot \frac{5!}{3! \cdot 2!} \cdot \frac{2!}{2! \cdot 0!} \\
          &= \frac{17! \cdot \cancel{10!} \cdot \cancel{5!} \cdot \cancel{2!}}{7! \cdot \cancel{10!} \cdot \cancel{5!} \cdot 5! \cdot 3! \cdot \cancel{2!} \cdot 2! \cdot 0!} \\
          &= 49 \ 008 \ 960
\end{align}
</math>
<i>Sol.</i>: In <math>49 \ 008 \ 960</math> ways.
|}<br/>

<div style="margin-left: 5%; margin-right: 5%; background: #FAFBFC; border: 1px solid rgb(153, 153, 153); padding: 1em; text-align:left;">
<strong>Question CT5.</strong> (2.5 points).<br/>
Use a combinatorial reasoning to respond.
<ul style = "list-style: none; margin-left: 5; padding-left: 1.35em; text-indent: -1.35em;">
<li>a) A number is palindrome if it reads the same from left to right and from right to left. In base ten, how many seven-digit numbers are palindromes? (1.25 p.)</li>
<li>b) Let us assume a <math>n</math> sided polygon network (<math>n</math>-gon network). Calculate <math>n</math>, the number of nodes (vertices) of the network, knowing that the number of line segments (sides + diagonals) is <math>253</math>. (1.25 p.)</li>
</ul>
</div>
{| class="mw-collapsible wikitable {{#ifeq: {{{plegada|sí}}}|no||mw-collapsed}}" style="margin-left: 5%; margin-right: 5%; text-align:left; padding:0px; background-color:white;"
| align="left"   | <span style="text-align:left; font-weight:bold;">Solution:</span>
|-
| style="text-align:left; font-size:100%; padding:6px" | <ul style = "list-style: none; margin-left: 5; padding-left: 1.35em; text-indent: -1.35em;">
<li>a) A seven digit palindrome fits into the model <math>abcdcba</math>, where <math>a \neq 0</math>. There are <math>9</math> possibilities for <math>a</math>, <math>10</math> for <math>b</math>, <math>10</math> for <math>c</math> and <math>10</math> other possibilities for <math>d</math>. Because of the [[rule of product]], there is a total of <math>9 \cdot 10 \cdot 10 \cdot 10 = 9000</math> seven digit palindromes.</li>
<li>b) If <math>n</math> is the number of nodes, then the number of line segments is the number of subsets of two elements (each line segment can be viewed as a subset of two elements, as it can be abstracted from the fact that it joins two nodes) from a set of <math>n</math> elements (the <math>n</math> nodes), and this number is, by definition of combination, <math>\binom{n}{2}</math>. Then, <math>\binom{n}{2} = 253 \Rightarrow \frac{n(n-1)}{2} = 253 \Rightarrow n = -22 \vee n = 23</math>. Thus, <math>n = 23</math>.</li>
</ul>
|}<br/>

==== Theme 4.- Finite difference equations (recurrence relations) ====

<div style="margin-left: 5%; margin-right: 5%; background: #FAFBFC; border: 1px solid rgb(153, 153, 153); padding: 1em; text-align:left;">
<strong>Question RR1.</strong> (2.5 points)<br/>
Let the following be the definition of the sum of two natural numbers <math>n</math> and <math>m</math>:
<math display=block>
\begin{align}
  S(n,0) &= n \\
  S(n,m) &= S(n,m-1) + 1
\end{align}
</math>
Prove that the solution of this recurrence is <math>S(n) = n + m</math>.
</div>
{| class="mw-collapsible wikitable {{#ifeq: {{{plegada|sí}}}|no||mw-collapsed}}" style="margin-left: 5%; margin-right: 5%; text-align:left; padding:0px; background-color:white;"
| align="left"   | <span style="text-align:left; font-weight:bold;">Solution:</span>
|-
| style="text-align:left; font-size:100%; padding:6px" | Let us note that <math>n</math> is alien to recursion. Thus, in an easier but equivalent way, denoting <math>S(n,m)</math> by <math>f(m)</math>, we get a linear non homogeneous recurrence relation with constant coefficients and with a constant function as the function on the RHS of the equation:
<math display=block>
\begin{align}
  f(0) &= n \\
  f(m) - f(m-1) &= 1
\end{align}
</math>
<ul style = "list-style: none; margin-left: 5; padding-left: 1.35em; text-indent: -1.35em;">
<li>a) General solution of the homogeneous:
<br />
The characteristic polynomial is:
<math display=block>
P(x) = x - 1
</math>
thus, <math>1</math> is a simple characteristic root.
<br />
The general solution of the homogeneous is:
<math display=block>
f(m) = c_1 1^m; \;\; c_1 \in \mathbb{R}
</math>
</li>
<li>b) Particular solution of the non homogeneous:
<br />
As the function on the RHS is constant, let us try out a general constant (real number) as a possible particular solution:
<math display=block>
k - k = 1
</math>
but this is a contradiction, so we have to increase the degree of the polynomial. Let us try out with a first degree polynomial, <math>P(m) = Am</math>. Substituting:
<math display=block>
Am - A(m-1) = 1 \Rightarrow Am - Am + A = 1 \Rightarrow A = 1
</math>
Thus:
<math display=block>
f(m) = m
</math>
is a particular solution of the non homogeneous.
</li>
<li>c) General solution of the non homogeneus:
<math display=block>
f(m) = c_1 + m; \;\; c_1 \in \mathbb{R} \;\;\;\;\; {\scriptstyle(1)}
</math>
</li>
<li>d) Considering the initial conditions:
<br />
From the initial condition, <math>f(0) = n</math>, we have:
<math display=block>
n = f(0) = c_1 + 0 \Rightarrow c_1 = n
</math>
Substituting in (1), we get the desired solution:
<math display=block>
f(m) = n + m
</math>
or, in other words:
<math display=block>
S(n,m) = n + m
</math>
</li>
</ul>
|}<br/>

<div style="margin-left: 5%; margin-right: 5%; background: #FAFBFC; border: 1px solid rgb(153, 153, 153); padding: 1em; text-align:left;">
<strong>Question RR2.</strong> (2.5 points)<br/>
Let <math>x(t)</math> and <math>y(t)</math> be the numbers of malicious software belonging to two malware types, in the day <math>t</math>, that coexist in a certain insecure wide area network (WAN) under malware evolution daily control. Let us assume that the original memberships were of <math>x(0)=3</math> and <math>y(0)=7</math> and that the coexistence evolution is as follows:
* every day, the growth in malware type <math>x</math> is the sum of the triple of the growth in type <math>x</math> on the previous day and the growth in type <math>y</math> also on the previous day plus seven new malware (that were classified as type <math>x</math>),
* and also every day, the growth in malware type <math>y</math> is the result of subtracting the growth in type <math>x</math> on the previous day from the growth in type <math>y</math> on the previous day, plus three new malware (that were classified as type <math>y</math>).
Find out and solve the system of recurrence equations of the evolution of the malware.
</div>
{| class="mw-collapsible wikitable {{#ifeq: {{{plegada|sí}}}|no||mw-collapsed}}" style="margin-left: 5%; margin-right: 5%; text-align:left; padding:0px; background-color:white;"
| align="left"   | <span style="text-align:left; font-weight:bold;">Solution:</span>
|-
| style="text-align:left; font-size:100%; padding:6px" | Let us analyze the evolutions of the two types of malware on their own growths (that these evolutions had to be calculated in terms of the populations or not is not specified by the wording and, on the other side, calculating them on their growths is easier because the order of the recurrence relation has decreased in one time unit). Let <math>X_{t}</math> and <math>Y_{t}</math> denote the growths from time <math>t</math> to time <math>t+1</math>, i.e. <math>X_{t} = x(t+1) - x(t)</math> and <math>Y_{t} = y(t+1) - y(t)</math>. The system of linear recurrence equations that correponds to this situation is:
<math display=block>
\left\lbrace
\begin{align}
X_{t+1} &= 3 X_t &+ Y_t &+ 7 \;\;\;\;\; {\scriptstyle(1)} \\
Y_{t+1} &= Y_t &- X_t &+ 3 \;\;\;\;\; {\scriptstyle(2)}
\end{align}
\right.
</math>
<ul style = "list-style: none; margin-left: 5; padding-left: 1.35em; text-indent: -1.35em;">
<li>a) Calculating <math>X_t</math>:
<br />
From the first equation we get:
<math display=block>
\begin{align}
Y_t &= X_{t+1} - 3 X_t - 7 \;\;\;\;\; {\scriptstyle(3)} \\
X_{t+2} &= 3 X_{t+1} + Y_{t+1} + 7 \;\;\;\;\; {\scriptstyle(4)}
\end{align}
</math>
Substituting (2) in (4):
<math display=block>
X_{t+2} = 3 X_{t+1} + Y_t - X_t + 3 + 7
</math>
Substituting (3) in the latter, and simplifying, grouping and sorting, we get a linear non homogeneous recurrence relation with constant coefficients and with a constant function as the function on the RHS (right-hand side) of the equation:
<math display=block>
X_{t+2} - 4 X_{t+1} + 4 X_t = 3
</math>
<ul style = "list-style: none; margin-left: 5; padding-left: 1.35em; text-indent: -1.35em;">
<li>a.1) General solution of the homogeneous:
<br />
The characteristic polynomial is:
<math display=block>
P(X) = X^2 - 4 X + 4
</math>
that is:
<math display=block>
P(X) = (X - 2)^2
</math>
thus, <math>2</math> double characteristic root.
<br />
The general solution of the homogeneous is:
<math display=block>
X_t = c_1 2^t + c_2 t 2^t; \;\; c_1, c_2 \in \mathbb{R}
</math>
</li>
<li>a.2) Particular solution of the non homogeneous:
<br />
As the function on the RHS is constant, let us try out a general constant (real number) as a possible particular solution:
<math display=block>
k - 4k + 4k = 3
</math>
then <math>k = 3</math>. Thus:
<math display=block>
X_t = 3
</math>
is a particular solution of the non homogeneous.
</li>
<li>a.3) General solution of the non homogeneous:
<math display=block>
X_t = c_1 2^t + c_2 t 2^t + 3; \;\; c_1, c_2 \in \mathbb{R} \;\;\;\;\; {\scriptstyle(5)}
</math>
</li>
</ul>
</li>
<li>b) Calculating <math>Y_t</math>:
<br />
Substituting (5) in (1), and simplifying, grouping and sorting, we get:
<math display=block>
Y_t = \left( 2c_2 - c_1 \right) 2^t - c_2 t 2^t - 13; \;\; c_1, c_2 \in \mathbb{R}
</math>
</li>
<li>c) Considering the initial conditions:
<br />
From the initial conditions, <math>x_0 = 3</math> and <math>y_0 = 7</math> we have:
<math display=block>
X_0 = x_1 - x_0 = x_1 - 3 = c_1 2^0 + c_2 \cdot 0 \cdot 2^0 + 3 = c_1 + 3 \Rightarrow c_1 = x_1 - 6  \;\;\;\;\; {\scriptstyle(6)}
</math>
On the other side:
<math display=block>
Y_0 = y_1 - y_0 = y_1 - 7 = \left( 2c_2 - c_1 \right) 2^0 - c_2 \cdot 0 \cdot 2^0 - 13 = 2c_2 - c_1 - 13  \;\;\;\;\; {\scriptstyle(7)}
</math>
and now, substituting (6) in (7):
<math display=block>
y_1 - 7 = 2c_2 - x_1 + 6 - 13 \Rightarrow y_1 = 2c_2 - x_1 \Rightarrow c_2 = \frac{x_1 + y_1}{2}
</math>
</li>
<li>d) The solution to the situation under the wording of the question (evolution of these two types of malware on their own growths) is:
<math display=block>
\left\lbrace
\begin{align}
X_t &= \left( x_1 - 6 \right) 2^t + \left( x_1 + y_1 \right) t 2^{t-1} + 3 \\
Y_t &= \left( y_1 + 6 \right) 2^t - \left( x_1 + y_1 \right) t 2^{t-1} - 13
\end{align}
\right.
</math>
where <math>x_1</math> and <math>y_1</math> are the populations of both types of malware by the end of the first hour (data not provided in the wording).
</li>
</ul>
|}<br/>

==== Apéndices ====

===== Graphs =====

<div style="margin-left: 5%; margin-right: 5%; background: #FAFBFC; border: 1px solid rgb(153, 153, 153); padding: 1em; text-align:left;">
<table>
<td width=66% style="vertical-align:top;">
<strong>Question G1.</strong> (2.5 points)<br/>
The accompanying graph shows the connections among four tram stations. You may:
<ul style = "list-style: none; margin-left: 5; padding-left: 1.35em; text-indent: -1.35em;">
<li>a) Write the adjacency matrix <math>G</math> of that graph.</li>
<li>b) Interpret the matrices <math>G^2</math> and <math>G^3</math> (reason what situations they represent).</li>
<li>c) Reason, using those matrix representations, if it is a strongly connected graph or not.</li>
<li>d) Reason, using those matrix representations, what is the length of the shortest path from <math>A</math> to <math>D</math> and how many paths may be considered as the "shortest" ones.</li>
</ul>
</td>
<td width=33% style="vertical-align:top;">
<!-- https://en.wikipedia.org/wiki/Box-drawing_character -->
<div style="text-align: center;">
<pre>
+—+       +—+
|D| <———> |C|
+—+       +—+
    \      ⋀
     \     |
      \    |
       \   |
        ╶┘ |
+—+       +—+
|A| <———> |B|
+—+       +—+
</pre>
</div>
</td>
</table>
</div>
{| class="mw-collapsible wikitable {{#ifeq: {{{plegada|sí}}}|no||mw-collapsed}}" style="margin-left: 5%; margin-right: 5%; text-align:left; padding:0px; background-color:white;"
| align="left"   | <span style="text-align:left; font-weight:bold;">Solution:</span>
|-
| style="text-align:left; font-size:100%; padding:6px" | <ul style = "list-style: none; margin-left: 5; padding-left: 1.35em; text-indent: -1.35em;">
        <li>a) The adjacency matrix <math>G</math> of this graph is:
	<math display=block>
	G =
	\begin{pmatrix}
	0 & 1 & 0 & 0 \\
	1 & 0 & 1 & 0 \\
	0 & 0 & 0 & 1 \\
	0 & 1 & 1 & 0
	\end{pmatrix}
	</math>
	We draw up the adjacency matrix of the graph, by matching the positional subscripts <math>1</math>, <math>2</math>, <math>3</math> and <math>4</math> of its elements with the labels <math>A</math>, <math>B</math>, <math>C</math> and <math>D</math>, so that, for example, <math>g_{23}</math> corresponds to a possible path from <math>B</math> to <math>C</math>, <math>B \rightsquigarrow C</math>. Thus, the element <math>g_{ij}</math> of <math>G</math> is the number of direct connections --- with no intermediate station (paths of length one in the graph) --- between the tram stations corresponding to <math>i</math> and <math>j</math>, in the direction <math>i \rightsquigarrow j</math>. In this way, we interpret <math>g_{23} = 1</math> as the existence of one direct connection from <math>2</math> to <math>3</math>, <math>2 \rightarrow 3</math>, this is, <math>B \rightarrow C</math>, whilst <math>g_{32} = 0</math> corresponds to the non existence of a direct connection from <math>3</math> to <math>2</math>, <math>3 \nrightarrow 2</math>, this is, <math>C \nrightarrow B</math>. </li>
	<li>b) The powers <math>2</math> and <math>3</math> of <math>G</math> are:
	<math display=block>
	G^2 =
	\begin{pmatrix}
	1 & 0 & 1 & 0 \\
	0 & 1 & 0 & 1 \\
	0 & 1 & 1 & 0 \\
	1 & 0 & 1 & 1
	\end{pmatrix}
	\quad \quad
	G^3 =
	\begin{pmatrix}
	0 & 1 & 0 & 1 \\
	1 & 1 & 2 & 0 \\
	1 & 0 & 1 & 1 \\
	0 & 2 & 1 & 1
	\end{pmatrix}
	</math>
	The element <math>g^{(2}_{ij}</math> of <math>G^2</math> is the number of connections with exactly one station in the middle (length two paths in the graph) from the corresponding station to <math>i</math> to the corresponding station to <math>j</math>, under the previous formalization. Similarly, the element <math>g^{(3}_{ij}</math> of <math>G^3</math> represents the number of connections with two intermediate stations (three length paths in the graph) from the corresponding station to <math>i</math> to the corresponding station to <math>j</math>, again according to the previous formalization. </li>
	<li>c) A graph is strongly-connected precisely if there exists a path from any vertex to any vertex. On the other side, given a graph <math>G</math>, with <math>n</math> vertices, it is possible to know if there exists a path from the vertex <math>i</math> to the vertex <math>j</math>, regardless of the length but depending on the element <math>p_{ij}</math> of the matrix <math>P = G + G^2 + G^3 + \cdots + G^{n-1}</math> as it is the total number of paths from <math>i</math> to <math>j</math> (if there were <math>i, j</math> such that <math>p_{ij} = 0</math> then no path would be possible from <math>i</math> to <math>j</math> and the graph would not be strongly connected). <br />The graph under our study, <math>G</math>, with <math>4</math> vertices, is strongly connected because <math>P = G + G^2 + G^3</math> has not zero elements:
	<math display=block>
	P = G + G^2 + G^3
	=
	\begin{pmatrix}
	1 & 2 & 1 & 1 \\
	2 & 2 & 3 & 1 \\
	1 & 1 & 2 & 2 \\
	1 & 3 & 3 & 2
	\end{pmatrix}
	</math>
	which mean that any two stations are connected each other, either directly or indirectly via one or two intermediate stops. Indeed, for this particular graph, <math>G^2 + G^3</math> neither has zero elements:
	<math display=block>
	G^2 + G^3
	=
	\begin{pmatrix}
	1 & 1 & 1 & 1 \\
	1 & 2 & 2 & 1 \\
	1 & 1 & 2 & 1 \\
	1 & 2 & 2 & 2
	\end{pmatrix}
	</math>
	that is to say, any two stations are connected each other via one or two intermediate stops. </li>
        <li>d) Denoting by <math>g_{ij}^{(k)}</math> the element at position <math>(i,j)</math> of the matrix <math>G^{k}</math>, we note that <math>g_{14} = g_{14}^{(2)} = 0</math> and that <math>g_{14}^{(3)} = 1</math>, this being the first non-zero digit, so the shortest path has two intermediate stops and there is only one shortest path (since the value of <math>g_{14}^{(3)}</math> is one).
	</ul>
|}<br/>

===== Numerical algebra and calculus =====

<div style="margin-left: 5%; margin-right: 5%; background: #FAFBFC; border: 1px solid rgb(153, 153, 153); padding: 1em; text-align:left;">
<strong>Question NAC1.</strong> (2.5 points)<br/>
Find a possible general formula for computing the ''n''th term, that is, <math>a_n</math>, of the [[Sequence|sequence]] <math>a_1 = 1, a_2 = 2, a_3 = 4, a_4 = 7</math> using the [[Newton polynomial|Newton's divided differences interpolation polynomial]].
</div>

{| class="mw-collapsible wikitable {{#ifeq: {{{plegada|sí}}}|no||mw-collapsed}}" style="margin-left: 5%; margin-right: 5%; text-align:left; padding:0px; background-color:white;"
| align="left"   | <span style="text-align:left; font-weight:bold;">Solution:</span>
|-
| style="text-align:left; font-size:100%; padding:6px" | Because of custom and maybe tradition we begin considering <math>n = 0</math>, and so we start dealing this question with <math>b_0 = 1, b_1 = 2, b_2 = 4, b_3 = 7</math> and then adjust to match what was required. Let <math>f(n)</math> be <math>b_n</math>. Then, the table of divided differences is:
<math display=block>
\begin{array}{cc|ccc|cccc|cc}
i & j & x_i & f(x_i) & f(x_i,x_j) & i & j & k & l & f(x_i,x_j,x_k) & f(x_i,x_j,x_k,x_l) \\
\hline
0 &   & 0 & 1 &   &   &   &   &   &     &   \\
0 & 1 &   &   & 1 &   &   &   &   &     &   \\
1 &   & 1 & 2 &   & 0 & 1 & 2 &   & 1/2 &   \\
1 & 2 &   &   & 2 & 0 & 1 & 2 & 3 &     & 0 \\
2 &   & 2 & 4 &   & 1 & 2 & 3 &   & 1/2 &   \\
2 & 3 &   &   & 3 &   &   &   &   &     &   \\
3 &   & 3 & 7 &   &   &   &   &   &     &   \\
\end{array}
</math>
where:
<math display=block>
\begin{align}
f(x_i,x_j) &= \frac{f(x_j) - f(x_i)}{x_j - x_i} \\
f(x_i,x_j,x_k) &= \frac{f(x_j,x_k) - f(x_i,x_j)}{x_k - x_i} \\
&\;\;\vdots \\
f(x_0,x_1,\ldots,x_n) &= \frac{f(x_1,x_2,\ldots,x_n) - f(x_0,x_1,\ldots,x_{n-1})}{x_n - x_0}
\end{align}
</math>
The interpolating polynomial is:
<math display=block>
\begin{align}
f_n(x) = f(x_0) &+ (x - x_0) f(x_0, x_1) \\
                &+ (x - x_0) (x - x_1) f(x_0, x_1, x_2) \\
                &+ \cdots \\
                &+ (x - x_0) (x - x_1) \ldots (x - x_{n-1}) f(x_0, x_1, \dots, x_n)
\end{align}
</math>
as well as in recurrence form:
<math display=block>
f_n(x) = f_{n-1}(x) + (x - x_0) (x - x_1) \ldots (x - x_{n-1}) f(x_0, x_1, \dots, x_n)
</math>
Thus:

* <math>f_0(x) = f(x_0) = f(0) = 1</math>, that is satisfied by <math>x_0</math>, but no longer by <math>x_1</math> since <math>f_0(x_1) = f_0(1) = 1 \neq 2 = f(1)</math>.

* <math>f_1(x) = f_0(x) + (x - x_0) f(x_0, x_1) = 1 + (x - 0) \cdot 1 = 1 + x</math>, that is satisfied by <math>x_0</math> and <math>x_1</math>, but no longer by <math>x_2</math> since <math>f_1(x_2) = f_1(2) = 3 \neq 4 = f(2)</math>.

* <math>f_2(x) = f_1(x) + (x - x_0) (x - x_1) f(x_0, x_1, x_2) = 1 + x + (x - 0) (x - 1) \cdot 1/2 = 1 + \frac{x}{2} + \frac{x^2}{2}</math>, which is satisfied by all of the points, even by <math>x_3</math> since <math>f_2(x_3) = f_2(3) = 7 = f(3)</math>.

* The next difference is zero, which confirms that the interpolating polynomial, suggested by this method, is a second degree polynomial:
<math display=block>
f_2(x) = 1 + \frac{x}{2} + \frac{x^2}{2}
</math>

In summary, starting with <math>n = 0</math>, the general term is <math>b_n = 1 + \frac{n}{2} + \frac{n^2}{2}</math>, and adjusting to match the <math>n = 1</math> beginning as required, the general term is:
<math display=block>
a_n = 1 + \frac{n-1}{2} + \frac{(n-1)^2}{2}
</math>
<i>Sol.</i>: <math>a_n = 1 + \frac{n-1}{2} + \frac{(n-1)^2}{2}</math>.
|}<br/>

</div>

==== Sample preparatory exams ====
{{ambox
|nocat=true
| type  = notice
| text  =
My intention is to having two practice tests for training the final exam, the '''mid-course and end-course preparatory exams''' (another two assessment tools). Based on what we have worked with so far, both preparatory exams will replicate the final exam in level, content and format. These exams will be performed as '''homework''' and should be expected to be closed-book exams and to last two hours (the same as the final exam). This 'controlled self-assessment' mechanism intend to stimulate the personal work of the student, detect mistakes and weakness and make aware of possible comprehension gaps, and correct them. '''Two large-group classes are dedicated to share ideas and solutions in a whole-class discussion, one for each exam'''. These exams are for training and self-study use. They are not graded by the teacher and they are '''not included in the calculation of the final grade'''. As the final exam is individual-based, I recommend that you do them individually rather than in groups, anyway the choice is yours.
}}

===== Academic year 2016-2017 =====
(The questions and their identifiers are placed from the next section onwards).

*Part 1: Themes 1 and 2.
**Sample preparatory exam, 1: L2, SRF1, ACN1, NT1.
**Sample preparatory exam, 2: L3, C1, NT5, NT3.
*Part 2: Themes 3 and 4.
**Sample preparatory exam, 1: CT1, CT2, RR1, AE1.
**Sample preparatory exam, 2: CT3, CT4, RR2, G1.

===== Academic year 2017-2018 =====
* The [[Wikipedia:School_and_university_projects/Discrete_and_numerical_mathematics/Learning_plan#Past_qualifying_activities_and_real_exams_with_some_solutions|midterm qualifying activity]] was the first sample preparatory exam.

===== Academic year 2018-2019 =====
* [https://drive.google.com/open?id=17g0ewJ58p-OCpbdttvo3-5pM7nIJ73aA Mid course preparatory exam]
* [https://drive.google.com/open?id=1DZsVWqEl7GH_As9_lOdMn-75dL8_oB0Y End course preparatory exam]

===== Academic year 2019-2020 =====

== Past qualifying activities and real exams with some solutions <!--<span style="font-variant: small-caps; background:yellow;">(new)</span>-->==
Most of them are '''bilingual documents''' with side by side texts in a double column format, the left column being the Spanish text and the right column being the corresponding English text. (Why? Please read for instance what [https://www.gse.harvard.edu/news/07/03/mathematics-language Maria Martinello says]).

=== Academic year 2016-2017 ===
* [https://zurbaran.unex.es/cloud/index.php/s/XJPCYdon7qYCeqO Final exam, 1 June, 2017] (A resolution approach to the first question has been published here: [[Wikipedia:School and university projects/Discrete and numerical mathematics/Learning plan#Logic_2|L1]]).
* [https://zurbaran.unex.es/cloud/index.php/s/ELFZ8uY8GZpV1nt Resit final exam, 7 July, 2017] (Resolution approaches to the second, third and fifth questions have been published here: [[Wikipedia:School and university projects/Discrete and numerical mathematics/Learning plan#Cardinality_2|C2]], [[Wikipedia:School and university projects/Discrete and numerical mathematics/Learning plan#Number theory_2|NT4]] y [[Wikipedia:School and university projects/Discrete and numerical mathematics/Learning plan#Combinatorial_theory_2|CT5]], respectively).

=== Academic year 2017-2018 ===
* [https://zurbaran.unex.es/cloud/index.php/s/i7YhPGblU6gscBt Midterm qualifying activity, White, 10 April, 2018]
* [https://zurbaran.unex.es/cloud/index.php/s/wS6JMyE5thGADF6 Midterm qualifying activity, Green, 10 April, 2018]
* [https://zurbaran.unex.es/cloud/index.php/s/SYxGlu8WhdZCDLN Final exam, 24 May, 2018]
* [https://zurbaran.unex.es/cloud/index.php/s/edXVk0uUNuzpnTF Resit final exam, 29 June, 2018]

=== Academic year 2018-2019 ===
* Midterm Qualifying Activity (10 April, 2019) [https://drive.google.com/open?id=1NwfF_gQOfc_It9MO4MEiV-pbC9xQ_Vva Questions] [https://drive.google.com/file/d/10u2P1ehX0c2xvUk_Uxc83-IbgpqEI82j Resolution approach]
* [https://drive.google.com/open?id=1IuUgVKFNUaooMz4UxLT5mkMJHrlFrbIr Final Exam (3 June, 2019)]
* [https://drive.google.com/open?id=19nZEH6y3UH1b1DfEPrztDqTG4fFWAux1 Resit Final Exam (25 June, 2019)]

=== Academic year 2019-2020 ===
* [https://drive.google.com/open?id=jmlr Two-Week Take-Home Final Exam (due to the COVID-19 pandemic) (before 12 June, 2020)]
* [https://drive.google.com/open?id=jmlr Two-Week Take-Home Resit Final Exam (due to the COVID-19 pandemic) (before 14 July, 2020)]

== Tentative course outline (chronogram for the 2019-2020 academic year) ==
<!--{{Being translated|es|Wikipedia:Proyecto educativo/Matemática discreta y numérica/Plan de aprendizaje}}-->

'''Important:''' Let us remember that exercises from Rosen's books and many others are available under the [[all rights reserved]] regime. However, their study and work on are an endless source of ideas for making contributions to Wikipedia.
<!--
{| style="margin:auto; width:90%; color:black;"
 | style="width:0%; text-align:right; background-color:White;" | [[File:Ambox warning green construction.svg|center|46px]]
 | style="width:100%; text-align:center; background-color:#f1fadf; border-style: solid; border-width: 0px 0px 0px 0px" | <hr style="border-width: 1px;">This tentative course outline, which corresponds to the [https://zurbaran.unex.es/cloud/index.php/s/9ZYJep76byeHsKB approved course program (ficha12a)] is '''under construction''' and therefore '''subject to change''', possibly until a few days before the first day of classes in second academic semester. Please excuse the inconvenience.<hr style="border-width: 1px;">
|}
-->
{{ambox
|nocat=true
| type  = notice
| text  =
This tentative course outline, which corresponds to [https://zurbaran.unex.es/cloud/index.php/s/9ZYJep76byeHsKB the approved course program (ficha12a)] is <strong>orientative and dynamic</strong>
(the planned development of the course may vary according to the particularities of the students involved).
}}
<!-- † <small>[https://zurbaran.unex.es/cloud/index.php/s/9ZYJep76byeHsKB Click here to retrieve the approved course program (ficha12a)].</small> -->

=== Calendar of activities ===

<!-- GRAN TABLA -->
<div style="column-count:1;-moz-column-count:1;-webkit-column-count:1; font-size:85%;">
{| class="wikitable"
| colspan="8" style="text-align: center;" |
[[File:UN emblem blue.svg|link=|46x46px]]

'''4 January'''
<br />
[[World Braille Day]]

'''24 January'''
<br />
[[International Day of Education]]

'''27 January'''
<br />
[[International Holocaust Remembrance Day]]
|-
! colspan="8" style="text-align: center; background: #4b92db; border-left:solid 1px black;border-right:solid 1px black;border-top:solid 3px black;border-bottom:solid 3px black;" |<span style="font-variant: small-caps; font-size: large;">'''Further Mathematics'''</span>

<span style="font-variant: small-caps; font-size: 120%;">'''Academic Year 2019-2020'''</span>

<span style="font-variant: small-caps; font-size: 120%;">'''2<sup>nd</sup> semester'''</span>

<span style="font-variant: small-caps; font-size: 120%;">'''Class (large group) meetings (48 <span style="font-variant: normal;">h</span>) and seminar/laboratory meetings (12 <span style="font-variant: normal;">h</span>)'''</span>
|-
! rowspan="2" style="text-align: center; background: LightSteelBlue; border-left:solid 1px black;border-right:solid 1px black;border-top:solid 1px black;border-bottom:solid 1px black;" |'''Dates'''
! rowspan="2" style="text-align: center; background: LightSteelBlue; border-left:solid 1px black;border-right:solid 1px black;border-top:solid 1px black;border-bottom:solid 1px black;" |'''Topics''' 
! colspan="3" style="text-align: center; background: LightSteelBlue; border-left:solid 1px black;border-right:solid 1px black;border-top:solid 1px black;border-bottom:solid 1px black;" |'''Basic <span style="background:yellow">texts</span> readings and study'''
! colspan="3" style="text-align: center; background: LightSteelBlue; border-left:solid 1px black;border-right:solid 1px black;border-top:solid 1px black;border-bottom:solid 1px black;" |'''Following is a selection of training <span style="background:yellow">exercises</span>,''' '''essentially instrumental'''. Do not forget those that are worked in class and seminar/lab meetings. <!--<br />([http://zurbaran.unex.es Una selección más amplia], en breve).--><small>The concreteness of the examples implies in no case a course content cut. It is strongly recommended to solve exercises and other questions, the more the better. There is a more than enough bibliography to which to consult.</small>
|-
| style="text-align: center; background: LightSteelBlue; border-left:solid 1px black;border-right:solid 1px black;border-top:solid 1px black;border-bottom:solid 1px black;" |'''Rosen 5<sup>th</sup> ed. Spain/USA'''
| style="text-align: center; background: LightSteelBlue; border-left:solid 1px black;border-right:solid 1px black;border-top:solid 1px black;border-bottom:solid 1px black;" |'''Rosen 7<sup>th</sup> ed. USA'''
| style="text-align: center; background: LightSteelBlue; border-left:solid 1px black;border-right:solid 1px black;border-top:solid 1px black;border-bottom:solid 1px black;" |'''Others'''
| style="text-align: center; background: LightSteelBlue; border-left:solid 1px black;border-right:solid 1px black;border-top:solid 1px black;border-bottom:solid 1px black;" |'''Rosen 5<sup>th</sup> ed. Spain/USA'''
| style="text-align: center; background: LightSteelBlue; border-left:solid 1px black;border-right:solid 1px black;border-top:solid 1px black;border-bottom:solid 1px black;" |'''Rosen 7<sup>th</sup> ed. USA'''
| style="text-align: center; background: LightSteelBlue; border-left:solid 1px black;border-right:solid 1px black;border-top:solid 1px black;border-bottom:solid 1px black;" |'''Others'''
|-
| colspan="8" style="box-sizing: border-box; border-bottom-width: 3px; border-bottom-style: solid; border-bottom-color: rgb(0, 173, 239); border-top-width: 3px; border-top-style: solid; border-top-color: rgb(0, 173, 239); text-align: center; background: #E6ECFF;" |<p style="text-transform: uppercase; font-size: 120%;">'''Theme 1'''</p>
<!--  line-height: 1.1; margin-top: 26px; margin-bottom: 9px; padding-left: 1px; padding-bottom: 7px; -->
<p style="text-transform: uppercase;">'''Fundamentals'''</p>

<p>(16 h LG and 5 h S/L)</p>
|-
| style="text-align: center;"|
'''Wed'''
<br />
'''29/1'''
<br />
'''Start date of classes'''
|Symbolic logic, I: Propositional logic

* Propositional logic;
* The method of truth tables as a verification strategy.
|
* § 1.1.
|
* § 1.1;
* § 1.2.
|
* [[:en:Wikipedia:School_and_university_projects/Discrete_and_numerical_mathematics/Learning_plan#Logic|WP+ (Logic)]].
|
* § 1.1 (8, 15, 30, 48, 42, 51-55, 59).
|
* § 1.1 (12, 21, 38);
* § 1.2 (12, 16, 19-23, 35).
|
* [[:en:Wikipedia:School_and_university_projects/Discrete_and_numerical_mathematics/Learning_plan#Logic|WP+ (Logic)]].
|-
| rowspan="2" style="text-align: center;"|
Thu
<br />
30/1
|Symbolic logic, I: Propositional logic

* Propositional equivalences;
* Formal derivation.
|
* § 1.2.
|
* § 1.3.
|
* [[:en:Wikipedia:School_and_university_projects/Discrete_and_numerical_mathematics/Learning_plan#Logic|WP+ (Logic)]].
|
* § 1.2 (8, 10, 29, 51).
|
* § 1.3 (10, 12, 29, 57).
|
* [[:en:Wikipedia:School_and_university_projects/Discrete_and_numerical_mathematics/Learning_plan#Logic|WP+ (Logic)]].
|- style="vertical-align:top;"
| colspan="7" style="text-align: left; background: #FFF5FA;" |[[File:Twenty sided dice.svg|32px|link=]] [[Wikipedia:School and university projects/Discrete and numerical mathematics|University project Discrete and numerical mathematics (optional out-of-class activity)]]: '''Beginning date''' of the academic component of the project in the 2nd semester of the academic year 2018-2019. You should read its descriptive web page, [[Wikipedia:School and university projects/Discrete and numerical mathematics]]. '''Once you have read''' that web page, '''and if''' you are interested in the project '''and only if''' you have queries '''or''' need help to do what you have been told (on that web page) to do '''or''' want to help your colleagues to do it '''or''' want to share questions, concerns '''or''' suggestions about the project, you could attend '''at 4:00 p.m., to Room O5''' (meeting will finish at no later than 5:30 p.m.). (Bring a computer if you need help). (This meeting will be in Spanish).
|-
| style="text-align: center;"|
'''Group B'''
<br />
Fri
<br />
31/1
<hr>
'''Groups A and E'''
<br />
Mon
<br />
3/2
| style="text-align: left; background: #EEFFEE;" |'''Seminar/Laboratory No. 1''':<br />Proofs and refutations, I

* (Occasionally computer-assisted) hands-on [[word problem|word-problem]]-solving on issues related to [[Formal language|formal]] and [[Natural language|informal]] [[argument|arguments]], essentially using:
** truth tables,
**''reductio ad absurdum'', or
** normal forms.
| style="text-align: left; background: #EEFFEE;" |
* § 1.1, 1.2;
* § 1.5.7;
* —.
| style="text-align: left; background: #EEFFEE;" |
* § 1.1, 1.2, 1.3;
* § 1.7.7;
* —.
| style="text-align: left; background: #EEFFEE;" |
*Rosen 7<sup>th</sup> Global Edition: § 1.7 (normal forms);
*[[Wikipedia:School_and_university_projects/Discrete_and_numerical_mathematics/Learning_plan#Logic|WP+ (Logic)]].
| style="text-align: left; background: #EEFFEE;" |
*[[iarchive:KDEM.DA.HD.70301|León-Rojas, J. M., KDEM DA/HD – 703-01]].
| style="text-align: left; background: #EEFFEE;" |
*[[iarchive:KDEM.DA.HD.70301|León-Rojas, J. M., KDEM DA/HD – 703-01]].
| style="text-align: left; background: #EEFFEE;" |
*Rosen 7<sup>th</sup> Global Edition: § 1.7 (1, 2, 3, 4, 5, 6);
*[[Wikipedia:School_and_university_projects/Discrete_and_numerical_mathematics/Learning_plan#Logic|WP+ (Logic)]];
* [https://drive.google.com/open?id=1FyeHoQXRIFNm8ItLFpW2jIepzFL7pklt León-Rojas, J. M., Question selection no. 1 and 1<sup>st</sup> seminar/laboratory. Proofs and refutations, I.]
|-
| style="text-align: center;"|
Tue
<br />
4/2
<br />
[[File:UN emblem blue.svg|link=|46x46px]]
<br />
[[World Cancer Day]]
|Symbolic logic, I: Propositional logic

* ''Reductio ad absurdum'' (proof by contradiction);
* Normal forms.

Symbolic logic, II: Predicate logic

* Predicates, variables, quantifiers, negation of quantifiers and logical equivalences.
|
* § 1.5.7;
* —
* § 1.3.
|
*§ 1.7.7;
* —
* § 1.4.
|
* Rosen 7th Global Edition: § 1.7 (normal forms);
* [[Wikipedia:School_and_university_projects/Discrete_and_numerical_mathematics/Learning_plan#Logic|WP+ (Logic)]].
|
* —
* § 1.3 (8, 10, 22, 41, 48, 55);
*[[iarchive:KDEM.DA.HD.70301|León-Rojas, J. M., KDEM DA/HD – 703-01]].
|
* —
* § 1.4 (8, 10, 24, 43, 52, 59);
*[[iarchive:KDEM.DA.HD.70301|León-Rojas, J. M., KDEM DA/HD – 703-01]].
|
* Rosen 7th Global Edition: § 1.7 (normal forms) (1, 2, 3, 4, 5, 6);
* [[Wikipedia:School_and_university_projects/Discrete_and_numerical_mathematics/Learning_plan#Logic|WP+ (Logic)]].
|-
| style="text-align: center;"|
Wed
<br />
5/2
|Symbolic logic, II: Predicate logic

* Reverse translation (from English to predicate logic).
|
* § 1.3.
|
* § 1.4.
|
*[[Wikipedia:School_and_university_projects/Discrete_and_numerical_mathematics/Learning_plan#Logic|WP+ (Logic)]].
|
* § 1.3 (8, 10, 22, 41, 48, 55).
|
* § 1.4 (8, 10, 24, 43, 52, 59).
|
*[[Wikipedia:School_and_university_projects/Discrete_and_numerical_mathematics/Learning_plan#Logic|WP+ (Logic)]].
|-
| style="text-align: center;"|
Thu
<br />
6/2
<br />
[[File:UN emblem blue.svg|link=|46x46px]]
<br />
[[International Day of Zero Tolerance for Female Genital Mutilation]]
|Symbolic logic, II: Predicate logic

* Nested quantifiers; order of quantifiers; negating nested quantifiers;
*Translation from predicate logic to English;
*Reverse translation (from English to predicate logic).
|
* § 1.4.
|
* § 1.5.
|
*[[Wikipedia:School_and_university_projects/Discrete_and_numerical_mathematics/Learning_plan#Logic|WP+ (Logic)]].
|
* § 1.4 (5, 8, 13, 18, 21, 28, 37, 44).
|
* § 1.5 (5, 8, 13, 18, 21, 28, 39, 48).
|
*[[Wikipedia:School_and_university_projects/Discrete_and_numerical_mathematics/Learning_plan#Logic|WP+ (Logic)]].
|-
| style="text-align: center;"|
'''Group B'''
<br />
Fri
<br />
7/2
<hr>
'''Groups A and E'''
<br />
Mon
<br />
10/2
<br />
[[File:UN emblem blue.svg|link=|46x46px]]
<br />
[[World Pulses Day]]
| style="text-align: left; background: #EEFFEE;" |'''Seminar/Laboratory No. 2''':<br />Proofs and refutations, II

* (Occasionally computer-assisted) hands-on [[word problem|word-problem]]-solving on issues related to [[Formal language|formal]] and [[Natural language|informal]] [[argument|arguments]], essentially using:
** natural deduction calculus (Gentzen-style).
| style="text-align: left; background: #EEFFEE;" |
* § 1.5 (particularly, § 1.5.3, 1.5.6).
| style="text-align: left; background: #EEFFEE;" |
* § 1.6 (particularly, § 1.6.4, 1.6.7).
| style="text-align: left; background: #EEFFEE;" |
*[[Wikipedia:School_and_university_projects/Discrete_and_numerical_mathematics/Learning_plan#Logic|WP+ (Logic)]].
| style="text-align: left; background: #EEFFEE;" |
*[[iarchive:KDEM.DA.HD.70301|León-Rojas, J. M., KDEM DA/HD – 703-01]]
| style="text-align: left; background: #EEFFEE;" |
*[[iarchive:KDEM.DA.HD.70301|León-Rojas, J. M., KDEM DA/HD – 703-01]]
| style="text-align: left; background: #EEFFEE;" |
*[[Wikipedia:School_and_university_projects/Discrete_and_numerical_mathematics/Learning_plan#Logic|WP+ (Logic)]];
* [https://drive.google.com/open?id=1TwpCBInPE5ELIw0PFKvQZUwUNixFCSYp León-Rojas, J. M., Question selection no. 2 and 2<sup>nd</sup> seminar/laboratory. Proofs and refutations, II.]
|-
| style="text-align: center;"|
Tue
<br />
11/2
<br />
[[File:UN emblem blue.svg|link=|46x46px]]
<br />
[[International Day of Women and Girls in Science]]
|Symbolic logic, III: Proofs

* Valid arguments and rules of inference.
|
* § 1.5.1, 1.5.2, 1.5.3, 1.5.4, 1.5.5, 1.5.6.
|
* § 1.6.
|
*[[Wikipedia:School_and_university_projects/Discrete_and_numerical_mathematics/Learning_plan#Logic|WP+ (Logic)]].
|
* § 1.5 (10, 12).
|
* § 1.6 (14, 16).
|
*[[Wikipedia:School_and_university_projects/Discrete_and_numerical_mathematics/Learning_plan#Logic|WP+ (Logic)]].
|-
| style="text-align: center;"|
Wed
<br />
12/2
|Symbolic logic, III: Proofs

* Introduction to proofs;
* Proof methods and strategy.
|
* § 1.5.7, 1.5.8, 1.5.9, 1.5.10;
* § 3.1.
|
* § 1.7;
* § 1.8.
|
*[[Wikipedia:School_and_university_projects/Discrete_and_numerical_mathematics/Learning_plan#Logic|WP+ (Logic)]].
|
* § 1.5 (20, 22, 30, 32, 35, 46, 58);
* § 3.1 (11, 13, 14, 19, 20, 27, 32, 38, 44, 49, 51).
|
* § 1.7 (14, 18, 24);
* § 1.8 (3, 7, 20, 23, 25, 26, 27, 42).
|
*[[Wikipedia:School_and_university_projects/Discrete_and_numerical_mathematics/Learning_plan#Logic|WP+ (Logic)]].
|-
| style="text-align: center;"|
Thu
<br />
13/2
<br />
[[File:UN emblem blue.svg|link=|46x46px]]
<br />
[[World Radio Day]]
|Sets

* Sets;
* Set operations; Boolean algebra; partitions.
|
* § 1.6;
* § 1.7.
|
* § 2.1;
* § 2.2.
|
*[[Wikipedia:School_and_university_projects/Discrete_and_numerical_mathematics/Learning_plan#Sets, relations and functions|WP+ (Sets, relations and functions)]].
|
* § 1.6 (5, 6, 17, 24, 27, 30);
* § 1.7 (2, 10, 12, 20, 29, 34).
|
* § 2.1 (7, 8, 23, 32, 41, 46);
* § 2.2 (2, 14, 16, 26, 37, 42).
|
*[[Wikipedia:School_and_university_projects/Discrete_and_numerical_mathematics/Learning_plan#Sets, relations and functions|WP+ (Sets, relations and functions)]].
|-
| style="text-align: center;"|
'''Group B'''
<br />
Fri
<br />
14/2
<hr>
'''Groups A and E'''
<br />
Mon
<br />
17/2
| style="text-align: left; background: #EEFFEE;" |'''Seminar/Laboratory No. 3''':<br />Proofs and refutations, III

* (Occasionally computer-assisted) hands-on [[word problem|word-problem]]-solving on issues related to [[Formal language|formal]] and [[Natural language|informal]] [[argument|arguments]], essentially using:
** Beth and Hintikka semantic tableaux (Smullyan&Jeffrey-style).
| style="text-align: left; background: #EEFFEE;" | —
| style="text-align: left; background: #EEFFEE;" | —
| style="text-align: left; background: #EEFFEE;" |
*Antón and Casañ, 1987: § 3.2;
*Manzano and Huertas, 2006: § 4 (for Propositional logic), § 11 (for First-order logic);
*[[Wikipedia:School_and_university_projects/Discrete_and_numerical_mathematics/Learning_plan#Logic|WP+ (Logic)]].
| style="text-align: left; background: #EEFFEE;" | —
| style="text-align: left; background: #EEFFEE;" | —
| style="text-align: left; background: #EEFFEE;" |
*[[Wikipedia:School_and_university_projects/Discrete_and_numerical_mathematics/Learning_plan#Logic|WP+ (Logic)]];
* [https://drive.google.com/open?id=11Blhk5sZ-0WvTV_b2bT66jYxQ7d0si1R León-Rojas, J. M., Question selection no. 3 and 3<sup>rd</sup> seminar/laboratory. Proofs and refutations, III.]
|-
| style="text-align: center;"|
Tue
<br />
18/2
|Functions

* Correspondences, functions and mappings; injectivity, surjectivity and bijectivity; inverse function.
|
* § 1.8.
|
* § 2.3.
|
*[[Wikipedia:School_and_university_projects/Discrete_and_numerical_mathematics/Learning_plan#Sets, relations and functions|WP+ (Sets, relations and functions)]].
|
* § 1.8 (6, 8, 12, 13, 16, 17, 27, 29, 36, 45, 69).
|
* § 2.3 (6, 8, 12, 13, 20, 21, 35, 37, 44, 53, 77).
|
*[[Wikipedia:School_and_university_projects/Discrete_and_numerical_mathematics/Learning_plan#Sets, relations and functions|WP+ (Sets, relations and functions)]].
|-
| rowspan="2" style="text-align: center;"|
Wed
<br />
19/2
|Relations, I

* Relations and their properties (mainly: reflexivity, irreflexivity, symmetry, asymmetry, antisymmetry, transitivity, intransitivity and connexity);
* Representing relations (mainly using: correspondences, sets, cartesian diagrams, binary matrices and directed graphs [digraphs]).
|
* § 7.1;
* § 7.3.
|
* § 9.1;
* § 9.3.
|
*[[Wikipedia:School_and_university_projects/Discrete_and_numerical_mathematics/Learning_plan#Sets, relations and functions|WP+ (Sets, relations and functions)]].
|
* § 7.1 (6, 8, 13, 20, 32, 34, 38);
* § 7.3 (10, 14, 26, 36).
|
* § 9.1 (6, 10, 15, 22, 34, 36, 40);
* § 9.3 (10, 14, 26, 36).
|
*[[Wikipedia:School_and_university_projects/Discrete_and_numerical_mathematics/Learning_plan#Sets, relations and functions|WP+ (Sets, relations and functions)]].
|-
| colspan="7" style="text-align: left; background: #FFF5FA;" |[[File:Twenty sided dice.svg|32px|link=]] [[Wikipedia:School and university projects/Discrete and numerical mathematics|University project Discrete and numerical mathematics (optional out-of-class activity)]]: ('''First checkpoint'''). Due date for having joined the English-language Wikipedia, if not yet, and for having chosen the articles of which you become responsible (follow the indications on the [[Wikipedia:School and university projects/Discrete and numerical mathematics|project page]] and on the [[Wikipedia:School_and_university_projects/Discrete_and_numerical_mathematics/Participants_and_major_contributions|contributions page]]).
|-
| style="text-align: center;"|
Thu
<br />
20/2
<br />
[[File:UN emblem blue.svg|link=|46x46px]]
<br />
[[World Day of Social Justice]]
|Relations, II

* Equivalence relations;
* Tolerance (or compatibility) relations;
* Partial, linear and strict preorders and orders; Hasse diagrams.
* Preference and indiference relations.
|
* § 7.5;
*§ —;
* § 7.6;
* § —.
|
* § 9.5;
*§ —;
* § 9.6;
* § —.
|
*[[Wikipedia:School_and_university_projects/Discrete_and_numerical_mathematics/Learning_plan#Sets, relations and functions|WP+ (Sets, relations and functions)]].
|
* § 7.5 (3, _, 7, 8, 10, 18, 26, 29, 31, 46, 48);
* § 7.6 (2, 3, 4, 5, 10, 13, 16, 28, 32, 36, 49, 51, 56, 59).
|
* § 9.5 (3, 8, 11, 12, 16, 24, 36, 41, 43, 60, 62);
* § 9.6 (8, 9, 10, 11, 16, 19, 22, 34, 38, 42, 55, 57, 62, 67).
|
*[[Wikipedia:School_and_university_projects/Discrete_and_numerical_mathematics/Learning_plan#Sets, relations and functions|WP+ (Sets, relations and functions)]].
|-
| style="text-align: center;"|
'''Group B'''
<br />
Fri
<br />
21/2
<br />
[[File:UN emblem blue.svg|link=|46x46px]]
<br />
[[International Mother Language Day]]
<hr>
'''Groups A and E'''
<br />
Mon
<br />
24/2
| style="text-align: left; background: #EEFFEE;" |'''Seminar/Laboratory No. 4''':<br />Induction and recursion

* (Occasionally computer-assisted) hands-on [[word problem|word-problem]]-solving on issues related to:
** mathematical induction;
** strong induction and well order;
** structural induction.
| style="text-align: left; background: #EEFFEE;" |
* (§ 1.5.7, 1.5.8, 1.5.9, 1.5.10; § 3.1);
* § 3.3;
*§ 3.3;
*§ 3.4.
| style="text-align: left; background: #EEFFEE;" |
* (§ 1.7; § 1.8);
* § 5.1;
*§ 5.2;
*§ 5.3.
| style="text-align: left; background: #EEFFEE;" |
* G. Polya. How to solve it. Princeton, New Jersey (US-NJ), USA: Princeton University Press;
*[[Wikipedia:School_and_university_projects/Discrete_and_numerical_mathematics/Learning_plan#Logic|WP+ (Logic)]].
| style="text-align: left; background: #EEFFEE;" |
* § 3.3 (8, 10, 6, 14, 11, 20, 25, 50, 38, 60);
*§ 3.3 (_, 34, _);
*§ 3.4 (__, __, __);
*[[iarchive:KDEM.DA.HD.70301|León-Rojas, J. M., KDEM DA/HD – 703-01]].
| style="text-align: left; background: #EEFFEE;" |
* ([[iarchive:KDEM.DA.HD.70301|León-Rojas, J. M., KDEM DA/HD – 703-01]]);
* § 5.1 (4, 6, 10, 18, 25, 32, 45, 50, 54, 62); § 5.2 (4, 10, 26).
| style="text-align: left; background: #EEFFEE;" |
* G. Polya. How to solve it. Princeton, New Jersey (US-NJ), USA: Princeton University Press;
*[[Wikipedia:School_and_university_projects/Discrete_and_numerical_mathematics/Learning_plan#Logic|WP+ (Logic)]];
* [https://drive.google.com/open?id=198r8Ens9n9GawthPyBJ6Ft5HyJu9XvBi León-Rojas, J. M., Question selection no. 4 and 4<sup>th</sup> seminar/laboratory. Induction and recursion.]
|-
| rowspan="2" style="text-align: center;" |
Tue
<br />
25/2
|Relations, III

* Solving questions on relations.
|
—
|
—
|
*[[Wikipedia:School_and_university_projects/Discrete_and_numerical_mathematics/Learning_plan#Sets, relations and functions|WP+ (Sets, relations and functions)]].
|
—
|
—
|
* [[Wikipedia:School_and_university_projects/Discrete_and_numerical_mathematics/Learning_plan#Sets, relations and functions|WP+ (Sets, relations and functions)]].
* [https://drive.google.com/open?id=198r8Ens9n9GawthPyBJ6Ft5HyJu9XvBi León-Rojas, J. M., Question selection no. 4 and 4<sup>th</sup> seminar/laboratory. Induction and recursion.]
|-
|Is there something greater than infinity? (Cardinality, I)

* Countable sets: <math>\mathbb{N}</math>, <math>\mathbb{Z}</math> and <math>\mathbb{Q}</math> are countable sets.
|
* § 3.2.5.
|
* § 2.5.1, 2.5.2.
|
* [[Wikipedia:School_and_university_projects/Discrete_and_numerical_mathematics/Learning_plan#Cardinality|WP+ (Cardinality)]].
|
* § 3.2.5 (31, 32, 34, 38);
|
* § 2.5 (1, 4, 16, 28).
|
* [[Wikipedia:School_and_university_projects/Discrete_and_numerical_mathematics/Learning_plan#Cardinality|WP+ (Cardinality)]].
|-
| style="text-align: center;" |
Wed
<br />
26/2
|Is there something greater than infinity? (Cardinality, II)

*<math>\mathbb{R}</math> is an uncountable set;
*Computability;
*Cantor's Theorem and the Continuum Hypothesis.
|
* § 3.2.5;
*§ 3.2.5: exercises 41, 42, 43;
*—.
|
* § 2.5.3;
*§ 2.5.3 and exercises 37, 38, 39;
*§ 2.5.3.
|
*[[Wikipedia:School_and_university_projects/Discrete_and_numerical_mathematics/Learning_plan#Cardinality|WP+ (Cardinality)]].
|
* § 3.2.5 (31, 32, 34, 38);
|
* § 2.5 (1, 4, 16, 28).
|
*[[Wikipedia:School_and_university_projects/Discrete_and_numerical_mathematics/Learning_plan#Cardinality|WP+ (Cardinality)]].
|-
| style="text-align: center;"|
Thu
<br />
27/2
|Algebraic structures, I

* Algebraic structures;
* Semigroups, monoids and groups;
|—
|—
|
*Rosen 7<sup>th</sup> Global Edition: § 12.1;
*Rosen 7<sup>th</sup> Global Edition: § 12.2;
*[[Wikipedia:School_and_university_projects/Discrete_and_numerical_mathematics/Learning_plan#Algebraic structures|WP+ (Algebraic structures)]].
|—
|—
|
*Rosen 7<sup>th</sup> Global Edition: § 12.1 (1, 2);
*Rosen 7<sup>th</sup> Global Edition: § 12.2 (2, 4, 5, 8, 12, 17, 18, 19, 20, 26, 31, 36, 40);
*[[Wikipedia:School_and_university_projects/Discrete_and_numerical_mathematics/Learning_plan#Algebraic structures|WP+ (Algebraic structures)]].
|-
| style="text-align: center;"|
'''Group B'''
<br />
Fri
<br />
28/2
<hr>
'''Groups A and E'''
<br />
Mon
<br />
2/3
| style="text-align: left; background: #EEFFEE;" |'''Seminar/Laboratory No. 5''':<br />Cardinality and Algebraic Structures

* (Occasionally computer-assisted) hands-on [[word problem|word-problem]]-solving on issues related to:
** cardinality;
** algebraic structures.
| style="text-align: left; background: #EEFFEE;" |—
| style="text-align: left; background: #EEFFEE;" |—
| style="text-align: left; background: #EEFFEE;" |
*Rosen 7<sup>th</sup> Global Edition: § 12.1;
*Rosen 7<sup>th</sup> Global Edition: § 12.2;
*Rosen 7<sup>th</sup> Global Edition: § 12.3;
*Rosen 7<sup>th</sup> Global Edition: § 12.4;
*[[Wikipedia:School_and_university_projects/Discrete_and_numerical_mathematics/Learning_plan#Algebraic structures|WP+ (Algebraic structures)]].
| style="text-align: left; background: #EEFFEE;" |—
| style="text-align: left; background: #EEFFEE;" |—
| style="text-align: left; background: #EEFFEE;" |
*Rosen 7<sup>th</sup> Global Edition: § 12.1 (1, 2);
*Rosen 7<sup>th</sup> Global Edition: § 12.2 (2, 4, 5, 8, 12, 17, 18, 19, 20, 26, 31, 36, 40);
*Rosen 7<sup>th</sup> Global Edition: § 12.3 (4, 5, 7, 8);
*Rosen 7<sup>th</sup> Global Edition: § 12.4 (2, 3, 4, 5);
*[[Wikipedia:School_and_university_projects/Discrete_and_numerical_mathematics/Learning_plan#Algebraic structures|WP+ (Algebraic structures)]];
* [https://drive.google.com/open?id=1DWxssdLHX55eyzjkR3WCCKFBX_QeiwG9 León-Rojas, J. M., Question selection no. 5 and 5<sup>th</sup> seminar/laboratory. Cardinality and Algebraic structures.]
|-
| style="text-align: center;"|
Sun
<br />
1/3
<br />
[[File:UN emblem blue.svg|link=|46x46px]]
<br />
[[Zero Discrimination Day]]
| colspan=7 style="background: none;" |
<!--
|
|
|
|
|
|
|
-->
|-
| style="text-align: center;"|
Tue
<br />
3/3
<br />
[[File:UN emblem blue.svg|link=|46x46px]]
<br />
[[World Wildlife Day]]
|Algebraic structures, II

* Homomorphisms;
* Rings, integral domains and fields.
|—
|—
|
*Rosen 7<sup>th</sup> Global Edition: § 12.3;
*Rosen 7<sup>th</sup> Global Edition: § 12.4;
*[[Wikipedia:School_and_university_projects/Discrete_and_numerical_mathematics/Learning_plan#Algebraic structures|WP+ (Algebraic structures)]].
|—
|—
|
*Rosen 7<sup>th</sup> Global Edition: § 12.3 (4, 5, 7, 8);
*Rosen 7<sup>th</sup> Global Edition: § 12.4 (2, 3, 4, 5);
*[[Wikipedia:School_and_university_projects/Discrete_and_numerical_mathematics/Learning_plan#Algebraic structures|WP+ (Algebraic structures)]].
|-
| style="text-align: center;"|
Wed
<br />
4/3
|Algebraic structures, III

* Solving some questions on algebraic structures.
|—
|—
|
*Rosen 7<sup>th</sup> Global Edition: § 12.1;
*Rosen 7<sup>th</sup> Global Edition: § 12.2;
*Rosen 7<sup>th</sup> Global Edition: § 12.3;
*Rosen 7<sup>th</sup> Global Edition: § 12.4;
*[[Wikipedia:School_and_university_projects/Discrete_and_numerical_mathematics/Learning_plan#Algebraic structures|WP+ (Algebraic structures)]].
|—
|—
|
*Rosen 7<sup>th</sup> Global Edition: § 12.1 (1, 2);
*Rosen 7<sup>th</sup> Global Edition: § 12.2 (2, 4, 5, 8, 12, 17, 18, 19, 20, 26, 31, 36, 40);
*Rosen 7<sup>th</sup> Global Edition: § 12.3 (4, 5, 7, 8);
*Rosen 7<sup>th</sup> Global Edition: § 12.4 (2, 3, 4, 5);
*[[Wikipedia:School_and_university_projects/Discrete_and_numerical_mathematics/Learning_plan#Algebraic structures|WP+ (Algebraic structures)]].
|-
| colspan="8" style="box-sizing: border-box; line-height: 1.1; border-bottom-width: 3px; border-bottom-style: solid; border-bottom-color: rgb(0, 173, 239); border-top-width: 3px; border-top-style: solid; border-top-color: rgb(0, 173, 239); margin-top: 26px; margin-bottom: 9px; padding-left: 1px; padding-bottom: 7px; text-align: center; background: #E6ECFF;" |<p style="text-transform: uppercase; font-size: 120%;">'''Theme 2'''</p>

<p style="text-transform: uppercase;">'''Number theory'''</p>

<p>(9 h LG and 3 h S/L)</p>

<p>(1h LG Solving the mid-course preparatory exam)</p>
|-
| style="text-align: center;"|
Thu
<br />
5/3
|Divisibility and modular arithmetic

* Divisibility;
* The division algorithm;
* Modular arithmetic.
|
* § 2.4.1, 2.4.2;
* § 2.4.4;
* § 2.4.6.
|
* § 4.1.1, 4.1.2;
* § 4.1.3;
* § 4.1.4, 4.1.5.
|
*[http://www2.uca.es/matematicas/Docencia/2005-2006/ESI/1710003/Apuntes/Leccion10.pdf González Gutiérrez, 2004a, § 10.1, 10.4];
*[http://www2.uca.es/matematicas/Docencia/2005-2006/ESI/1710003/Apuntes/Leccion13.pdf González Gutiérrez, 2004d, § 13.1];
*[[Wikipedia:School_and_university_projects/Discrete_and_numerical_mathematics/Learning_plan#Number theory|WP+ (Number theory)]].
|
* § 2.4 (5, 6, 7);
* § 2.4 (10, 22, 34, 36);
* § 2.4 (38, 42, 44).
|
* § 4.1 (5, 6, 7);
* § 4.1 (10, 16, 18, 20);
* § 4.1 (26, 34, 36).
|
*[http://www2.uca.es/matematicas/Docencia/2005-2006/ESI/1710003/Apuntes/Leccion10.pdf González Gutiérrez, 2004a, § 10.1, 10.4];
*[http://www2.uca.es/matematicas/Docencia/2005-2006/ESI/1710003/Apuntes/Leccion13.pdf González Gutiérrez, 2004d, § 13.1];
*[[Wikipedia:School_and_university_projects/Discrete_and_numerical_mathematics/Learning_plan#Number theory|WP+ (Number theory)]].
|-
| style="text-align: center;"|
'''Group B'''
<br />
Fri
<br />
6/3
<hr>
'''Groups A and E'''
<br />
Mon
<br />
9/3
| style="text-align: left; background: #EEFFEE;" |'''Seminar/Laboratory No. 6''':<br />Divisibility, modular arithmetic, primes, GCD and congruences

* (Occasionally computer-assisted) hands-on [[word problem|word-problem]]-solving on issues related to:
** divisibility;
** modular arithmetic;
** primes;
** GCD;
** congruences.
| style="text-align: left; background: #EEFFEE;" |
* (Everything we have studied on the subject).
| style="text-align: left; background: #EEFFEE;" |
* (Everything we have studied on the subject).
| style="text-align: left; background: #EEFFEE;" |
* (Everything we have studied on the subject);
*[[Wikipedia:School_and_university_projects/Discrete_and_numerical_mathematics/Learning_plan#Number theory|WP+ (Number theory)]].
| style="text-align: left; background: #EEFFEE;" |
* (Everything we have studied on the subject).
| style="text-align: left; background: #EEFFEE;" |
* (Everything we have studied on the subject).
| style="text-align: left; background: #EEFFEE;" |
* (Everything we have studied on the subject);
*[[Wikipedia:School_and_university_projects/Discrete_and_numerical_mathematics/Learning_plan#Number theory|WP+ (Number theory)]];
* [https://drive.google.com/open?id=1I-zIrN5x7-6NDrlFJhWFmjuITaG96O86 León-Rojas, J. M., Question selection no. 6 and 6<sup>th</sup> seminar/laboratory. Divisibility, modular arithmetic, primes, GCD and congruences.]
|-
| style="text-align: center;"|
Sun
<br />
8/3
<br />
[[File:UN emblem blue.svg|link=|46x46px]]
<br />
[[International Women's Day]]
| colspan=7 style="background: none;" |
<!--
|
|
|
|
|
|
|
-->
|-
| style="text-align: center;"|
Tue
<br />
10/3
|Primes

* Prime numbers;
* The fundamental theorem of arithmetic.
|
* § 2.4.3.
|
* § 4.3.1, 4.3.2, 4.3.3, 4.3.4, 4.3.5;
* § 4.3.2.
|
*[http://www2.uca.es/matematicas/Docencia/2005-2006/ESI/1710003/Apuntes/Leccion11.pdf González Gutiérrez, 2004b];
*[[Wikipedia:School_and_university_projects/Discrete_and_numerical_mathematics/Learning_plan#Number theory|WP+ (Number theory)]].
|
* § 2.4 (8, 12, 14, 15, 20, 24, 25, 26, 27).
|
* § 4.3 (2, 4, 6, 11, 18, 20, 21, 22, 23).
|
*[http://www2.uca.es/matematicas/Docencia/2005-2006/ESI/1710003/Apuntes/Leccion11.pdf González Gutiérrez, 2004b];
*[[Wikipedia:School_and_university_projects/Discrete_and_numerical_mathematics/Learning_plan#Number theory|WP+ (Number theory)]].
|-
| style="text-align: center;"|
Wed
<br />
11/3
|Greatest common divisor (GCD)

* GCD and LCM;
* The Euclidean algorithm;
*Bézout's theorem and the extended Euclidean algorithm.
|
* § 2.4.5;
* § 2.5.5;
* § 2.6.2 and p. 180.
|
* § 4.3.6;
* § 4.3.7;
* § 4.3.8 and p. 273.
|
*[http://www2.uca.es/matematicas/Docencia/2005-2006/ESI/1710003/Apuntes/Leccion10.pdf González Gutiérrez, 2004a, § 10.6, 10.7, 10.8];
*[[Wikipedia:School_and_university_projects/Discrete_and_numerical_mathematics/Learning_plan#Number theory|WP+ (Number theory)]].
|
* § 2.4 (17, 28);
* § 2.5 (21, 22);
* § 2.5 (2, 50).
|
* § 4.3 (15, 24);
* § 4.3 (33, 32);
* § 4.3 (40, 44).
|
*[http://www2.uca.es/matematicas/Docencia/2005-2006/ESI/1710003/Apuntes/Leccion10.pdf González Gutiérrez, 2004a, § 10.6, 10.7, 10.8];
*[[Wikipedia:School_and_university_projects/Discrete_and_numerical_mathematics/Learning_plan#Number theory|WP+ (Number theory)]].
|-
| style="text-align: center;"|
Thu
<br />
12/3
|Solving congruences, I

* Linear congruences;
* The Chinese remainder theorem;
* Computer arithmetic with large integers.
|
* § 2.6.3;
* § 2.6.4;
* § 2.6.5.
|
* § 4.4.2;
* § 4.4.3;
* § 4.4.4.
|
*[[Wikipedia:School_and_university_projects/Discrete_and_numerical_mathematics/Learning_plan#Number theory|WP+ (Number theory)]].
|
* § 2.6 (4, 5, 6, 7, 8, ...);
* § 2.6 (... ... ...);
* § 2.6 (... ... ...).
|
* § 4.4 (2, 5a, 6a, 5b, 6c, ...);
* § 4.4 (... ... ...);
* § 4.4 (... ... ...).
|
*[[Wikipedia:School_and_university_projects/Discrete_and_numerical_mathematics/Learning_plan#Number theory|WP+ (Number theory)]].
|-
| style="text-align: center;"|
'''Group B'''
<br />
Fri
<br />
13/3
<hr>
'''Groups A and E'''
<br />
Mon
<br />
16/3
| style="text-align: left; background: #EEFFEE;" |'''Seminar/Laboratory No. 7''':<br />Diophantine and congruence equations, I

* (Occasionally computer-assisted) hands-on [[word problem|word-problem]]-solving on issues related to:
** diophantine equations;
** congruence equations.
| style="text-align: left; background: #EEFFEE;" |
* —;
* § 2.6.8, 2.6.9, 2.6.10.
| style="text-align: left; background: #EEFFEE;" |
* —;
* § 4.6.4, 4.6.5, 4.6.6, 4.6.7, 4.6.8.
| style="text-align: left; background: #EEFFEE;" |
*[http://www2.uca.es/matematicas/Docencia/2005-2006/ESI/1710003/Apuntes/Leccion12.pdf González Gutiérrez, 2004c];
*[[Wikipedia:School_and_university_projects/Discrete_and_numerical_mathematics/Learning_plan#Number theory|WP+ (Number theory)]].
| style="text-align: left; background: #EEFFEE;" |
* —;
* § 2.6 (46, 47, 45*).
| style="text-align: left; background: #EEFFEE;" |
* —;
* § 4.6 (24, 27, 23*).
| style="text-align: left; background: #EEFFEE;" |
*[http://www2.uca.es/matematicas/Docencia/2005-2006/ESI/1710003/Apuntes/Leccion12.pdf González Gutiérrez, 2004c];
*[[Wikipedia:School_and_university_projects/Discrete_and_numerical_mathematics/Learning_plan#Number theory|WP+ (Number theory)]];
* [https://drive.google.com/open?id=1DpVGomTJ0yY7Ml8wkC7Z2PGLA2KZL9eI León-Rojas, J. M., Question selection no. 7 and 7<sup>th</sup> seminar/laboratory. Diophantine and congruence equations, I.]
|-
| style="text-align: center;"|
Tue
<br />
17/3
|Solving congruences, II

*Fermat's little theorem and pseudoprimes; Euler's theorem and Wilson's theorem.
| 
* § 2.6.6; p. 179.
|
* § 4.4.5 and § 4.4.6; p. 285.
|
*[[Wikipedia:School_and_university_projects/Discrete_and_numerical_mathematics/Learning_plan#Number theory|WP+ (Number theory)]].
|
* § 2.6 (17, 28, 32, 34, 43, 44, 52, 56).
|
* § 4.4 (19, 38, 46, 48,41, 42, 58, 62).
|
*[[Wikipedia:School_and_university_projects/Discrete_and_numerical_mathematics/Learning_plan#Number theory|WP+ (Number theory)]].<!--
* [https://www.youtube.com/watch?v=Z8m0vfnkYQw Soto, Jesús. Ecuaciones diofánticas. UCAM]
* [https://www.youtube.com/watch?v=RGovvYZV7Eo Hervás, Antonio. Ecuaciones diofánticas (1). UPV] -->
|-
| style="text-align: center;"|
Wed
<br />
18/3
|Divisibility rules

* Power residues and divisibility rules.
| —
| —
|
*[http://www2.uca.es/matematicas/Docencia/2005-2006/ESI/1710003/Apuntes/Leccion10.pdf González Gutiérrez, 2004a, § 10.5];
*[http://ocw.upm.es/matematica-aplicada/matematica-discreta/contenidos/material-de-clase/tema-3 Sendra, Martín, Méndez and Ortiz, 2011];
*[[Wikipedia:School_and_university_projects/Discrete_and_numerical_mathematics/Learning_plan#Number theory|WP+ (Number theory)]].
| —
| —
|
*[http://www2.uca.es/matematicas/Docencia/2005-2006/ESI/1710003/Apuntes/Leccion10.pdf González Gutiérrez, 2004a, § 10.5];
*[http://ocw.upm.es/matematica-aplicada/matematica-discreta/contenidos/material-de-clase/tema-3 Sendra, Martín, Méndez and Ortiz, 2011];
*[[Wikipedia:School_and_university_projects/Discrete_and_numerical_mathematics/Learning_plan#Number theory|WP+ (Number theory)]].
|-
| style="text-align: center;"|
Thu
<br />
19/3
|Diophantine equations

* Diophantine equations.
| —
| —
|
*[http://www2.uca.es/matematicas/Docencia/2005-2006/ESI/1710003/Apuntes/Leccion12.pdf González Gutiérrez, 2004c];
*[[Wikipedia:School_and_university_projects/Discrete_and_numerical_mathematics/Learning_plan#Number theory|WP+ (Number theory)]].
| —
| —
|
*[http://www2.uca.es/matematicas/Docencia/2005-2006/ESI/1710003/Apuntes/Leccion12.pdf González Gutiérrez, 2004c];
*[[Wikipedia:School_and_university_projects/Discrete_and_numerical_mathematics/Learning_plan#Number theory|WP+ (Number theory)]].
|-
| style="text-align: center;"|
'''Group B'''
<br />
Fri
<br />
20/3
<br />
[[File:UN emblem blue.svg|link=|46x46px]]
<br />
[[International Francophonie Day]]
<br />
[[International Day of Happiness]]
<hr>
'''Groups A and E'''
<br />
Mon
<br />
23/3
<br />
[[File:UN emblem blue.svg|link=|46x46px]]
<br />
[[World Meteorological Day]]
| style="text-align: left; background: #EEFFEE;" |'''Seminar/Laboratory No. 8''':<br />Diophantine and congruence equations, II

* (Occasionally computer-assisted) hands-on [[word problem|word-problem]]-solving on issues related to:
** diophantine equations;
** congruence equations;
** applications of congruences.
| style="text-align: left; background: #EEFFEE;" |
* (Everything we have studied on the subject).
| style="text-align: left; background: #EEFFEE;" |
* (Everything we have studied on the subject).
| style="text-align: left; background: #EEFFEE;" |
* (Everything we have studied on the subject).
| style="text-align: left; background: #EEFFEE;" |
* (Everything we have studied on the subject).
| style="text-align: left; background: #EEFFEE;" |
* (Everything we have studied on the subject).
| style="text-align: left; background: #EEFFEE;" |
* (Everything we have studied on the subject);
* [https://drive.google.com/open?id=1spOIr9ekEGcwxrmwr7Si2ANlV0CKDRdz León-Rojas, J. M., Question selection no. 8 and 8<sup>th</sup> seminar/laboratory. Diophantine and congruence equations, II.]
|-
|style="text-align: center;"|
Sat
<br />
21/3
<br />
[[File:UN emblem blue.svg|link=|46x46px]]
<br />
[[World Poetry Day]]
<br />
[[International Day for the Elimination of Racial Discrimination]]
<br />
[[International Nowruz Day]] <sup>([[:es:Día Internacional del Novruz|es]])</sup>
<br />
[[World Down Syndrome Day]]
<br />
[[International Day of Forests]]
| colspan=7 style="background: none;" |
<!--
|
|
|
|
|
|
|
-->
|-
|style="text-align: center;"|
Sun
<br />
22/3
<br />
[[File:UN emblem blue.svg|link=|46x46px]]
<br />
[[World Water Day]]
| colspan=7 style="background: none;" |
<!--
|
|
|
|
|
|
|
-->
|-
| style="text-align: center;"|
Tue
<br />
24/3
<br />
[[File:UN emblem blue.svg|link=|46x46px]]
<br />
[[International Day for the Right to the Truth Concerning Gross Human Rights Violations and for the Dignity of Victims]] <sup>([[:es:Día Internacional del Derecho a la Verdad en relación con Violaciones Graves de los Derechos Humanos y de la Dignidad de las Víctimas|es]])</sup>
<br />
[[World Tuberculosis Day]]
|Applications of congruences, I

* Hashing functions (optional);
* Pseudorandom numbers (optional);
* Cryptography, I.
|
* § 2.4.7;
*§ 2.4.7;
*§ 2.6.7, 2.6.8, 2.6.9, 2.6.10.
|
* § 4.5.1;
* § 4.5.2;
*§ 4.6.4, 4.6.5, 4.6.6, 4.6.7.
|
*[[Wikipedia:School_and_university_projects/Discrete_and_numerical_mathematics/Learning_plan#Number theory|WP+ (Number theory)]].
|
* § 2.4 (48, 49);
* § 2.4 (50, 51, 52);
*§ 2.6 (45, 46, 47).
|
* § 4.5 (2, 3);
* § 4.5 (6, 7, 8);
*§ 4.6 (23, 24, 27).
|
*[[Wikipedia:School_and_university_projects/Discrete_and_numerical_mathematics/Learning_plan#Number theory|WP+ (Number theory)]].
|-
| style="text-align: center;"|
Wed
<br />
25/3
<br />
[[Annunciation]]
<br />
[[File:UN emblem blue.svg|link=|46x46px]]
<br />
[[International Day of Solidarity with Detained and Missing Staff Members]]
<br />
[[International Day of Remembrance of the Victims of Slavery and the Transatlantic Slave Trade]]
|Applications of congruences, II

* Cryptography, II. RSA.
|
* § 2.6.7, 2.6.8, 2.6.9, 2.6.10.
|
*§ 4.6.4, 4.6.5, 4.6.6, 4.6.7.
|
*[[Wikipedia:School_and_university_projects/Discrete_and_numerical_mathematics/Learning_plan#Number theory|WP+ (Number theory)]].
|
*§ 2.6 (45, 46, 47).
|
* § 4.6 (23, 24, 27).
|
*[[Wikipedia:School_and_university_projects/Discrete_and_numerical_mathematics/Learning_plan#Number theory|WP+ (Number theory)]].
|-
| style="text-align: center;"|
Thu
<br />
26/3
| colspan="7" style="text-align: left; background: #ffddff;" |[[File:Group full edit.svg|32px|link=]] '''Exam review''': large-group class dedicated to share ideas and solutions in a whole-class discussion about the '''mid-course preparatory exam''' (done as homework).
|-
| colspan="8" style="box-sizing: border-box; line-height: 1.1; border-bottom-width: 3px; border-bottom-style: solid; border-bottom-color: rgb(0, 173, 239); border-top-width: 3px; border-top-style: solid; border-top-color: rgb(0, 173, 239); margin-top: 26px; margin-bottom: 9px; padding-left: 1px; padding-bottom: 7px; text-align: center; background: #E6ECFF;" |<p style="text-transform: uppercase; font-size: 120%;">'''Theme 3'''</p>

<p style="text-transform: uppercase;">'''Combinatorics'''</p>

<p>(8 h LG and 3 h S/L)</p>
|-
| style="text-align: center;"|
'''Group B'''
<br />
Fri
<br />
27/3
<hr>
'''Groups A and E'''
<br />
Mon
<br />
30/3
| style="text-align: left; background: #EEFFEE;" |'''Seminar/Laboratory No. 9''':<br />Combinatorics, I

* (Occasionally computer-assisted) hands-on [[word problem|word-problem]]-solving on issues related to:
** combinatorics.
| style="text-align: left; background: #EEFFEE;" |
* (Everything we have studied on the subject).
| style="text-align: left; background: #EEFFEE;" |
* (Everything we have studied on the subject).
| style="text-align: left; background: #EEFFEE;" |
* (Everything we have studied on the subject);
*[[#FrancoBranas2008|Franco, Espinel and Almeida, 2008]];
*[[#Sedgewick1977|Sedgewick, 1977]];
*[[Wikipedia:School_and_university_projects/Discrete_and_numerical_mathematics/Learning_plan#Combinatorics|WP+ (Combinatorics)]].
| style="text-align: left; background: #EEFFEE;" |
* (Everything we have studied on the subject).
| style="text-align: left; background: #EEFFEE;" |
* (Everything we have studied on the subject).
| style="text-align: left; background: #EEFFEE;" |
* (Everything we have studied on the subject);
*[[#FrancoBranas2008|Franco, Espinel and Almeida, 2008]];
*[[#Sedgewick1977|Sedgewick, 1977]];
*[[Wikipedia:School_and_university_projects/Discrete_and_numerical_mathematics/Learning_plan#Combinatorics|WP+ (Combinatorics)]];
* [https://drive.google.com/open?id=1Jqvg6319nWAsfi7ae5QidEoLZ33jaLcR León-Rojas, J. M., Question selection no. 9 and 9<sup>th</sup> seminar/laboratory. Combinatorics, I.]
|-
| style="text-align: center;"|
Tue
<br />
31/3
|Combinatorics

* The basics of counting.
|
* § 4.1.
|
* § 6.1.
|
*[[#FrancoBranas2008|Franco, Espinel and Almeida, 2008, § 1, § 3.1, § 3.1.1]];
*[[Wikipedia:School_and_university_projects/Discrete_and_numerical_mathematics/Learning_plan#Combinatorics|WP+ (Combinatorics)]].
|
* § 4.1 (6, 16, 22, 28, 33, 37, 41, 51, 52).
|
* § 6.1 (8, 16, 26, 32, 37, 41, 49, 67, 68).
|
*[[#FrancoBranas2008|Franco, Espinel and Almeida, 2008, § 1, § 3.1, § 3.1.1]];
*[[Wikipedia:School_and_university_projects/Discrete_and_numerical_mathematics/Learning_plan#Combinatorics|WP+ (Combinatorics)]].
|-
| style="text-align: center;"|
Wed
<br />
1/4
|Combinatorics

* The pigeonhole principle.
|
* § 4.2.
|
* § 6.2.
|
*[[#FrancoBranas2008|Franco, Espinel and Almeida, 2008, § 1, § 3.1, § 3.1.1]];
*[[Wikipedia:School_and_university_projects/Discrete_and_numerical_mathematics/Learning_plan#Combinatorics|WP+ (Combinatorics)]].
|
* § 4.2 (9, 10, 16, 17, 18, 20, 24, 25, 26, 36).
|
* § 6.2 (9, 10, 16, 17, 18, 20, 26, 27, 28, 40).
|
*[[#FrancoBranas2008|Franco, Espinel and Almeida, 2008, § 1, § 3.1, § 3.1.1]];
*[[Wikipedia:School_and_university_projects/Discrete_and_numerical_mathematics/Learning_plan#Combinatorics|WP+ (Combinatorics)]].
|-
| rowspan="2" style="text-align: center;"|
Thu
<br />
2/4
<br />
[[File:UN emblem blue.svg|link=|46x46px]]
<br />
[[World Autism Awareness Day]]
|Combinatorics

* Permutations and combinations;
* Binomial coefficients and identities.
|
* § 4.3;
* § 4.4
|
* § 6.3;
* § 6.4.
|
*[[#FrancoBranas2008|Franco, Espinel and Almeida, 2008, § 4, § 5]];
*[[Wikipedia:School_and_university_projects/Discrete_and_numerical_mathematics/Learning_plan#Combinatorics|WP+ (Combinatorics)]].
|
* § 4.3 (5, 12, 18, 22, 23, 35, 36, 37);
* § 4.4 (4, 8, 20, 22, 24, 33, 34).
|
* § 6.3 (5, 12, 18, 22, 23, 35, 36, 37);
* § 6.4 (4, 8, 20, 22, 24, 33, 34).
|
*[[#FrancoBranas2008|Franco, Espinel and Almeida, 2008, § 4, § 5]];
*[[Wikipedia:School_and_university_projects/Discrete_and_numerical_mathematics/Learning_plan#Combinatorics|WP+ (Combinatorics)]].
|-
| colspan="7" style="text-align: left; background: #FFF5FA;" |[[File:Twenty sided dice.svg|32px|link=]] [[Wikipedia:School and university projects/Discrete and numerical mathematics|University project Discrete and numerical mathematics (optional out-of-class activity)]]: ('''Second checkpoint'''). You should have continually been working in your contributions, publishing each update, along with the corresponding themes to which they belong are worked in class, and linking each new major contribution on the [[Wikipedia:School_and_university_projects/Discrete_and_numerical_mathematics/Participants_and_major_contributions|contributions page]] of the project. Furthermore, you must publish, also on an ongoing basis, in your logbook (sandbox), the part of your [[Wikipedia:School and university projects/Discrete and numerical mathematics#Self-report|self-report]] that deals with what you have developed so far.
|-
| style="text-align: center;"|
'''Group B'''
<br />
Fri
<br />
3/4
<br />
[[Lent]]
<br />
[[Friday of Sorrows]]
<hr>
'''Groups A and E'''
<br />
Mon
<br />
20/4
| style="text-align: left; background: #EEFFEE;" |'''Seminar/Laboratory No. 10''':<br />Combinatorics, II

* (Occasionally computer-assisted) hands-on [[word problem|word-problem]]-solving on issues related to:
** combinatorics.
| style="text-align: left; background: #EEFFEE;" |
* (Everything we have studied on the subject).
| style="text-align: left; background: #EEFFEE;" |
* (Everything we have studied on the subject).
| style="text-align: left; background: #EEFFEE;" |
* (Everything we have studied on the subject);
*[[#FrancoBranas2008|Franco, Espinel and Almeida, 2008]];
*[[Wikipedia:School_and_university_projects/Discrete_and_numerical_mathematics/Learning_plan#Combinatorics|WP+ (Combinatorics)]].
| style="text-align: left; background: #EEFFEE;" |
* (Everything we have studied on the subject).
| style="text-align: left; background: #EEFFEE;" |
* (Everything we have studied on the subject).
| style="text-align: left; background: #EEFFEE;" |
* (Everything we have studied on the subject);
*[[#FrancoBranas2008|Franco, Espinel and Almeida, 2008]];
*[[Wikipedia:School_and_university_projects/Discrete_and_numerical_mathematics/Learning_plan#Combinatorics|WP+ (Combinatorics)]];
* [https://drive.google.com/open?id=1QXJ3A_66bnq6aUhrME1Hu3lIh8CdFyH2 León-Rojas, J. M., Question selection no. 10 and 10<sup>th</sup> seminar/laboratory. Combinatorics, II.]
|-
| style="text-align: center;"|
Sat
<br />
4/4
<br />
[[Lazarus Saturday]]
<br />
[[File:UN emblem blue.svg|link=|46x46px]]
<br />
[[International Day for Mine Awareness and Assistance in Mine Action]] <sup>([[:es:Día Internacional de Información sobre el Peligro de las Minas y de Asistencia para las Actividades Relativas a las Minas|es]])</sup>
| colspan=7 style="background: none;" |
<!--
|
|
|
|
|
|
|
-->
|-
| style="text-align: center;"|
Mon
<br />
6/4
<br />
[[Holy Week]]
<br />
[[Holy Monday]]
<br />
[[File:UN emblem blue.svg|link=|46x46px]]
<br />
[[International Day of Sport for Development and Peace]]
| colspan=7 style="background: none;" |
<!--
|
|
|
|
|
|
|
-->
|-
| style="text-align: center;"|
Tue
<br />
7/4
<br />
[[Holy Week]]
<br />
[[Holy Tuesday]]
<br />
[[File:UN emblem blue.svg|link=|46x46px]]
<br />
[[World Health Day]]
| colspan=7 style="background: none;" |
<!--
|
|
|
|
|
|
|
-->
|-
| style="text-align: center;"|
Wed
<br />
8/4
<br />
[[Holy Week]]
<br />
[[Holy Wednesday]]
| colspan=7 style="background: none;" |
<!--
|
|
|
|
|
|
|
-->
|-
| style="text-align: center;"|
Thu
<br />
9/4
<br />
[[Holy Week]]
<br />
[[Maundy Thursday]]
| colspan=7 style="background: none;" |
<!--
|
|
|
|
|
|
|
-->
|-
| style="text-align: center;"|
Fri
<br />
10/4
<br />
[[Holy Week]]
<br />
[[Good Friday]]
| colspan=7 style="background: none;" |
<!--
|
|
|
|
|
|
|
-->
|-
| style="text-align: center;"|
Sun
<br />
12/4
<br />
[[Holy Week]]
<br />
[[Resurrection Sunday]]
<br />
[[File:UN emblem blue.svg|link=|46x46px]]
<br />
[[International Day of Human Space Flight]]
| colspan=7 style="background: none;" |
<!--
|
|
|
|
|
|
|
-->
|-
| style="text-align: center;"|
Mon
<br />
13/4
<br />
[[Eastertide]]
<br />
[[Easter Monday]]
| colspan=7 style="background: none;" |
<!--
|
|
|
|
|
|
|
-->
|-
| style="text-align: center;"|
Tue
<br />
14/4
|Combinatorics

* Generalised permutations and combinations (variations, combinations and permutations, with repetition).
|
* § 4.5.1, 4.5.2, 4.5.3, 4.5.4.
|
* § 6.5.1, 6.5.2, 6.5.3, 6.5.4.
|
*[[#FrancoBranas2008|Franco, Espinel and Almeida, 2008, § 4.1.3, § 4.3, § 4.4]];
*[[Wikipedia:School_and_university_projects/Discrete_and_numerical_mathematics/Learning_plan#Combinatorics|WP+ (Combinatorics)]].
|
* § 4.5 (10, 15, 16, 34, 56).
|
* § 6.5 (10, 15, 16, 34, 66).
|
*[[#FrancoBranas2008|Franco, Espinel and Almeida, 2008, § 4.1.3, § 4.3, § 4.4]];
*[[Wikipedia:School_and_university_projects/Discrete_and_numerical_mathematics/Learning_plan#Combinatorics|WP+ (Combinatorics)]].
|-
| style="text-align: center;"|
Wed
<br />
15/4
|Combinatorics

* Distributing objects to boxes when the order of objects in each box does not matter and both the objects and the boxes may be distinguishable or not.
|
* § 4.5.5 (~).
|
* § 6.5.5.
|
*[[#FrancoBranas2008|Franco, Espinel and Almeida, 2008, § 7.1]];
*[[Wikipedia:School_and_university_projects/Discrete_and_numerical_mathematics/Learning_plan#Combinatorics|WP+ (Combinatorics)]].
|
* § 4.5.5 (22, 47, 50).
|
* § 6.5.5 (22, 47, 50).
|
*[[#FrancoBranas2008|Franco, Espinel and Almeida, 2008, § 7.1]];
*[[Wikipedia:School_and_university_projects/Discrete_and_numerical_mathematics/Learning_plan#Combinatorics|WP+ (Combinatorics)]].
|-
| style="text-align: center;"|
Thu
<br />
16/4
|Combinatorics

* Distributing objects to boxes when the order of objects in each box matters and both the objects and the boxes may be distinguishable or not.
|
* § 4.5.5 (~).
|
* § 6.5.5.
|
*[[#FrancoBranas2008|Franco, Espinel and Almeida, 2008, § 7.1]];
*[[Wikipedia:School_and_university_projects/Discrete_and_numerical_mathematics/Learning_plan#Combinatorics|WP+ (Combinatorics)]].
|
* § 4.5.5 (22, 47, 50).
|
* § 6.5.5 (22, 47, 50).
|
*[[#FrancoBranas2008|Franco, Espinel and Almeida, 2008, § 7.1]];
*[[Wikipedia:School_and_university_projects/Discrete_and_numerical_mathematics/Learning_plan#Combinatorics|WP+ (Combinatorics)]].
|-
| style="text-align: center;"|
'''Group B'''
<br />
Fri
<br />
17/4
<hr>
'''Groups A and E'''
<br />
Mon
<br />
4/5
| style="text-align: left; background: #EEFFEE;" |'''Seminar/Laboratory No. 11''':<br />Combinatorics, III

* (Occasionally computer-assisted) hands-on [[word problem|word-problem]]-solving on issues related to:
** combinatorics.
| style="text-align: left; background: #EEFFEE;" |
* (Everything we have studied on the subject).
| style="text-align: left; background: #EEFFEE;" |
* (Everything we have studied on the subject).
| style="text-align: left; background: #EEFFEE;" |
* (Everything we have studied on the subject);
*[[#FrancoBranas2008|Franco, Espinel and Almeida, 2008]];
*[[Wikipedia:School_and_university_projects/Discrete_and_numerical_mathematics/Learning_plan#Combinatorics|WP+ (Combinatorics)]].
| style="text-align: left; background: #EEFFEE;" |
* (Everything we have studied on the subject).
| style="text-align: left; background: #EEFFEE;" |
* (Everything we have studied on the subject).
| style="text-align: left; background: #EEFFEE;" |
* (Everything we have studied on the subject);
*[[#FrancoBranas2008|Franco, Espinel and Almeida, 2008]];
*[[Wikipedia:School_and_university_projects/Discrete_and_numerical_mathematics/Learning_plan#Combinatorics|WP+ (Combinatorics)]];
* [https://drive.google.com/open?id=1rn5-JtQKLxHjdCBnykReqMjFRUkc-V2s León-Rojas, J. M., Question selection no. 11 and 11<sup>st</sup> seminar/laboratory. Combinatorics, III.]
|-
| style="text-align: center;"|
Sun
<br />
19/4
<br />
[[File:UN emblem blue.svg|link=|46x46px]]
<br />
[[UN Chinese Language Day]]
| colspan=7 style="background: none;" |
<!--
|
|
|
|
|
|
|
-->
|-
| style="text-align: center;"|
Tue
<br />
21/4
<br />
[[File:UN emblem blue.svg|link=|46x46px]]
<br />
[[World Creativity and Innovation Day]]
|Combinatorics

* Partitions of a set.
|
* 
|
* 
|
*[[Wikipedia:School_and_university_projects/Discrete_and_numerical_mathematics/Learning_plan#Combinatorics|WP+ (Combinatorics)]];
|
* 
|
* 
|
*[[Wikipedia:School_and_university_projects/Discrete_and_numerical_mathematics/Learning_plan#Combinatorics|WP+ (Combinatorics)]];
|-
| style="text-align: center;"|
Wed
<br />
22/4
<br />
[[File:UN emblem blue.svg|link=|46x46px]]
<br />
[[Earth Day]]
|Combinatorics

* Additive decompositions of numbers.
|
* 
|
* 
|
*[[Wikipedia:School_and_university_projects/Discrete_and_numerical_mathematics/Learning_plan#Combinatorics|WP+ (Combinatorics)]];
|
* 
|
* 
|
*[[Wikipedia:School_and_university_projects/Discrete_and_numerical_mathematics/Learning_plan#Combinatorics|WP+ (Combinatorics)]];
|-
| style="text-align: center;"|
Thu
<br />
23/4
<br />
[[Saint George's Day]]
<br />
[[File:UN emblem blue.svg|link=|46x46px]]
<br />
[[World Book and Copyright Day]]
<br />
[[UN English Language Day]]
<br />
[[UN Spanish Language Day]]
<br />
[[International Girls in ICT Day]] <sup>([[:es:Día Internacional de las Niñas en las TIC|es]])</sup>
| colspan=7 style="background: none;" |
<!--
|
|
|
|
|
|
|
-->
|-
| colspan="8" style="box-sizing: border-box; line-height: 1.1; border-bottom-width: 3px; border-bottom-style: solid; border-bottom-color: rgb(0, 173, 239); border-top-width: 3px; border-top-style: solid; border-top-color: rgb(0, 173, 239); margin-top: 26px; margin-bottom: 9px; padding-left: 1px; padding-bottom: 7px; text-align: center; background: #E6ECFF;" |<p style="text-transform: uppercase; font-size: 120%;">'''Theme 4'''</p>

<p style="text-transform: uppercase;">'''Difference equations'''</p>

<p>(8 h LG and 2 h S/L)</p>

<p>(1h LG Solving the end-course preparatory exam)</p>
|-
| style="text-align: center;"|
'''Group B'''
<br />
Fri
<br />
24/4
<br />
[[File:UN emblem blue.svg|link=|46x46px]]
<br />
[[International Day of Multilateralism and Diplomacy for Peace]] <sup>([https://www.un.org/en/events/diplomacyday/index.shtml ref])</sup>
<hr>
'''Groups A and E'''
<br />
Mon
<br />
27/4
| style="text-align: left; background: #EEFFEE;" |'''Seminar/Laboratory No. 12''':<br />Difference equations, I

* (Occasionally computer-assisted) hands-on [[word problem|word-problem]]-solving on issues related to:
** linear finite difference equations.
| style="text-align: left; background: #EEFFEE;" |
* (Everything we have studied on the subject);
* § 6.3.
| style="text-align: left; background: #EEFFEE;" |
* (Everything we have studied on the subject);
* § 8.3.
| style="text-align: left; background: #EEFFEE;" |
* (Everything we have studied on the subject);
*[[#NavasEstebanQuesada2009a|Navas, Esteban and Quesada, 2009a]];
*[[#NavasEstebanQuesada2009b|Navas, Esteban and Quesada, 2009b]];
*[[Wikipedia:School_and_university_projects/Discrete_and_numerical_mathematics/Learning_plan#Finite difference equations (recurrence relations)|WP+ (Finite difference equations (recurrence relations))]].
| style="text-align: left; background: #EEFFEE;" |
* (Everything we have studied on the subject);
* § 6.2 (45, 46, 47);
* § 6.3 (10, 11, 14, 15, 16).
| style="text-align: left; background: #EEFFEE;" |
* (Everything we have studied on the subject);
* § 8.2 (45, 46, 47);
* § 8.3 (10, 11, 14, 15, 16).
| style="text-align: left; background: #EEFFEE;" |
* (Everything we have studied on the subject);
*[[#NavasEstebanQuesada2009a|Navas, Esteban and Quesada, 2009a]];
*[[#NavasEstebanQuesada2009b|Navas, Esteban and Quesada, 2009b]];
*[[Wikipedia:School_and_university_projects/Discrete_and_numerical_mathematics/Learning_plan#Finite difference equations (recurrence relations)|WP+ (Finite difference equations (recurrence relations))]];
* [https://drive.google.com/open?id=1ZX7_j6RzDw3I79ySpO57ZVLztSLavJGP León-Rojas, J. M., Question selection no. 12 and 12<sup>nd</sup> seminar/laboratory. Difference equations, I.]
|-
| style="text-align: center;"|
Sat
<br />
25/4
<br />
[[File:UN emblem blue.svg|link=|46x46px]]
<br />
[[World Malaria Day]]
<br />
[[International Delegate's Day]] <sup>([https://www.un.org/en/events/delegatesday/ ref])</sup>
| colspan=7 style="background: none;" |
<!--
|
|
|
|
|
|
|
-->
|-
| style="text-align: center;"|
Sun
<br />
26/4
<br />
[[File:UN emblem blue.svg|link=|46x46px]]
<br />
[[World Intellectual Property Day]]
<br />
[[International Chernobyl Disaster Remembrance Day]] <sup>([https://www.un.org/en/events/chernobylday/index.shtml ref])</sup>
| colspan=7 style="background: none;" |
<!--
|
|
|
|
|
|
|
-->
|-
| style="text-align: center;"|
Tue
<br />
28/4
<br />
[[File:UN emblem blue.svg|link=|46x46px]]
<br />
[[World Day for Safety and Health at Work]]
|Finite difference equations (recurrence relations)

* Linear finite difference equations, models and applications.
|
* (§ 3.2.1, 3.2.2, 3.2.3, 3.2.4);
* § 6.1.
|
* (§ 2.4);
* § 8.1.
|
*[[#NavasEstebanQuesada2009a|Navas, Esteban and Quesada, 2009a, § 9.1, 9.2]];
*[[#NavasEstebanQuesada2009b|Navas, Esteban and Quesada, 2009b]];
*[[Wikipedia:School_and_university_projects/Discrete_and_numerical_mathematics/Learning_plan#Finite difference equations (recurrence relations)|WP+ (Finite difference equations (recurrence relations))]].
|
* § 6.1 (17, 22, 23, 25, 27, 36, 37, 42, 46).
|
* § 8.1 (1, 6, 7, 9, 11, 20, 21, 26, 30).
|
*[[#NavasEstebanQuesada2009a|Navas, Esteban and Quesada, 2009a, § 9.1, 9.2]];
*[[#NavasEstebanQuesada2009b|Navas, Esteban and Quesada, 2009b]];
*[[Wikipedia:School_and_university_projects/Discrete_and_numerical_mathematics/Learning_plan#Finite difference equations (recurrence relations)|WP+ (Finite difference equations (recurrence relations))]].
|-
| style="text-align: center;"|
Wed
<br />
29/4
|Finite difference equations (recurrence relations)

* Homogeneous linear finite difference equations with constant coefficients, I.
|
* § 6.2.1, 6.2.2.
|
* § 8.2.1, 8.2.2.
|
*[[#NavasEstebanQuesada2009a|Navas, Esteban and Quesada, 2009a, § 9.1, 9.2, 9.3]];
*[[#NavasEstebanQuesada2009b|Navas, Esteban and Quesada, 2009b]];
*[[Wikipedia:School_and_university_projects/Discrete_and_numerical_mathematics/Learning_plan#Finite difference equations (recurrence relations)|WP+ (Finite difference equations (recurrence relations))]].
|
* § 6.2 (2, 3, 4, 7, 8, 11, 12, 13, 14, 15, 17, 18).
|
* § 8.2 (2, 3, 4, 7, 8, 11, 12, 13, 14, 15, 17, 18).
|
*[[#NavasEstebanQuesada2009a|Navas, Esteban and Quesada, 2009a, § 9.1, 9.2, 9.3]];
*[[#NavasEstebanQuesada2009b|Navas, Esteban and Quesada, 2009b]];
*[[Wikipedia:School_and_university_projects/Discrete_and_numerical_mathematics/Learning_plan#Finite difference equations (recurrence relations)|WP+ (Finite difference equations (recurrence relations))]].
|-
| style="text-align: center;"|
Thu
<br />
30/4
<br />
[[File:UN emblem blue.svg|link=|46x46px]]
<br />
[[International Jazz Day]]
|Finite difference equations (recurrence relations)

* Homogeneous linear finite difference equations with constant coefficients, II.
|
* § 6.2.1, 6.2.2.
|
* § 8.2.1, 8.2.2.
|
*[[#NavasEstebanQuesada2009a|Navas, Esteban and Quesada, 2009a, § 9.1, 9.2, 9.3]];
*[[#NavasEstebanQuesada2009b|Navas, Esteban and Quesada, 2009b]];
*[[Wikipedia:School_and_university_projects/Discrete_and_numerical_mathematics/Learning_plan#Finite difference equations (recurrence relations)|WP+ (Finite difference equations (recurrence relations))]].
|
* § 6.2 (2, 3, 4, 7, 8, 11, 12, 13, 14, 15, 17, 18).
|
* § 8.2 (2, 3, 4, 7, 8, 11, 12, 13, 14, 15, 17, 18).
|
*[[#NavasEstebanQuesada2009a|Navas, Esteban and Quesada, 2009a, § 9.1, 9.2, 9.3]];
*[[#NavasEstebanQuesada2009b|Navas, Esteban and Quesada, 2009b]];
*[[Wikipedia:School_and_university_projects/Discrete_and_numerical_mathematics/Learning_plan#Finite difference equations (recurrence relations)|WP+ (Finite difference equations (recurrence relations))]].
|-
| style="text-align: center;"|
Fri
<br />
1/5
<br />
[[International Workers' Day]]
| colspan=7 style="background: none;" |
<!--
|
|
|
|
|
|
|
-->
|-
| style="text-align: center;"|
Sat
<br />
2/5
<br />
[[File:UN emblem blue.svg|link=|46x46px]]
<br />
[[World Tuna Day]] <sup>([[:es:Día Mundial del Atún|es]])</sup>
| colspan=7 style="background: none;" |
<!--
|
|
|
|
|
|
|
-->
|-
| style="text-align: center;"|
Sun
<br />
3/5
<br />
[[File:UN emblem blue.svg|link=|46x46px]]
<br />
[[World Press Freedom Day]]
| colspan=7 style="background: none;" |
<!--
|
|
|
|
|
|
|
-->
|-
| style="text-align: center;"|
Tue
<br />
5/5
<br />
[[File:UN emblem blue.svg|link=|46x46px]]
<br />
[[African World Heritage Day]]
|Finite difference equations (recurrence relations)

* Non-homogeneous linear finite difference equation with constant coefficients, I.
|
* § 6.2.3.
|
* § 8.2.3.
|
*[[#NavasEstebanQuesada2009a|Navas, Esteban and Quesada, 2009a, § 9.1, 9.2, 9.3 (esp. 9.3.2)]];
*[[#NavasEstebanQuesada2009b|Navas, Esteban and Quesada, 2009b]];
*[[Wikipedia:School_and_university_projects/Discrete_and_numerical_mathematics/Learning_plan#Finite difference equations (recurrence relations)|WP+ (Finite difference equations (recurrence relations))]].
|
* § 6.2 (23, 24, 26, 31).
|
* § 8.2 (23, 24, 26, 31).
|
*[[#NavasEstebanQuesada2009a|Navas, Esteban and Quesada, 2009a, § 9.1, 9.2, 9.3 (esp. 9.3.2)]];
*[[#NavasEstebanQuesada2009b|Navas, Esteban and Quesada, 2009b]];
*[[Wikipedia:School_and_university_projects/Discrete_and_numerical_mathematics/Learning_plan#Finite difference equations (recurrence relations)|WP+ (Finite difference equations (recurrence relations))]].
|-
| style="text-align: center;"|
Wed
<br />
6/5
|Finite difference equations (recurrence relations)

* Non-homogeneous linear finite difference equation with constant coefficients, II.
|
* § 6.2.3.
|
* § 8.2.3.
|
*[[#NavasEstebanQuesada2009a|Navas, Esteban and Quesada, 2009a, § 9.1, 9.2, 9.3 (esp. 9.3.2)]];
*[[#NavasEstebanQuesada2009b|Navas, Esteban and Quesada, 2009b]];
*[[Wikipedia:School_and_university_projects/Discrete_and_numerical_mathematics/Learning_plan#Finite difference equations (recurrence relations)|WP+ (Finite difference equations (recurrence relations))]].
|
* § 6.2 (23, 24, 26, 31).
|
* § 8.2 (23, 24, 26, 31).
|
*[[#NavasEstebanQuesada2009a|Navas, Esteban and Quesada, 2009a, § 9.1, 9.2, 9.3 (esp. 9.3.2)]];
*[[#NavasEstebanQuesada2009b|Navas, Esteban and Quesada, 2009b]];
*[[Wikipedia:School_and_university_projects/Discrete_and_numerical_mathematics/Learning_plan#Finite difference equations (recurrence relations)|WP+ (Finite difference equations (recurrence relations))]].
|-
| rowspan="2" style="text-align: center;"|
Thu
<br />
7/5
<br />
[[File:UN emblem blue.svg|link=|46x46px]]
<br />
[[Vesak Day]]
<br />
(Held on the/a full moon day of May each year)
|Finite difference equations (recurrence relations)

* Systems of linear finite difference equations, I.
|—
|—
|
*[[#NavasEstebanQuesada2009a|Navas, Esteban and Quesada, 2009a, § 9.4]];
*[[#NavasEstebanQuesada2009b|Navas, Esteban and Quesada, 2009b]];
*[[Wikipedia:School_and_university_projects/Discrete_and_numerical_mathematics/Learning_plan#Finite difference equations (recurrence relations)|WP+ (Finite difference equations (recurrence relations))]].
|—
|—
|
*[[#NavasEstebanQuesada2009a|Navas, Esteban and Quesada, 2009a, § 9.4]];
*[[#NavasEstebanQuesada2009b|Navas, Esteban and Quesada, 2009b]];
*[[Wikipedia:School_and_university_projects/Discrete_and_numerical_mathematics/Learning_plan#Finite difference equations (recurrence relations)|WP+ (Finite difference equations (recurrence relations))]].
|-
| colspan="7" style="text-align: left; background: #FFF5FA;" |[[File:Twenty sided dice.svg|32px|link=]] [[Wikipedia:School and university projects/Discrete and numerical mathematics|University project Discrete and numerical mathematics (optional out-of-class activity)]]: ('''Third and last checkpoint'''). You should have continually been working in your contributions, publishing each update, along with the corresponding themes to which they belong are worked in class, and linking each new major contribution on the [[Wikipedia:School_and_university_projects/Discrete_and_numerical_mathematics/Participants_and_major_contributions|contributions page]] of the project. Furthermore, you must publish, also on an ongoing basis, in your logbook (sandbox), the part of your [[Wikipedia:School and university projects/Discrete and numerical mathematics#Self-report|self-report]] that deals with what you have developed so far (in this case all you have done). Starting from now until the ending date, you can '''review all what you have done''' and you can '''correct minor errors''' and '''complete other small details'''.
|-
| style="text-align: center;"|
Fri
<br />
8/5
<br />
Academic celebration, [[Cáceres School of Technology]] <sup>([[:es:Escuela Politécnica de Cáceres|es]])</sup>
<br />
[[File:UN emblem blue.svg|link=|46x46px]]
<br />
[[Time of Remembrance and Reconciliation for Those Who Lost Their Lives during the Second World War]]
| colspan=7 style="background: none;" |
<!--
|
|
|
|
|
|
|
-->
|-
| style="text-align: center;"|
Sat
<br />
9/5
<br />
[[File:UN emblem blue.svg|link=|46x46px]]
<br />
[[Time of Remembrance and Reconciliation for Those Who Lost Their Lives during the Second World War]]
<br />
[[World Migratory Bird Day]] <sup>([[:es:Día Mundial de las Aves Migratorias|es]])</sup>
| colspan=7 style="background: none;" |
<!--
|
|
|
|
|
|
|
-->
|-
| style="text-align: center;"|
'''Group B'''
<br />
Mon
<br />
11/5
<br />
(please come and attend as far as possible)
<hr>
'''Groups A and E'''
<br />
Mon
<br />
11/5
| style="text-align: left; background: #EEFFEE;" |'''Seminario/Laboratorio N.º 13''':<br />Difference equations, II

* (Occasionally computer-assisted) hands-on [[word problem|word-problem]]-solving on issues related to:
** finite difference equations.
| style="text-align: left; background: #EEFFEE;" |
* (Everything we have studied on the subject).
| style="text-align: left; background: #EEFFEE;" |
* (Everything we have studied on the subject).
| style="text-align: left; background: #EEFFEE;" |
* (Everything we have studied on the subject);
*[[#NavasEstebanQuesada2009a|Navas, Esteban and Quesada, 2009a]];
*[[#NavasEstebanQuesada2009b|Navas, Esteban and Quesada, 2009b]];
*[[Wikipedia:School_and_university_projects/Discrete_and_numerical_mathematics/Learning_plan#Finite difference equations (recurrence relations)|WP+ (Finite difference equations (recurrence relations))]].
| style="text-align: left; background: #EEFFEE;" |
* (Everything we have studied on the subject).
| style="text-align: left; background: #EEFFEE;" |
* (Everything we have studied on the subject).
| style="text-align: left; background: #EEFFEE;" |
* (Everything we have studied on the subject);
*[[#NavasEstebanQuesada2009a|Navas, Esteban and Quesada, 2009a]];
*[[#NavasEstebanQuesada2009b|Navas, Esteban and Quesada, 2009b]];
*[[Wikipedia:School_and_university_projects/Discrete_and_numerical_mathematics/Learning_plan#Finite difference equations (recurrence relations)|WP+ (Finite difference equations (recurrence relations))]];
* [https://drive.google.com/open?id=1k8zpe9vPG0259ZtrvOQHExEXz9XSt3Vu León-Rojas, J. M., Question selection no. 13 and 13<sup>rd</sup> seminar/laboratory. Difference equations, II.]
|-
| style="text-align: center;"|
Tue
<br />
12/5
|Finite difference equations (recurrence relations)

* Systems of linear finite difference equations, II.
|—
|—
|
*[[#NavasEstebanQuesada2009a|Navas, Esteban and Quesada, 2009a, § 9.4]];
*[[#NavasEstebanQuesada2009b|Navas, Esteban and Quesada, 2009b]];
*[[Wikipedia:School_and_university_projects/Discrete_and_numerical_mathematics/Learning_plan#Finite difference equations (recurrence relations)|WP+ (Finite difference equations (recurrence relations))]].
|—
|—
|
*[[#NavasEstebanQuesada2009a|Navas, Esteban and Quesada, 2009a, § 9.4]];
*[[#NavasEstebanQuesada2009b|Navas, Esteban and Quesada, 2009b]];
*[[Wikipedia:School_and_university_projects/Discrete_and_numerical_mathematics/Learning_plan#Finite difference equations (recurrence relations)|WP+ (Finite difference equations (recurrence relations))]].
|-
| style="text-align: center;"|
Wed
<br />
13/5
|Finite difference equations (recurrence relations)

* Systems of linear finite difference equations, III.
|—
|—
|
*[[#NavasEstebanQuesada2009a|Navas, Esteban and Quesada, 2009a, § 9.4]];
*[[#NavasEstebanQuesada2009b|Navas, Esteban and Quesada, 2009b]];
*[[Wikipedia:School_and_university_projects/Discrete_and_numerical_mathematics/Learning_plan#Finite difference equations (recurrence relations)|WP+ (Finite difference equations (recurrence relations))]].
|—
|—
|
*[[#NavasEstebanQuesada2009a|Navas, Esteban and Quesada, 2009a, § 9.4]];
*[[#NavasEstebanQuesada2009b|Navas, Esteban and Quesada, 2009b]];
*[[Wikipedia:School_and_university_projects/Discrete_and_numerical_mathematics/Learning_plan#Finite difference equations (recurrence relations)|WP+ (Finite difference equations (recurrence relations))]].
|-
| rowspan="2" style="text-align: center;"|
'''Thu'''
<br />
'''14/5'''
<br />
'''End of classes'''
| colspan="7" style="text-align: left; background: #ffddff;" |[[File:Group full edit.svg|32px|link=]] '''Exam review''': large-group class dedicated to share ideas and solutions in a whole-class discussion about the '''end-course preparatory exam''' (done as homework).
|-
| colspan="7" style="text-align: left; background: #FFF5FA;" |[[File:Twenty sided dice.svg|32px|link=]] [[Wikipedia:School and university projects/Discrete and numerical mathematics|University project Discrete and numerical mathematics (optional out-of-class activity)]]: '''Ending date''' of the academic component in the 2<sup>nd</sup> semester of the academic year 2019-2020.
|-
| colspan="8" style="text-align: center;" |
[[File:UN emblem blue.svg|link=|46x46px]]

'''15 May'''
<br />
[[International Day of Families]]

'''16 May'''
<br />
[[International Day of Living Together in Peace]] <sup>([[:es:Día Internacional de la Convivencia en Paz|es]])</sup>
<br />
[[International Day of Light]] <sup>([[:es:Día Internacional de la Luz|es]])</sup>

'''17 May'''
<br />
[[World Telecommunication and Information Society Day]]
|-
| style="text-align: center;"|
Mon
<br />
20/5
<br />
'''Start of June exam period'''
|-
| colspan="8" style="text-align: center;" |
[[File:UN emblem blue.svg|link=|46x46px]]

'''20 May'''
<br />
[[World Bee Day]]

'''21 May'''
<br />
[[World Day for Cultural Diversity for Dialogue and Development]]
<br />
[[International Tea Day (UN)]] <sup>([https://undocs.org/en/A/74/386 ref])</sup>

'''22 May'''
<br />
[[International Day for Biological Diversity]]

'''23 May'''
<br />
[[ International Day to End Obstetric Fistula ]] <sup>([[:es:Día Internacional para la Erradicación de la Fístula Obstétrica|es]])</sup>

'''29 May'''
<br />
[[International Day of United Nations Peacekeepers]]
<br />

'''31 May'''
<br />
[[World No Tobacco Day]]
|-
|___ __/_
| colspan="7" style="text-align: left; background: #ff88ff; color: black;" |[[File:Hand-33988.svg|32px|link=]] '''2019-2020 Final exam'''.
|-
| colspan="8" style="text-align: center;" |
[[File:UN emblem blue.svg|link=|46x46px]]

...
|-
| style="text-align: center;"|
Sat
<br />
6/7
<br />
'''End of June exam period'''
|-
| colspan="8" style="text-align: center;" |
[[File:UN emblem blue.svg|link=|46x46px]]

...
|-
| style="text-align: center;"|
Mon
<br />
22/6
<br />
'''Start of July exam period'''
|-
| colspan="8" style="text-align: center;" |
[[File:UN emblem blue.svg|link=|46x46px]]

...
|-
|___ __/_
| colspan="7" style="text-align: left; background: #ffbbff; color: black;" |[[File:Hand-33988.svg|32px|link=]] '''2019-2020 Resit final exam'''.
|-
| colspan="8" style="text-align: center;" |
[[File:UN emblem blue.svg|link=|46x46px]]

...
|-
| style="text-align: center;"|
Fri
<br />
10/7
<br />
'''End of July exam period'''
|-
| colspan="8" style="text-align: center;" |
[[File:UN emblem blue.svg|link=|46x46px]]

...
|-
| style="text-align: center;"|
Mon
<br />
20/7
<br />
'''End of term'''
|-
| colspan="8" style="text-align: center;" |
[[File:UN emblem blue.svg|link=|46x46px]]

...

(See: [https://www.un.org/en/sections/observances/international-days/index.html International days currently observed by the United Nations]).
<!-- INICIO PARTES DE OTROS AÑOS, POR EJEMPLO, GRAFOS
|-
|
|'''Seminario/Laboratorio N.º 9''':<br />Combinatoria

* Métodos de generación de permutaciones y combinaciones.
|
* § 4.6
|
* § 6.6
|
*[[Usuario:Jmleonrojas/Taller#FrancoBranas2008|Franco, Espinel y Almeida, 2008]];
*[[Usuario:Jmleonrojas/Taller#Sedgewick1977|Sedgewick, 1977]];
*[[wikipedia:Proyecto_educativo/Matemática_discreta_y_numérica/Plan_de_aprendizaje#Combinatoria|WP+ (Combinatoria)]].
|
* § 4.6 (4, 2, 8, 9, 10, 11).
|
* § 6.6 (2, 6, 10, 11, 12, 13).
|
*[[Usuario:Jmleonrojas/Taller#FrancoBranas2008|Franco, Espinel y Almeida, 2008]];
*[[Usuario:Jmleonrojas/Taller#Sedgewick1977|Sedgewick, 1977]];
*[[wikipedia:Proyecto_educativo/Matemática_discreta_y_numérica/Plan_de_aprendizaje#Combinatoria|WP+ (Combinatoria)]].
|-
|Mié 8/5
|Ecuaciones en diferencias finitas y combinatoria

* Funciones generatrices
|
* § 6.4.
|
* § 8.4.
|
*[[Usuario:Jmleonrojas/Taller#FrancoBranas2008|Franco, Espinel y Almeida, 2008, § 8]];
*[[wikipedia:Proyecto_educativo/Matemática_discreta_y_numérica/Plan_de_aprendizaje#Combinatoria|WP+ (Combinatoria)]].
|
* § 6.4 (30, 32, 34, 39).
|
* § 8.4 (30, 32, 34, 39).
|
*[[Usuario:Jmleonrojas/Taller#FrancoBranas2008|Franco, Espinel y Almeida, 2008, § 8]];
*[[wikipedia:Proyecto_educativo/Matemática_discreta_y_numérica/Plan_de_aprendizaje#Combinatoria|WP+ (Combinatoria)]].
|-
|Mié 8/5
|Combinatoria: el principio de inclusión-exclusión

* El principio de inclusión-exclusión;
* Aplicaciones del principio de inclusión-exclusión.
|
* § 6.5.
* § 6.6.1, 6.6.2, 6.6.3, 6.6.4.
|
* § 8.5;
* § 8.6.1, 8.6.2, 8.6.3, 8.6.4.
|
* [[Usuario:Jmleonrojas/Taller#FrancoBranas2008|Franco, Espinel y Almeida, 2008, § 3.1.2]];
* [[wikipedia:Proyecto_educativo/Matemática_discreta_y_numérica/Plan_de_aprendizaje#Combinatoria|WP+ (Combinatoria)]].
|
* § 6.5 (8, 11, 24);
* § 6.6 (1, 2, 3, 4, 6, 8, 10).
|
* § 8.5 (8, 11, 24);
* § 8.6 (1, 2, 3, 4, 6, 8, 10).
|
* [[Usuario:Jmleonrojas/Taller#FrancoBranas2008|Franco, Espinel y Almeida, 2008, § 3.1.2]];
* [[wikipedia:Proyecto_educativo/Matemática_discreta_y_numérica/Plan_de_aprendizaje#Combinatoria|WP+ (Combinatoria)]].
|-
|
|Combinatoria: el principio de inclusión-exclusión

* Desórdenes.
|
* § 6.6.5.
|
* § 8.6.5.
|
* [[Usuario:Jmleonrojas/Taller#FrancoBranas2008|Franco, Espinel y Almeida, 2008, § 4.6.4]];
* [[wikipedia:Proyecto_educativo/Matemática_discreta_y_numérica/Plan_de_aprendizaje#Combinatoria|WP+ (Combinatoria)]].
|
* § 6.6 (13, 14, 16, 17, 18, 20, 26).
|
* § 8.6 (13, 14, 16, 17, 18, 20, 26).
|
* [[Usuario:Jmleonrojas/Taller#FrancoBranas2008|Franco, Espinel y Almeida, 2008, § 4.6.4]];
* [[wikipedia:Proyecto_educativo/Matemática_discreta_y_numérica/Plan_de_aprendizaje#Combinatoria|WP+ (Combinatoria)]].
|-
|
|'''Seminario/Laboratorio N.º 13''':<br />Combinatoria: el principio de inclusión-exclusión

* Polinomios de torre. Disposiciones con posiciones prohibidas.
|—
|—
|
*Grimaldi, § 8.4, 8.5;
* John G. Michaels y Kenneth H. Rosen. Applications of Discrete Mathematics. NY: McGraw-Hill, Updated Edition, 2007. (Ch. 9: Arrangements with forbidden positions);
*[[wikipedia:Proyecto_educativo/Matemática_discreta_y_numérica/Plan_de_aprendizaje#Combinatoria|WP+ (Combinatoria)]].
|—
|—
|
*Grimaldi, § 8.4, 8.5;
* John G. Michaels y Kenneth H. Rosen. Applications of Discrete Mathematics. NY: McGraw-Hill, Updated Edition, 2007. (Ch. 9: Arrangements with forbidden positions);
*[[wikipedia:Proyecto_educativo/Matemática_discreta_y_numérica/Plan_de_aprendizaje#Combinatoria|WP+ (Combinatoria)]].
|-
|Vie 22/3, Lun 25/3 [[Cuaresma]]: [[Viernes de Dolores]]
| colspan="7" style="text-align: left; background: #EEFFEE;" |[[Archivo:High-contrast-input-keyboard.svg|vínculo=|32x32px]] [[wikipedia:Editatón|Editatón]]: Inicio de la lectura y redacción optativa de un breve análisis matemático y computacional, reflexivo, crítico y analítico, sobre el texto [[iarchive:tinkertoycomputer00dewd|A. K. Dewdney (1993) The Tinkertoy Computer and other machinations. Chapter 17: Automated Math''. Nueva York: W. H. Freeman.'']] (En inglés). (Juegos de ordenador. De cómo un par de programas obtusos pasan por genios en los tests de inteligencia. ''Investigación y Ciencia'', N. 116, mayo 1986, pp. 94-98, Prensa Científica, S. A.,https://issuu.com/joshidelhampfert/docs/niyc116 [En español]). ('''Nota:''' Los contenidos concretos de esta actividad de desarrollo personal no serán parte obligatoria de ningún examen).
|-
|
|Grafos y sus aplicaciones

* Caminos y circuitos eulerianos y hamiltonianos.
|
* § 8.5.
|
* § 10.5.
|
*[[wikipedia:Proyecto_educativo/Matemática_discreta_y_numérica/Plan_de_aprendizaje#Grafos|WP+ (Grafos)]].
|
* § 8.5 (3, 5, 13, 14, 26, 28, 44, 45, 49).
|
* § 10.5 (3, 5, 13, 14, 26, 28, 44, 45, 49).
|
*[[wikipedia:Proyecto_educativo/Matemática_discreta_y_numérica/Plan_de_aprendizaje#Grafos|WP+ (Grafos)]].
|-
|Jue 3/5
|Grafos y sus aplicaciones

* Grafos y modelos basados en grafos.
|
* § 8.1.
|
* § 10.1.
|
* [[wikipedia:Proyecto_educativo/Matemática_discreta_y_numérica/Plan_de_aprendizaje#Grafos|WP+ (Grafos)]].
|
* § 8.1 (_, _, 20, 22).
|
* § 10.1 (12, 28, 22, 24).
|
* [[wikipedia:Proyecto_educativo/Matemática_discreta_y_numérica/Plan_de_aprendizaje#Grafos|WP+ (Grafos)]].
|-
|Vie 4/5, Lun 7/5
| style="text-align: left; background: #EEFFEE;" |Grafos y sus aplicaciones

* Teoría de emparejamiento. El problema de asignación.
| style="text-align: left; background: #EEFFEE;" |—
| style="text-align: left; background: #EEFFEE;" |—
| style="text-align: left; background: #EEFFEE;" |
* Grimaldi, § 13.4;
* John G. MichaelsMichaels y Kenneth H. Rosen. Applications of Discrete Mathematics. NY: McGraw-Hill, Updated Edition, 2007. (Ch. 17: The assignment problem);
* [[wikipedia:Proyecto_educativo/Matemática_discreta_y_numérica/Plan_de_aprendizaje#Grafos|WP+ (Grafos)]].
| style="text-align: left; background: #EEFFEE;" |—
| style="text-align: left; background: #EEFFEE;" |—
| style="text-align: left; background: #EEFFEE;" |
* Grimaldi, § 13.4;
* John G. MichaelsMichaels y Kenneth H. Rosen. Applications of Discrete Mathematics. NY: McGraw-Hill, Updated Edition, 2007. (Ch. 17: The assignment problem);
* [[wikipedia:Proyecto_educativo/Matemática_discreta_y_numérica/Plan_de_aprendizaje#Grafos|WP+ (Grafos)]].
|-
|Mar 8/5
|Grafos y sus aplicaciones

* Terminología y tipos de grafos.
|
* § 8.2.
|
* § 10.2.
|
* [[wikipedia:Proyecto_educativo/Matemática_discreta_y_numérica/Plan_de_aprendizaje#Grafos|WP+ (Grafos)]].
|
* § 8.2 (4, 5, _, 18, 20, 25, 31, 35, 42).
|
* § 10.2 (4, 5, 18, 20, 22, 35, 49, 53, 60).
|
* [[wikipedia:Proyecto_educativo/Matemática_discreta_y_numérica/Plan_de_aprendizaje#Grafos|WP+ (Grafos)]].
|-
|Mié 9/5
|Grafos y sus aplicaciones

* Isomorfismo de grafos;
* Conectividad.
|§ 8.3;

* § 8.4.
|
* § 10.3;
* § 10.4.
|
* [[wikipedia:Proyecto_educativo/Matemática_discreta_y_numérica/Plan_de_aprendizaje#Grafos|WP+ (Grafos)]].
|
* § 8.3 (6, 23, 28, 29, 32, 33, 37, 41, 55);
* § 8.4 (1, _, 13, 15, _, _, 21, 35).
|
* § 10.3 (6, 23, 28, 29, 32, 33, 37, 41, 55);
* § 10.4 (1, 11, 15, 19, 21, 23, 29, 43).
|
* [[wikipedia:Proyecto_educativo/Matemática_discreta_y_numérica/Plan_de_aprendizaje#Grafos|WP+ (Grafos)]].
FIN PARTE DEDICADA A GRAFOS -->
|}
</div>
<!--
Template:Importance/colour
 | top      = #ff00ff
 | high     = #ff88ff
 | mid      = #ffbbff
 | low      = #ffddff
 | bottom   = #ffeeff
 | no       = #ffffff
 | na       = #f5f5f5
 | core     = #ff00ff
 | unknown  = transparent
 | #default = transparent
-->

== Coda ==
<!--
{| style="margin:auto; width:90%; color:black;"
 | style="width:0%; text-align:right; background-color:White;" | [[File:Ambox warning green construction.svg|center|46px]]
 | style="width:100%; text-align:center; background-color:#f1fadf; border-style: solid; border-width: 0px 0px 0px 0px" | <hr style="border-width: 1px;">This coda is '''under construction'''. Please excuse the inconvenience.<hr style="border-width: 1px;">
|}
-->
=== Ex post I: Arts & Humanities Classroom 'Juanelo Turriano' ===
(For the time being, please see [https://es.wikipedia.org/wiki/Wikipedia:Proyecto_educativo/Matem%C3%A1tica_discreta_y_num%C3%A9rica/Plan_de_aprendizaje#Ex_post_I:_Aula_de_Humanidades_Juanelo_Turriano_(Segunda_edici%C3%B3n) Ex post I on the Spanish Wikipedia]).

=== <i>Ex post</i> II: Humour, entertainment and curiosities ===
{{Humorous essays|state=expanded}}
* [http://phdcomics.com/ Piled Higher and Deeper. © ARR.]
* [http://www.rudimentsofwisdom.com/default.htm Tim Hunkin and Shane Frazer's The Rudiments of Wisdom Cartoon Encyclopaedia. © ARR.]
* [https://abstrusegoose.com/253 Wizards. Abstruse Goose. © CC BY-NC.]

== Notes ==
<!--{{listaref|group="note"|colwidth=30em}}-->
{{reflist|group="note"}}

== References ==
<!--{{listaref|2}}-->
{{reflist}}

== See also ==

;Inner links
* Participants and major contributions of the university project, learning plan and sandbox of the learning plan
** [[Wikipedia:School_and_university_projects/Discrete_and_numerical_mathematics|University project 'Discrete and numerical mathematics']]
** [[Wikipedia:School_and_university_projects/Discrete_and_numerical_mathematics/Participants and major contributions|University project 'Discrete and numerical mathematics': Participants and major contributions]]
<!--** [[Wikipedia:School_and_university_projects/Discrete_and_numerical_mathematics/Learning_plan|Learning plan 'Discrete and numerical mathematics']]-->
** [[Wikipedia:School and university projects/Discrete and numerical mathematics/Learning plan/Sandbox|Sandbox of the learning plan 'Discrete and numerical mathematics']]

* Talk pages
** [[Wikipedia_talk:School_and_university_projects/Discrete_and_numerical_mathematics|Talk page of the university project 'Discrete and numerical mathematics']]
** [[Wikipedia_talk:School_and_university_projects/Discrete_and_numerical_mathematics/Participants and major contributions|Talk page of the page of the participants and major contributions of the university project 'Discrete and numerical mathematics']]
** [[Wikipedia_talk:School_and_university_projects/Discrete_and_numerical_mathematics/Learning_plan|Talk page of the learning plan 'Discrete and numerical mathematics']]
** [[Wikipedia_talk:School_and_university_projects/Discrete_and_numerical_mathematics/Learning_plan/Sandbox|Talk page of the sandbox of the learning plan 'Discrete and numerical mathematics']]

;Interwiki links
* University project, participants and major contributions, learning plan and sandbox of the learning plan
** [https://es.wikipedia.org/wiki/Wikipedia:Proyecto_educativo/Matemática_discreta_y_numérica University project 'Matemática discreta y numérica'] (equivalent university project on the Spanish Wikipedia)
** [https://es.wikipedia.org/wiki/Wikipedia:Proyecto_educativo/Matemática_discreta_y_numérica/Participantes_y_contribuciones_principales University project 'Matemática discreta y numérica': participant and major contributions] (equivalent university project [partipants and major contributions page] on the Spanish Wikipedia)
** [https://es.wikipedia.org/wiki/Wikipedia:Proyecto_educativo/Matemática_discreta_y_numérica/Plan_de_aprendizaje Learning plan 'Matemática discreta y numérica'] (equivalent learning plan on the Spanish Wikipedia)
** [https://es.wikipedia.org/wiki/Wikipedia:Proyecto_educativo/Matemática_discreta_y_numérica/Plan_de_aprendizaje/Taller Sandbox of the learning project 'Matemática discreta y numérica'] (sandbox of the equivalent learning plan on the Spanish Wikipedia) (editathons are here)

* Talk pages
** [https://es.wikipedia.org/wiki/Wikipedia_discusión:Proyecto_educativo/Matemática_discreta_y_numérica Talk page of the university project 'Matemática discreta y numérica'] (talk page of the equivalent university project on the Spanish Wikipedia)
** [https://es.wikipedia.org/wiki/Wikipedia_discusión:Proyecto_educativo/Matemática_discreta_y_numérica/Participantes_y_contribuciones_principales Talk page of the page of the participants and major contributions of the university project 'Matemática discreta y numérica'] (talk page of the page of the participants and major contributions of the equivalent university project on the Spanish Wikipedia)
** [https://es.wikipedia.org/wiki/Wikipedia_discusión:Proyecto_educativo/Matemática_discreta_y_numérica/Plan_de_aprendizaje Talk page of the learning plan 'Matemática discreta y numérica'] (talk page of the equivalent learning plan on the Spanish Wikipedia)
** [https://es.wikipedia.org/wiki/Wikipedia_discusión:Proyecto_educativo/Matemática_discreta_y_numérica/Plan_de_aprendizaje/Taller Talk page of the sandbox of the learning plan 'Matemática discreta y numérica'] (talk page of the sandbox of the equivalent learning plan on the Spanish Wikipedia) (editathons, in Spanish, are here)

== To keep track, know more or write a comment ==
Feel free to correct any '''typographical error''' you have detected in any of the project or plan pages (this is Wikipedia!).

Also all the '''feedback''' for doing better the next time, that is, all the '''comments''', '''impressions''', '''opinions''', '''sensations''' and '''advices''', '''wishes''', '''suggestions''' or '''proposals''' concerning how we might improve this initiative will be most welcomed and greatly appreciated. The talk page of the learning plan is an ideal place to write them on. Please do not hesitate to do so. It means a lot to us.

* [[Wikipedia talk:{{PAGENAME}}|Talk page of the learning plan 'Discrete and numerical mathematics']]

== Declaration of conformity ==
<div lang="en" dir="ltr" style="position:relative;margin:20px auto;width:66%;min-width:18em;border:1px solid #AAA;background:white;padding:2px;">
  <div style="position:absolute;top:2px;left:2px;">[[File:Skeletal-hand.jpg|100px]]</div>
  <div style="position:relative;margin:0;margin-left:91px;min-height:30px;">
    <div style="padding:5px; font-size: 90%">
[[User:Jmleonrojas|Juan Miguel León Rojas]] declares under his own responsibility that the learning plan specified here meets all the essential requirements of the [https://zurbaran.unex.es/cloud/index.php/s/9ZYJep76byeHsKB academic program (ficha12a)] corresponding to the course Further Mathematics taught at the School of Technology, University of Extremadura.
    </div>
  </div>
</div>

== About this page on the English Wikipedia ==
* {{Pagelinks|{{FULLPAGENAME}}}} {{Check views|{{FULLPAGENAME}}}} <small>([{{fullurl:{{FULLPAGENAME}}|action=info}} more information])</small>

<br />
{| style="margin:auto; width:55%; color:black;"
 | style="width:0%; text-align:right; background-color:White;" | [[File:Gftp.svg|center|55px]]
 | style="width:100%; text-align:center; background-color:#f1fadf; padding:1em 1em 1em 1em; border-style: solid; border-width: 0px 0px 0px 0px" | <!--<hr style="border-width: 1px;">--><span style="font-family:cursive;">Please '''contribute to the protection of the environment''': print this document only if you consider it absolutely necessary.</span><!--<hr style="border-width: 1px;">-->
|}

|}
| style="border:1px solid transparent;" |
<!--          INFORMACIÓN TRANSVERSA, 1 y 2 / TRANSVERSE INFORMATION, 1 and 2          -->
| class="MainPageBG" style="width:20%; border:1px solid #deecc8; background:#f7fffa; vertical-align:top;"|
{| id="pamdan-right" style="width:100%; vertical-align:top; background:#f7fffa;"
<!--          INFORMACIÓN TRANSVERSA 1 / TRANSVERSE INFORMATION 1          -->
| style="padding:2px;" | <h2 id="pamdan-it1-h2" style="margin:3px; background:#deecc8; font-family:inherit; font-size:small; font-weight:bold; border:1px solid #deecc8; text-align:left; color:rgb(0, 0, 0); padding:0.2em 0.4em;">Transverse information 1.- Some points about Free/Libre & Open Knowledge (FLOK)<hr style="width: 100%; height: 4px; background-color: rgb(0, 102, 0);" /></h2>
|-
| style="color:#000; padding:2px 5px 5px;" | <div id="pamdan-it1" style="font-size:small;">
<div style="margin-left: 0%; text-align: right; color: rgb(0, 102, 0); font-size:small;">
  <table width="100%" border="0" cellpadding="10px">
    <tr>
      <td style="background-color: transparent;">
      </td>
      <td>'True poems of <i>cante jondo</i> are attributable to no one at all<br />but float on the wind like golden thistledown<br />and each generation clothes them in its own distinctive color,<br />in releasing them to the future.'
      </td>
    </tr>
    <tr>
      <td colspan="3"><small>[[Federico García Lorca|Federico García Lorca]] (1898-1936): <em>Importancia histórica y artística del primitivo canto andaluz llamado «cante jondo»</em> (Historical and artistic importance of the primitive Andalusian song, that which is called deep song, <i>cante jondo</i>.). (Lecture given at «Centro Artístico» in Granada, 19th February 1922). Vid. http://gnawledge.com/pdf/granada/LorcaCanteJondo.pdf. Translated into English by A. S. (Tony) Kline, in <i>Poetry in Translation</i>, vid. http://www.poetryintranslation.com/PITBR/Spanish/DeepSong.htm.</small>
      </td>
    </tr>
  </table>
</div>
==== <span style="color: rgb(0, 0, 0); font-style: italic;">Some referencies about free/libre & open licenses and free knowledge and culture</span> ====
* [http://es.creativecommons.org/ Creative Commons España (CC)]
* [http://www.opendefinition.org/okd/espanol/ Definición de Conocimiento Abierto (OKD) (''Open Definition'')]
* [http://freedomdefined.org/Definition/Es Definición de Obras Culturales Libres (''freedomdefined.org'')]
* [http://fsfe.org/index.es.html Free Software Foundation Europe (FSFE)]
* [http://www.ocwconsortium.org/ Open CourseWare (OCW)]
* [http://en.wikipedia.org/wiki/Open_education Open Education]
* [http://en.wikipedia.org/wiki/Open_educational_resources Open Educational Resources (OER)]
* [http://okfn.org/ Open Knowledge Foundation (OKF)]
* [http://www.opensource.org/ Open Source Initiative (OSI)]
* [http://en.wikipedia.org/wiki/Open_textbook Open Textbooks]
* [[:es:Usuario:Jmleonrojas/Taller/Software Libre y Conocimiento Libre]]
* [https://xnet-x.net/es/ Xnet]

==== <span style="color: rgb(0, 0, 0); font-style: italic;">Practising lawyers who are specialised in intellectual property and computer law (IP & IT lawyers)</span> ====
* [http://davidbravo.es/ David Bravo]
* [http://javierdelacueva.es/ Javier de la Cueva]
* [http://derechoynormas.blogspot.com.es/ David Maeztu]

==== <span style="color: rgb(0, 0, 0); font-style: italic;">Some open repositories</span> ====
* [https://archive.org/ Internet Archive: Non-profit digital library offering free universal access to books, movies &amp; music, as well as 386 billion archived web pages (Wayback Machine).]
* [http://www.bitacoravirtual.cl/2011/03/05/repositorios-de-proyectos-de-arquitectura/ Algunos sobre proyectos de arquitectura (además de software libre para arquitectura) (CC BY-NC-SA)]
* [http://www.infotecarios.com/hackathon-repositorios-de-datos-abiertos-open-data-segunda-parte/ Infotecarios: Hackathon, Repositorios de Datos Abiertos – Open Data (segunda parte) (CC BY-NC-ND)]

==== <span style="color: rgb(0, 0, 0); font-style: italic;">Take into account the right of quotation</span> ====
* [http://ucinema.sim.ucm.es/video/ucm/demanda/3jornadacv_013.html Lema, Carlos (2006) Guía de buenas prácticas acordes con la Ley de Propiedad Intelectual; III Jornada Campus Virtual UCM (gratis OA).]
* [http://www.bufetalmeida.com/503/el-derecho-de-cita-con-fines-docentes.html Sánchez Almeida, Carlos (2009) El derecho de cita con fines docentes (CC BY-NC-SA).]
* [http://de.vroniplag.wikia.com/wiki/Home Universidad de Granada (2009) Derechos de autor en plataformas e-learning (CC BY-NC-SA)]

==== <span style="color: rgb(0, 0, 0); font-style: italic;">And the possible plagiarism</span> ====
* [https://scholarlyoa.com/category/plagiarism-2/ Beall, J. Plagiarism. ''Scholarly Open Access''. © gratis OA.]
* [http://copy-shake-paste.blogspot.com.es/ Copy, Shake, and Paste. A blog about plagiarism and scientific misconduct (CC BY-NC-SA)]
* [[:es:Usuario:Jmleonrojas/Taller/Anti-plagio|Usuario:Jmleonrojas/Taller/Anti-plagio]]
* [http://de.vroniplag.wikia.com/wiki/Home VroniPlag Wiki (CC BY-SA)] (more information on [http://en.wikipedia.org/wiki/VroniPlag_Wiki Wikipedia])
:---
* [[Plagiarism]]
* [[Plagiarism detection]]
* [[Wikipedia:Copying within Wikipedia]]

==== <span style="color: rgb(0, 0, 0); font-style: italic;">Libraries (texts, courses)</span> ====
* [http://alqua.org Alqua. (© FLOK).]
* [http://aimath.org/textbooks/approved-textbooks/ American Institute of Mathematics (AIM): Approved Textbooks (open textbooks). (© FLOK).]
* [https://www.wdl.org/es/ Biblioteca Digital Mundial] / [https://www.wdl.org/en/ World Digital Library (WDL) (UNESCO)]
* [http://www.cervantesvirtual.com/ Biblioteca Virtual Miguel de Cervantes (España)]
* [http://www.biblioteca.org.ar/ Biblioteca Virtual Universal (Argentina)]
* [http://bookboon.com/ Bookboon (© gratis OA)]
* [https://www.boundless.com/ Boundless (© CC BY-SA)]
* [http://www.ck12.org/ CK-12]
* [http://collegeopentextbooks.org/textbook-listings/textbooks-by-subject/math College Open Textbooks: Math]
* [http://oerconsortium.org/discipline-specific/ Community College Consortium for Open Educational Resources: Open Textbooks]
* [http://www.elastico.net/copyfight/listing.php?sec=0 CopyFight]
* [http://www.siyavula.com/work-oer.html Free High School Science Texts Project]
* [http://www.freetechbooks.com/numerical-methods-f66.html FreeTechBooks]
* Proyecto / Project LATIn, [http://www.proyectolatin.org/index.php/es/component/booklibrary/512/all_categories libros publicados], [http://comunidad.proyectolatin.org/groups/all comunidades] y [http://comunidad.proyectolatin.org/groups/writing_groups/ grupos de escritura].
* [https://www.merlot.org Merlot]
* [http://problems.magmath.com/math/textbooks MagMath]
* [http://ocw.mit.edu/courses/online-textbooks/ MIT OpenCourseWare: Online Textbooks]
* [https://www.oercommons.org/ OER Commons]
* [https://www.openbookpublishers.com Open Book Publishers]
* [http://opencourselibrary.org/ Open Course Library]
* [http://www.openculture.com/free_textbooks Open Culture: Free Textbooks]
* [https://oli.cmu.edu/ Open Learning Initiative (Carnegie Mellon University)]
* [http://textbooks.opensuny.org/ Open SUNY Textbooks]
* [http://www.opentextbookstore.com/catalog.php Open Textbook Store]
* [http://www.etnassoft.com/biblioteca/tag/matematicas/ OpenLibra]
* [https://openstax.org/ OpenStax]
* [http://cnx.org/ OpenStax CNX]
* [https://florida.theorangegrove.org/og/hierarchy.do?topic=d37c6ed5-3822-84a6-721c-6d9033a88541&page=1 Orange Grove Texts]
* [http://www.oreilly.com/openbook/ O'Reilly Open Books Project]
* [http://www.gutenberg.org/ Project Gutenberg]
* [https://sites.google.com/site/redmatematicaantioquia/home/libros-de-la-red Red Matematica Antioquia (© gratisOA)]
* [http://www.saylor.org/ Saylor Academy (© CC BY-SA)]
* [http://scholarcommons.usf.edu/oa_textbooks/ Scholar Commons (University of South Florida)]
* [http://www.studentpirgs.org/open-textbooks/catalog Student PIRGS: affordable textbooks initiative], conjuntamente con la Universidad de Minesota creando la [http://open.umn.edu/opentextbooks/ Open Textbook Library]
* [http://textbookequity.org/ Textbook Equity LLC]
* [http://textbookrevolution.org/index.php/Main_Page Textbook Revolution]
* [http://textosmareaverde.blogspot.com.es/ Textos Marea Verde]
* [http://www.theassayer.org/ The Assayer (© OPL, without A or B)]
* [http://onlinebooks.library.upenn.edu/ The Online Books Page]. (Véase, también: [http://onlinebooks.library.upenn.edu/licenses.html Copyrights and licenses] y [http://onlinebooks.library.upenn.edu/archives.html Archives and Indexes]).
* [https://unedabierta.uned.es/ Universidad Nacional de Educación a Distancia (UNED) Abierta]
* [https://wiki.creativecommons.org/wiki/Free_to_Learn_Guide/Index_of_OER_Resources Más...]
</div>
|-
<!--          INFORMACIÓN TRANSVERSA 2 / TRANSVERSE INFORMATION 2          -->
| style="padding:2px;" | <h2 id="pamdan-it1-h2" style="margin:3px; background:#deecc8; font-family:inherit; font-size:small; font-weight:bold; border:1px solid #deecc8; text-align:left; color:rgb(0, 0, 0); padding:0.2em 0.4em;">Transverse information 2.- About more topics of interest<hr style="width: 100%; height: 4px; background-color: rgb(0, 102, 0);" /></h2>
|-
| style="color:#000; padding:2px 5px 5px;" | <div id="pamdan-it2" style="font-size:small;">

==== <span style="color: rgb(0, 0, 0); font-style: italic;">Accessibility and usability</span> ====
* [http://alistapart.com/ A List Apart]
* [https://www.nngroup.com/articles/usability-101-introduction-to-usability/ Nielsen, Jacob (2012) Usability 101: Introduction to Usability.''Nielsen Norman Group Weekly Newsletter, January 4''. © ARR, Acceso Gratis).]
* [https://www.w3.org/ World Wide Web Consortium (W3C)]
* [http://www.w3schools.com/ W3Schools Online Web Tutorials]
:---
* [[Accessibility|Accessibility. In ''Wikipedia, The Free Encyclopedia''. (© CC BY-SA).]]
* [[Usability|Usability. In ''Wikipedia, The Free Encyclopedia''. (© CC BY-SA).]]
* [[Web usability|Web usability. In ''Wikipedia, The Free Encyclopedia''. (© CC BY-SA).]]

==== <span style="color: rgb(0, 0, 0); font-style: italic;">Library</span> ====
* [https://openlibrary.org/works/OL3457014W/Two_against_one Caplow, Theodore (1968) ''Two against one: Coallitions in Triads''. Prentice Hall, Inc., Englewood Cliffs, New Jersey, U.S.A.] (Versión española de Natividad Sánchez Sáinz-Trápaga: [http://www.worldcat.org/title/dos-contra-uno-teoria-de-coaliciones-en-las-triadas/oclc/48348679/editions?referer=di&editionsView=true ''Dos contra uno: Teoría de coaliciones en las tríadas''. Alianza Editorial, S. A., Madrid, España, 1974]).

==== <span style="color: rgb(0, 0, 0); font-style: italic;">Fair trade</span> ====

===== <span style="color: rgb(0, 0, 0); font-style: italic;">— Technology</span> =====
* [https://www.fairphone.com/es/ Fairphone]

==== <span style="color: rgb(0, 0, 0); font-style: italic;">Hacker ethic</span> ====
* [http://www.eldiario.es/turing/Libros-cultura_hacker-Wikileaks_0_284871542.html Alan Lazalde (2014) Libros imprescindibles para entender la cultura hacker, ''Diario Turing. Eldiario.es''. © CC BY-SA.]
* [http://www.elmundo.es/tecnologia/2014/10/27/544dea28ca474156028b456b.html Pablo Romero (2014) Richard Stallman: 'La RAE debe corregir la definición de hacker', ''El mundo''. © ARR.]
:---
* [[Hacker culture|Hacker culture. In ''Wikipedia, The Free Encyclopedia''. (© CC BY-SA).]]
* [[Hacker ethic|Hacker ethic. In ''Wikipedia, The Free Encyclopedia''. (© CC BY-SA).]]

==== <span style="color: rgb(0, 0, 0); font-style: italic;">Philosophy</span> ====
* [http://www.dmoz.org/World/Espa%C3%B1ol/Sociedad/Filosof%C3%ADa/ DMOZ (en varias lenguas)]
* [http://www.filosofia.org/pcero.htm '''Proyecto Filosofía en español (Fundación Gustavo Bueno)''']
* [http://www.iep.utm.edu/ Internet Encyclopedia of Philosophy (en inglés)]
* [http://www.philosophica.info/ Philosophica]
* [http://philosophy.stackexchange.com/ Philosophy Stack Exchange (en inglés)]
* [http://philpapers.org/ PhilPapers (en inglés)]
* [http://plato.stanford.edu/ '''Stanford Encyclopedia of Philosophy (en inglés)''']
* [http://www.informationphilosopher.com/ The Information Philosopher (en inglés)]
:---
* [[Philosophy|Philosophy. In ''Wikipedia, The Free Encyclopedia''. (© CC BY-SA).]]

==== <span style="color: rgb(0, 0, 0); font-style: italic;">Periodical library</span> ====
* [http://revistas.ucm.es/index.php/TEKN Teknokultura. (© CC BY-NC-ND)]

==== <span style="color: rgb(0, 0, 0); font-style: italic;">Tools</span> ====
* [[Atom_(text_editor)|ATOM (text editor hackable to the core) (© FLOSS).]]
* [[Etherpad#Etherpad Lite|Etherpad Lite (real-time collaborative online editor).]]
** [https://github.com/ether/etherpad-lite Código fuente.]
** [https://github.com/ether/etherpad-lite/wiki/Sites-that-run-Etherpad-Lite Sitios que lo proporcionan], por ejemplo, el colectivo [https://www.systemli.org/en/index.html Systemli.org] o el [http://www.piratpartiet.se/ Swedish Pirate Party / Partido Pirata sueco] con [http://piratepad.net/ PiratePad].

==== <span style="color: rgb(0, 0, 0); font-style: italic;">LaTeX</span> ====
* [http://cervantex.es/ '''CervanTeX''']
* [http://detexify.kirelabs.org/classify.html Detexify]
* [http://www.codecogs.com/latex/eqneditor.php Editor en línea de ecuaciones en LaTeX (CodeCogs)]
* [http://rinconmatematico.com/latexrender/ Editor en línea de ecuaciones en LaTeX (Rincón Matemático)]
* [https://www.overleaf.com/latex/examples Ejemplos (en Overleaf)]
* [http://alexeev.org/gmailtex.html LaTeX para Gmail]
* [http://tex.stackexchange.com LaTeX Stack Exchange]
* [http://papeeria.com/templates Modelos (en Papeeria)]
* [https://docs.moodle.org/32/en/Using_TeX_Notation Moodle LaTeX]
* [http://osl.ugr.es/CTAN/info/symbols/comprehensive/symbols-a4.pdf Pakin, Scott (2015). The Comprehensive LaTeX Symbol List.]
* [https://www.sharelatex.com/learn ShareLaTeX]
* [http://texample.net/ TeXample.net]
* [https://texblog.org/ TeXblog]
* [http://texnique.fr TeXnique]
* [https://www.latex-project.org// '''The LaTeX Project''']
* [http://makingtexwork.sourceforge.net/mtw/ Walsh, N. (2002) Making TeX Work. O'Reilly.]
* [http://asymptote.sourceforge.net/ Asymptote]
* [http://asy.marris.fr/asymptote/ Asymptote (en asy.marris.fr)]
* [http://www.piprime.fr/asymptote/ Asymptote (en piprime.fr)]
* [http://pgfplots.net/ PGFPlots.net]
:---
* [[LaTeX|LaTeX. In ''Wikipedia, The Free Encyclopedia''. (© CC BY-SA).]]
* [[Wikipedia:LaTeX_symbols|Wikipedia:LaTeX_symbols. In ''Wikipedia, The Free Encyclopedia''. (© CC BY-SA).]]

==== <span style="color: rgb(0, 0, 0); font-style: italic;">Spanish language</span> ====
:([[Spanish language]])
* [http://www.rae.es/ Real Academia Española (RAE) [Royal Spanish Academy]]
* [http://www.asale.org/ '''Asociación de Academias de la Lengua Española (ASALE).''' (© gratisOA).] (Para saber de otras asociaciones de la lengua española en el mundo, consúltese, por ejemplo: [https://es.wikipedia.org/wiki/Asociaci%C3%B3n_de_Academias_de_la_Lengua_Espa%C3%B1ola#Otras_asociaciones_de_lengua_espa.C3.B1ola_en_el_mundo Wikipedia [© CC BY-SA]]).
* [http://www.cilengua.es/ Centro Internacional de Investigación de la Lengua Española (Cilengua).]
* [http://cvc.cervantes.es/ Centro Virtual Cervantes (CVC).]
* [http://dle.rae.es Diccionario de la Lengua Española.]
* [http://lema.rae.es/dpd/ Diccionario panhispánico de dudas (DPD).]
* [http://diccionario.raing.es/ Diccionario español de ingeniería (DEI).]
* [https://virgulilla.wordpress.com/ Español (con virgulilla).]
* [http://www.caminodelalengua.com/ Fundación Camino de la Lengua Castellana.]
* [http://www.fundeu.es/ Fundación del Español Urgente.]
** [http://www.wikilengua.org/index.php/Portada Wikilengua.]
* [http://www.cervantes.es Instituto Cervantes.]
* [http://www.rae.es/sites/default/files/EL_NEOLOGISMO_EN_EL_DICCIONARIO.pdf Fernando Lázaro Carreter (2002): El neologismo en el Diccionario.]
* [http://www.lexipedia.com/spanish/Bienvenido Lexipedia.]
* [https://www.practicaespanol.com/ '''Practica Español''']
* [https://examendemo.siele.org/ '''SIELE (Servicio Internacional de Evaluación de la Lengua Española) (Práctica de examen)'''.]
* [http://www.rae.es/sites/default/files/Principales_novedades_de_la_Ortografia_de_la_lengua_espanola.pdf RAE (2010): Principales novedades de la ortografía de la lengua española.]
===== <span style="color: rgb(0, 0, 0); font-style: italic;">— U.S. Spanish</span> =====
* [http://www.anle.us/?sfl=es Academia Norteamericana de la Lengua Española (ANLE) [North American Academy of the Spanish Language]]
===== <span style="color: rgb(0, 0, 0); font-style: italic;">— Collocations dictionaries</span> =====
:([[Collocation]])
* [http://www.dicesp.com DICE]

===== <span style="color: rgb(0, 0, 0); font-style: italic;">— Spelling and grammar checkers</span> =====
* [http://spanishchecker.com/ SpanishChecker]

===== <span style="color: rgb(0, 0, 0); font-style: italic;">— Plain Spanish</span> =====
:([[Plain language]])
* [http://www.lecturafacil.net/es/ Asociación Lectura Fácil]
* [https://publications.europa.eu/es/publication-detail/-/publication/c2dab20c-0414-408d-87b5-dd3c6e5dd9a5 '''Cómo escribir con claridad, por la Comisión Europea''']
* [http://www.facillectura.es/ Extremadura vive la fácil lectura]
* [http://plainlanguagenetwork.org/plain-language/que-es-el-lenguaje-claro/ ¿Qué es el lenguaje claro?, por Plain Language Association International (PLAIN)]

==== <span style="color: rgb(0, 0, 0); font-style: italic;">English language</span> ====
:([[English language]])
* [https://learnenglish.britishcouncil.org/en/content '''Test and practise your English''' (by the British Council)]
===== <span style="color: rgb(0, 0, 0); font-style: italic;">— Academic English</span> =====
* [http://www.cambridge.org/gb/cambridgeenglish/catalog/grammar-vocabulary-and-pronunciation/academic-vocabulary-use-2nd-edition/academic-vocabulary-use-2nd-edition-edition-answers '''Michael McCarthy & Felicity O'Dell: Academic Vocabulary in Use, Cambridge University Press, 2nd ed., 2016. (© ARR).''']
* [http://www.uefap.com/ Using English for Academic Purposes: A Guide for Students in Higher Education (UEfAP)] y el / and the [http://www.uefap.com/test/ English Language Level Test].
* [http://writingguide.se/ Writing Guide.]
* '''[https://owl.purdue.edu/owl/purdue_owl.html Purdue Online Writing Lab]'''
===== <span style="color: rgb(0, 0, 0); font-style: italic;">— American English</span> =====
* [https://americanenglish.state.gov/resources-0 '''American English''' (U.S. Department of State)]
* [https://www.wikihow.com/Fake-a-Convincing-American-Accent How to Fake a Convincing American Accent. ''WikiHow''. (© gratisOA)]
* [http://www.merriam-webster.com/ '''Merriam-Webster''']
* [http://nws.merriam-webster.com/opendictionary/ '''Merriam-Webster: The Open Dictionary - New Words & Slang''']
* [http://www.thepunctuationguide.com/ The Punctuation Guide]
* [https://en.wikipedia.org/wiki/The_Elements_of_Style William Strunk (Jr.) (1918/1920), Elwin Brooks White (1959) ''The Elements of Style''. Harcourt, Brace &amp; Howe (1920), Macmillan (1959).]

===== <span style="color: rgb(0, 0, 0); font-style: italic;">— British English</span> =====
* [https://www.wikihow.com/Speak-in-a-British-Accent How to Speak in a British Accent. ''WikiHow''. (© gratisOA)]
* [http://queens-english-society.org/ The Queen's English Society. (© gratisOA)]

===== <span style="color: rgb(0, 0, 0); font-style: italic;">— Collocations dictionaries</span> =====
* [http://www.ozdic.com/ Ozdic]
===== <span style="color: rgb(0, 0, 0); font-style: italic;">— Dictionaries and thesauri</span> =====
* [http://dictionary.cambridge.org/ '''Cambridge''' dictionaries, grammar and thesaurus, (British & American)]
* [https://www.collinsdictionary.com/ Collins (multiple languages)]
* [http://www.linguee.es/ '''Linguee''' (multiple languages)]
* [http://www.onelook.com/ OneLook]
* [https://www.oxforddictionaries.com/ '''Oxford''' dictionaries, grammar and thesaurus, (British & American)]
* [https://www.urbandictionary.com/ '''Urban dictionary''']
* [http://www.wordreference.com/ '''WordReference.com''' (multiple languages)]

===== <span style="color: rgb(0, 0, 0); font-style: italic;">— English and mathematics</span> =====
* [http://www.oxfordreference.com/view/10.1093/acref/9780199679591.001.0001/acref-9780199679591 '''Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics, Oxford University Press, 5th ed., 2014. (© ARR).''']
* [https://eu.ixl.com/ IXL | Maths and English Practice.]
===== <span style="color: rgb(0, 0, 0); font-style: italic;">— Plain English</span> =====
* [https://www.linkedin.com/learning/writing-in-plain-english Judy Steiner-Williams: Writing in plain English (This course qualifies for professional development units (PDUs))]
* [http://www.bradford.ac.uk/external-affairs/web-team/web-style-guide-and-content-tips/plain-english/ '''University of Bradford (GB): Write in Plain English.''']
* [http://www.bristol.ac.uk/arts/exercises/grammar/grammar_tutorial/index.htm University of Bristol (GB): Improve your writing: grammar exercises]
* [http://sydney.edu.au/law/learning_teaching/legal_writing/plain_english.shtml '''University of Sidney (AU): Writing in Plain English.''']
* [https://global.oup.com/academic/product/oxford-guide-to-plain-english-9780199669172?cc=fr&lang=en& <strong>Martin Cutts. <i>Oxford Guide to Plain English</i>. Oxford University Press, Oxford, United Kingdom, , 4th edition, 2013. ISBN 978-0-19-966917-1</strong>]
* [http://ogden.basic-english.org/ Charles K. Ogden's Basic English], e. g., the [https://simple.wikipedia.org/wiki/Main_Page Simple English Wikipedia.]
* [http://www.clarity-international.net/plain-language-resources/general-resouces/ Clarity: General Resources]
* [http://plainlanguagenetwork.org/ Plain Language Association International (PLAIN)]
** [https://www.gov.uk/guidance/content-design/writing-for-gov-uk#plain-english Writing for GOV.UK]
** [http://www.plainlanguage.gov/resources/index.cfm Plain Language.gov (USA): Plain-Language Training Resources]
* [http://centerforplainlanguage.org/ Center for Plain Language]
** [http://centerforplainlanguage.org/plain-language-supports-science-communication/ Plain language supports science communication]
** [http://centerforplainlanguage.org/the-importance-of-plain-language-for-content-creation/ The importance of plain language for content creation]
* [http://www.plainenglish.co.uk/ '''Plain English Campaign (Fighting for crystal-clear communication since 1979).''']
** [http://www.plainenglish.co.uk/free-guides.html Free Guides]
** [http://www.plainenglish.co.uk/drivel-defence-text.html '''Drivel Defence for Text''']
* [http://hemingwayapp.com/ '''Hemingway Editor''']
* [https://www.plainenglishfoundation.com/free-writing-tools Plain English Foundation: Free Writing Tools]
* [http://www.clearest.co.uk/ Plain Language Commission]
** {{cite web
 |url=https://irp-cdn.multiscreensite.com/aaf9e928/files/uploaded/Plain_English_LEXICON_June_2011.pdf
 |title=Plain English Lexicon
 |last=Cutts
 |first=Martin
 |language=
 |date=
 |website=
 |publisher=
 |access-date=
 |quote= }}
** [http://www.clearest.co.uk/publications Publications]
* [https://www.globish.com Globish]

==== <span style="color: rgb(0, 0, 0); font-style: italic;">Mathematics</span> ====
===== <span style="color: rgb(0, 0, 0); font-style: italic;">— Web sites</span> =====
* [[:Category:Unsolved problems in mathematics]]
* [http://ztfnews.wordpress.com/ ::ZTFNews.org]
* [http://andrescaicedo.wordpress.com/ A kind of library]
* [http://www.youtube.com/user/AlgoRythmics AlgoRythmics]
* [http://www.ams.org/ '''American Mathematical Society (AMS)''']
* [http://artofproblemsolving.com/wiki/ AoPS: Art of Problem Solving]
* [http://bit-player.org/ bit-player]
* [http://blocdemat.wordpress.com/ blocdemat]
* [http://blog.i-math.org/ Blog para anti-matemáticos]
* [http://cafematematico.com/ Café matemático]
* [http://cifrasyteclas.com/ Cifras y teclas]
* [http://www.divulgamat.net/ divulgaMAT]
* [http://www.matematicas.net/ El Paraíso de las Matemáticas]
* [https://www.encyclopediaofmath.org/index.php/Main_Page '''Encyclopedia of Mathematics''']
* [http://www.epsilones.com/paginas/a-mapa.html Epsilones]
* [http://www.arrakis.es/~mcj/index.htm Gacetilla matemática]
* [http://gaussianos.com/ Gausianos]
* [http://www.geogebra.org/cms/es/ GeoGebra]
* [http://www.blogseitb.com/matematicas/ Grafitti matemático]
* [http://estadistica.gobex.es Instituto de Estadística de Extremadura]
* [http://www.ine.es/ Instituto Nacional de Estadística (INE)]
* [http://topologia.wordpress.com/ Juegos topológicos]
* [http://www.mpdf.es/ lasmatematicas.es en pdf]
* [http://www-history.mcs.st-and.ac.uk/ '''MacTutor History of Mathematics archive''']
* [http://www.matematicalia.net/ Matematicalia (© gratis OA)]
* [http://matematicascercanas.com/ Matemáticas cercanas. (© CC BY-NC-ND).]
* [http://www.matematicasdigitales.com/ Matemáticas digitales]
* [http://catedu.es/matematicas_mundo/index.html Matemáticas en tu mundo]
* [http://i-matematicas.com/blog/ Matemáticas interactivas y manipulativas]
* [http://www.madrimasd.org/blogs/matematicas/ Matemáticas y sus fronteras]
* [http://matesmates.wordpress.com/ ¡Mates, Mates!]
* [http://www.matetam.com/ MaTeTaM]
* [http://www.mathematik.com/ Mathematik]
* [http://www.mathsfact.com/ Mathsfact]
* [http://mathsmovies.com/ Mathsmovies]
* [http://mathworld.wolfram.com/ MathWorld (© ARR)]
* [http://blogs.20minutos.es/mati-una-profesora-muy-particular/ Mati, una profesora muy particular]
* [http://pequenoldn.librodenotas.com/matiaventuras Mati y sus mateaventuras]
* [http://memorandummatematico.wordpress.com/ Memorandum Matemático]
* [http://mate.dm.uba.ar/~lechague/zona.htm Museo interactivo de Matemáticas]
* [http://naukas.com/ Naukas]
* [https://www.bbvaopenmind.com/en/topic/mathematics/ OpenMind (© ARR)]
* [http://cms.dm.uba.ar/material/paenza Adrián Paenza: Libros de divulgación]
* [http://www.planetamatematico.com/ Planeta Matemático]
* [http://planetmath.org/ PlanetMath (© CC BY-SA)]
* [http://www.fmat.cl/ Portal Matemático (fmat)]
* [http://recursostic.educacion.es/descartes/web/ Proyecto Descartes]
* [https://www.quantamagazine.org/computer-science/ Quanta Magazine: Computer Science (© ARR)]
* [https://www.quantamagazine.org/mathematics/ Quanta Magazine: Mathematics (© ARR)]
* [http://www.rsme.es/ '''Real Sociedad Matemática Española''']
* [http://www.tec-digital.itcr.ac.cr/revistamatematica/ Revista Digital Matemática, Educación e Internet]
* [http://rinconmatematico.com/ Rincón Matemático]
* [http://www-history.mcs.st-and.ac.uk/ The MacTutor History of Mathematics archive]
* [http://oeis.org/ The On-line Encyclopedia of Integer Sequences]
* [http://tiopetrus.blogia.com/ Tio Petros]
* [http://eliatron.blogspot.com.es/ Tito Eliatron Dixit]
* [http://mateturismo.wordpress.com/ Turismo Matemático]
* [http://portales.educared.net/wikillerato/Matematicas Wikillerato (© CC BY)]
* [http://www.wolframalpha.com/examples/ Wolfram|Alpha]
* [http://mathworld.wolfram.com/ WolframMathWorld]
* [http://yair.es/ Yair.es]

===== <span style="color: rgb(0, 0, 0); font-style: italic;">— Android apps</span> =====
* [http://www.malmath.com/ MalMath]
:---
* [[Mathematics|Mathematics. In ''Wikipedia, The Free Encyclopedia''. (© CC BY-SA).]]

==== <span style="color: rgb(0, 0, 0); font-style: italic;">Programming</span> ====
===== <span style="color: rgb(0, 0, 0); font-style: italic;">— Languages</span> =====
* [[Free Pascal]]
** [[:b:Pascal_Programming|Wikibook 'Pascal Programming']]
* [[Python (programming language)]]
** [https://www.python.org/shell/ Python Shell]
** [[:b:Category:Shelf:Python_programming_language|Shelf of wikibooks about 'Python programming language']]
** [https://www.w3schools.com/python/default.asp w3school Python Tutorial]
* [[SageMath]]
** [https://sagecell.sagemath.org/ SageMathCell]
** [https://cocalc.com/ CoCalc]

===== <span style="color: rgb(0, 0, 0); font-style: italic;">— Online interpreters</span> =====
* [http://ideone.com/ Ideone]
* [https://rextester.com/ Rextester]
* [https://tio.run Try It Online]

==== <span style="color: rgb(0, 0, 0); font-style: italic;">Digital preservation</span> ====
* [https://wiki.archivematica.org/Main_Page Archivematica]
* [http://daitss.fcla.edu/ DAITSS]
* [http://www.ifs.tuwien.ac.at/dp/hoppla/ Hoppla]
* [http://jhove.sourceforge.net/ JHOVE]
* [http://www.roda-community.org/ RODA]
:---
* [[Digital preservation|Digital preservation. In ''Wikipedia, The Free Encyclopedia''. (© CC BY-SA).]]

==== <span style="color: rgb(0, 0, 0); font-style: italic;">Knowledge representation</span> ====
* [http://skos.um.es/ Simple Knowledge Organization System (SKOS)]
* [http://www.jfsowa.com/krbook/ Sowa, John F. (2000) ''Knowledge Representation. Logical, Philosophical, and Computational Foundations''. Brooks/Cole, Thomson Learning, Pacific Grove, CA, USA.]
:---
* [[Knowledge representation and reasoning|Knowledge representation and reasoning. In ''Wikipedia, The Free Encyclopedia''. (© CC BY-SA).]]

==== <span style="color: rgb(0, 0, 0); font-style: italic;">Computer security</span> ====
* [http://www.kriptopolis.com/ Kriptópolis | Criptografía y Seguridad]
* [https://nmap.org/ Nmap], incluyendo el repositorio [https://nmap.org/book/nse-scripts.html NSE].
* [http://www.elladodelmal.com/ Un informático en el lado del mal.]
:---
* [[Computer security|Computer security. In ''Wikipedia, The Free Encyclopedia''. (© CC BY-SA).]]

==== <span style="color: rgb(0, 0, 0); font-style: italic;">Free and open-source software</span> ====
* [http://alternativeto.net/ AlternativeTO]
* [http://www.findbestopensource.com/home/ Best Open Source]
* [http://libreprojects.net Libre Projects]
* [http://www.linuxadictos.com/alternativas-linux-programas-windows.html Linux Adictos]
* [http://linuxappfinder.com/alternatives Linux App Finder]
* [https://www.openhub.net/ Open Hub]
* [http://www.opensourcealternative.org/ Open Source Alternative]
* [http://osliving.com/ Open Source Living]
* [http://www.osalt.com/ Osalt]
* [http://ostatic.com/ OStatic]
* [http://directory.fsf.org/ The Free Software Directory]
* [http://www.linuxalt.com/ The Linux Alternative Project]
* [http://www.zwodnik.com/ Zwodnik]
* [http://www.alternativepedia.com/ AlternativePedia]
:---
* [[Free and open-source software|Free and open-source software. En ''Wikipedia, The Free Encyclopedia''. (© CC BY-SA).]]
* [[Alternative terms for free software|Alternative terms for free software. En ''Wikipedia, The Free Encyclopedia''. (© CC BY-SA).]]

<span style="color: rgb(0, 0, 0);">— &amp; —</span>

==== <span style="color: rgb(0, 0, 0); font-style: italic;">Snoozing</span> ====
* [https://arstechnica.com/ Ars Technica (© ARR)]
* [http://barrapunto.com Barrapunto (© ARR)]
* [http://elastico.net/ Elástico 3.0 (© CC BY-NC)]
* [https://geekgazette.org/ Geek Gazette (© ARR)]
* [http://www.genbeta.com/ Genbeta (© ARR)]
* [http://www.genbetadev.com/ Genbeta Dev (© ARR)]
* [https://hackernoon.com/ Hackernoon (© ARR)]
* [https://medium.com/ Medium (© ARR)]
* [http://www.microsiervos.com/ Microsiervos (© ARR)]
* [https://www.bbvaopenmind.com/en/topic/computing/ OpenMind (© ARR)]
* [https://www.redeszone.net/ Redes Zone (© ARR)]
* [https://slashdot.org/ Slashdot (© ARR)]
* [https://www.wired.com/ Wired (© ARR)]

==== <span style="color: rgb(0, 0, 0); font-style: italic;">Bubbles</span> ====
* [http://www.cogsci.uci.edu/~ddhoff/ Donald D. Hoffman (en inglés)]
* [http://www.careercast.com/jobs-rated/best-jobs-2015 The Best Jobs of 2015 (CareerCast) (en inglés)]
* {{cite magazine
| last       = Britton
| first      = Jack
| date       = November 17, 2017
| title      = The degrees that make you rich... and the ones that don't
| url        = http://www.bbc.com/news/education-41693230
| magazine   = BBC News
| location   = 
| publisher  = 
| access-date=November 17, 2017
}}

==== <span style="color: rgb(0, 0, 0); font-style: italic;">Miscellanea</span> ====
* [http://101claves.com/ 101claves]
* [https://99percentinvisible.org/ 99 % invisible (© TDR)]
* [https://culturacolectiva.com/en/ Cultura Colectiva + (© TDR)]
* [http://www.domestika.org/ Domestika]
* [http://dondeestamiequipo.net/ Mario Kogan y José Ochoa: ¿Dónde está mi equipo? / Where is my team?]
* [http://jordiclaramonte.blogspot.com.es/ Estética y Teoría del Arte]
* [http://www.joshuadavis.com/ Joshua Davis Studios]
* [http://kunsthallewien.at/#/en Kunsthalle Wien]
* [https://www.bbvaopenmind.com/en/books/ OpenMind books (© ARR)]
* [https://www.trecebits.com/ TreceBits (© ARR)]
* [http://yugop.com/ Yugop]

==== <span style="color: rgb(0, 0, 0); font-style: italic;">Wikimedia, in English</span> ====

===== <span style="color: rgb(0, 0, 0); font-style: italic;">— Wikimedia</span> =====
* [[outreach:Education/Case Studies|Wikimedia outreach: Education: Case_Studies]]
* [[outreach:Coursework feedback|Wikimedia outreach: Coursework feedback]]

===== <span style="color: rgb(0, 0, 0); font-style: italic;">— Wikipedia</span> =====
* [[Wikipedia:Welcoming committee/Welcome to Wikipedia|Welcome to Wikipedia]]
* [[Wikipedia:Training|Wikipedia: Training]]
* [[Wikipedia:Education program|Wikipedia: Education program]]
* [[Wikipedia:School and university projects|Wikipedia: School and university projects]]
* [[Wikipedia:Requested articles|Wikipedia: Requested articles]]
* [[Wikipedia:WikiProject Mathematics|Wikipedia: WikiProject Mathematics]]
* [[Help:Cheatsheet|Help: Cheatsheet]]

===== <span style="color: rgb(0, 0, 0); font-style: italic;">— Wikibooks</span> =====
* [[b:Wikibooks:Welcome|Wikibooks: Welcome]]
* [[b:Wikibooks:Requested books|Wikibooks: Requested books]]

==== <span style="color: rgb(0, 0, 0); font-style: italic;">Wikimedia, in Spanish</span> ====

===== <span style="color: rgb(0, 0, 0); font-style: italic;">— Wikipedia</span> =====
* [https://es.wikipedia.org/wiki/Wikipedia:Bienvenidos Wikipedia: Bienvenidos]
* [https://es.wikipedia.org/wiki/Wikipedia:Bienvenidos_alumnos Wikipedia: Bienvenidos alumnos]
* [https://es.wikipedia.org/wiki/Wikipedia:Bienvenidos_profesores Wikipedia: Bienvenidos profesores]
* [https://es.wikipedia.org/wiki/Wikipedia:Programa_educativo Wikipedia: Programa educativo]
* [https://es.wikipedia.org/wiki/Wikipedia:Proyectos_educativos Wikipedia: Proyectos educativos]
* [https://es.wikipedia.org/wiki/Wikipedia:Artículos_solicitados Wikipedia: Artículos solicitados]
* [https://es.wikipedia.org/wiki/Wikiproyecto:Matemáticas Wikiproyecto: Matemáticas]
* [https://es.wikipedia.org/wiki/Wikipedia:Wikipedia_en_CD/2009/Matemáticas Wikipedia en CD (2009): Matemáticas]
* [https://es.wikipedia.org/wiki/Ayuda:Referencia_rápida Ayuda: Referencia rápida]

===== <span style="color: rgb(0, 0, 0); font-style: italic;">— Wikilibros</span> =====
* [https://es.wikibooks.org/wiki/Wikilibros:Bienvenida Wikilibros: Bienvenida]
* [https://es.wikibooks.org/wiki/Wikilibros:Wikilibros_solicitados Wikilibros: Wikilibros solicitados]
</div>
|}
|}