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

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This university learning plan consists of a primer on discrete mathematics and its applications including a brief introduction to a few numerical analysis.

It has a special focus on dialogic learning (learning through argumentation) and computational thinking, promoting the development and enhancement of:

My personal experience is the foundation of it. Wikipedia and other information sources, in English and Spanish, support it.

Course educational materials — including video — by others, help to improve this plan.

Using Wikipedia, bibliography, multimedia and others, stimulate and enhance learning through crossover learning, incidental learning, learning by doing, learning by teaching and microlearning, leaving renewed flavours of blended learning such as flipped learning.

Please, be free for suggesting improvements (see this section).


Ex ante Background information Specific information Project on Wikipedia Course outline Paths on Wikipedia Sample exams Real exams Study programme Ex post


Template:TOC limit

Introduction

'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.'
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).

Glossary of abbreviations

Template:Smalldiv

Ex ante I: Mathematics and Computing

(Just as an appetiser)

— Harangues

— Discrete mathematics

— Algorithms

Ex ante II: Theme wikis

— Wikipedia (Starting points)

— Others:

Ex ante III: Graduates in computing (Spain)

Ex ante IIII: Pre-university mathematical literacy

Ex ante V: General motivation

'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.'

Roald Dahl: My Uncle Oswald. London, England (GB-ENG), UK: Penguin Books Limited, 1980, p. 37.

---

Ex ante VI: Specific motivation

— Films

— Prove why it is so

Background information

Universities

— International
— Spain: Institutions, organizations, associations
— Spain: Legislation

University of Extremadura (Spain)

School of Technology (EPCC)

Specific information

— Professor

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 course programme (ficha12a) to find out where it is).

E-mail: jmleon@unex.es.

Office hours.

— Course description

This course is a primer on discrete mathematics and its applications including a very short introduction to a few numerical methods.

UEX code: 501272.

— Rationale

The recommendations included in the Computer Engineering Curricula 2016* and in the Computer Science Curricula 2013†, 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 academic program (ficha12a), 60 hours are programmed as can be seen in a synthetic way in the course outline and scheduled in the tentative course outline (chronogram for the 2019-2020 academic year).

* https://www.computer.org/cms/Computer.org/professional-education/curricula/ComputerEngineeringCurricula2016.pdf
† https://www.acm.org/education/CS2013-final-report.pdf

— Course objectives

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.

— Prerequisites

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*. You might find out your English level taking this free online English test and then you might improve your knowledge of the English language, for instance, practising your English skills at your level, and many more things available on these pages by the British Council (Prince of Asturias Award for Communication and Humanities 2005).

* Please keep in mind that it is enough to know the English language at a CEFR B1 level to apply for British citizenship or to settle in the UK and at a CEFR B2 level to study in the UK at a degree level or above.

— Course program

Academic year 2019-2020
Academic year 2018-2019

— School hours

— Textbooks

— Discrete mathematics
Textbook

For the discrete mathematics part of the course, students are encouraged to use the following book as a textbook:

(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 textbook and a workbook with lots of exercises and practical cases (computer projects, computations and explorations). It is even a guidebook including suggested readings, Despite its encyclopaedic spirit, it is also a handbook including lists of key terms and results and review questions.

Companion website

In addition, it has a companion website: 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:

All these companion websites include, among other material and resources, interactive demos, self assessments and extra examples.

Companion books describing solutions for each of the proposed exercises

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, Student's Solutions Guide (odd exercises) (5th ed., 2003, Template:ISBN) (7th ed., 2012, Template:ISBN, © ARR.
  • Kenneth H. Rosen, Jerrold Grossman, Instructor's Resource Guide (even exercises) (5th ed., 2003, Template:ISBN) (7th ed., 2012, Template:ISBN), © ARR.
Companion books exploring and discussing contents and solutions to the proposed 'computer projects' and 'computations and explorations'

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:

Companion book about applications of discrete mathematics

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:

— Numerical calculus

For the short numerical calculus part of the course, students are encouraged to use the following book as a textbook:

  • Chapra, Steven C., & Canale, Raymond P. (2006) Numerical Methods for Engineers (5th international edition). New York: The McGraw­Hill Companies, Inc. Template:ISBN. © ARR.

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

Please be aware that:

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


— To find out more, while course is running (or once it is finished)

In addition to the references that appear in the course outline and in the academic program (ficha12a), and to those that can be mentioned in the classroom (large group and seminar/lab meetings) or posted on the talk page of the learning plan or at the UEX online campus in the course private forum, and to those that are referenced in the 13 question selections that are used throughout the course, you should consider:

— Communicating

University project on the English Wikipedia

Participating in MATDIN is an optional continuous evaluation out-of-class practical activity which is worth a try for contributing to your personal developmentand because it might help you boost your course grade; furthermore, if you are thinking of grading with distinction ['matrícula de honor', in Spanish], your participation in this project is strongly recommended. Find out more on its descriptive web page and in the welcome message to the course.

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:
  • (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 —;
  • (b) be polite and respect diversity (please remember, diversity is a wealth, neither a problem nor a threat);
  • (c) comply with the rules and obligations laid down by the project coordination for this project (click and read them here), in particular the dynamic commitments (click and read them here);
  • (d) help the individuals involved in the project as much as possible;
  • (e) above all, commit yourself to you.

— On the English-language Wikipedia

— Communicating

To keep track of the project you have joined to, please follow the recommendations on its descriptive page, particularly on 'The basics' subsection.

— Equivalent project on the Spanish-language Wikipedia

(Only if you take the course in Spanish).

Contents and learning paths on Wikipedia

'Do I contradict myself?
Very well then I contradict myself,
(I am large, I contain multitudes.)'
Walt Whitmann (1819-1892): Song of Myself (in Leaves of Grass, 1855)

Considerations

Course outline

Contents: ► 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). ► 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. ► Maps and functions: types (injective, surjective and bijective), monotony, representation (cartesian, arrow-set, matrix-based and graph-based), composition, inverse; multiset. ► Relations: properties, representing relations using matrices and graphs; equivalence relations, equivalence classes and partitions; tolerance relations; orderings, Hasse diagrams; preference relations. ► Cardinality: infinite sets, countability, Cantor's diagonal argument, Cantor's theorem and the continuum hypothesis. ► Induction: weak, strong and structural; well ordering. ► Algebraic structures: magma, semigroup, monoid, group, ring, integral domain, field; homomorphism.
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.
Connections: ...
Contents: ► Divisibility and modular arithmetic: divisibility, division algorithm, modular arithmetic. ► 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. ► 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. ► Applications of congruences: cryptography. ► Divisibility rules: power residues, divisibility rules. ► Diophantine equations: linear equations, systems.
Seminars/Labs: ► [6]: Divisibility, modular arithmetic, primes, gcd and congruences; ► [7]: Diophantine and congruence equations, I; ► [8]: Diophantine and congruence equations, II.
Connections: ...
Contents: ► 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. ► Combinatorial proofs: bijective proofs and double counting proofs. ► Combinatorial modeling: 1st, sample selection and unit labelling with and without repetition; 2nd, grouping units (distribution, storage or placement of objects into recipients); 3rd, partitions of sets, and 4th, partitions of numbers.
Seminars/Labs: ► [9]: Combinatorics, I; ► [10]: Combinatorics, II; ► [11]: Combinatorics, III.
Connections: Game theory; Network science; Number theory; Topology.
Contents: ► Linear difference equations: homogeneous and non-homogeneous; with constant coefficients; direct; simple or multiple; indirect: systems of linear difference equations. ► Linear discrete dynamical systems: population dynamics, linear discrete dynamical models, BIDE models, Markov chains. ► Solving equations numerically: method of successive approximations (fixed point iteration); secant method.
Seminars/Labs: ► [12]: Difference equations, I; ► [13]: Difference equations, II.
Connections: ...
Contents: ► Graphs; Numerical calculus; Complimentary knowledge pills; Editathons.
Connections: ...

WP+: Paths on Wikipedia, bibliography (theory and proposed and solved exercises), multimedia and even more

Using Wikipedia, bibliography, multimedia and others, stimulate and enhance learning through crossover learning, incidental learning, learning by doing, learning by teaching and microlearning, leaving renewed flavours of blended learning such as flipped learning.

Very important warning

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Logic Sets, relations and functions Cardinality, induction and recursion Algebraic structures Number theory Combinatorics Difference equations Appendix: Graphs Appendix: Numerical calculus Appendix: More knowledge pills


Theme 1.- Fundamentals

Logic
File:The-impossibility-of-intent.png
Key concepts

Template:Main Template:Main category Template:See also — Propositional logic Template:Main Template:Main category

See also: Conditioned disjunction, Indicative conditional and Logical constant.

— Verification and rebuttal strategies, I Template:See also

— Predicate logic Template:Main Template:Main category

Template:See also — Translating English statements into the language of logic and vice versa

— Valid arguments and inference rules

See also:

— Direct and indirect proofs

Template:See also — Verification and rebuttal strategies, II

— Verification and rebuttal strategies, III

— Some unusual situations in Logic

Connections

— Automated reasoning Template:Main — Boolean algebra Template:Main

— Diagrammatic reasoning Template:Main — Logic gates Template:Main

Bibliography: theory and proposed and solved exercises
In English: In Spanish:
  • —¤— Kenneth H. Rosen. Discrete mathematics and its applications. New York, New York State (US-NY), United States: McGraw-Hill, 7th edition, 2012. Template:ISBN. (Chapter 1 and related exercises).
  • —¤— Amador Antón y Pascual Casañ, Lógica Matemática. Ejercicios. I. Lógica de enunciados. Valencia, Valencian Community (ES-VC), Spain: NAU llibres, 3rd edition, 1987. Template:ISBN
  • —¤— María Manzano y Antonia Huertas, Lógica para principiantes. Humanes de Madrid, Madrid, Community of Madrid (ES-MD), Spain: Alianza, 2006. Template:ISBN.
  • —¤— Kenneth A. Rosen. Matemática discreta y sus aplicaciones. Aravaca, Madrid, Community of Madrid (ES-MD), Spain: McGraw-Hill/Interamericana de España, S.A.U., 5th edition, 2004. Template:ISBN. (Sections 1.1, 1.2, 1.3, 1.4, 1.5, 3.1 and related exercises).
Software
In English: In Spanish:
  • —¤— Logisim (a graphical tool for designing and simulating logic circuits)] (in Spanish, English and more languages). © GNU GPL.
Multimedia
In English: In Spanish:
See also
In English: In Spanish:
To find out more
  1. Portal:Mathematics
  2. Portal:Philosophy
  3. And more:
    1. Outline of logic
    2. Category:Concepts in logic
    3. WikiProject Logic
    4. Logic alphabet
    5. Metamath. © Public domain (with some exceptions)
    6. Equational logic; for instance, chapter 5 (Equational Logic: Part 1) from Backhouse, Roland, Program Construction. The Correct Way, 2002.
  4. And even more:
    1. Index of logic articles
    2. List of logicians

Template:Logic Template:Classical logic Template:Mathematical logic Template:Metalogic Template:Non-classical logic Template:Philosophical logic Template:Common logical symbols Template:Navbox Template:Logical paradoxes Template:Logical truth

Sets, relations and functions
Error creating thumbnail:
Key concepts
— Sets

Template:Main Template:Main category

- Partition and cover

— Relations

Template:Main Template:Main category Template:See also

- Representation

- Outstanding types

- Tolerance relations
- Indiference and preference relations
- Well order
— Functions

Template:Main Template:Main category

— Paradoxes
Connections

— Extensive systems Template:Main Template:See also Template:Hatnote Template:Hatnote — Entity-relationship model Template:Main Template:See also

Bibliography: theory and proposed and solved exercises
In English: In Spanish:
  • —¤— Kenneth A. Rosen. Discrete mathematics and its applications. 7th edition. (Sections 2.1, 2.2, 2.3, Chapter 9 and related exercises). McGraw-Hill, New York, New York, United States, 2012, Template:ISBN
Software
In English: In Spanish:
Error creating thumbnail: Multimedia
In English: In Spanish:

— Sets

— Relations

— Functions

— Sets

— Relations

— Functions

To find out more
  1. Portal:Mathematics

Template:Set theory

Template:Large Template:Navbar
Template:Aligned table
Template:Aye indicates that the column's property is always true for the row's term (at the very left), while Template:N& indicates that the property is not guaranteed in general (it might, or might not, hold). For example, that every equivalence relation is symmetric, but not necessarily antisymmetric, is indicated by Template:Aye in the "Symmetric" column and Template:N& in the "Antisymmetric" column, respectively.

All definitions tacitly require the homogeneous relation R be transitive: for all a,b,c, if aRb and bRc then aRc.
A term's definition may require additional properties that are not listed in this table.

Cardinality, induction and recursion
Key concepts

— Cardinality Template:Main Template:Main category

- ℕ, ℤ and ℚ are countable sets

- ℝ is an uncountable set

- Cantor's Theorem and the Continuum Hypothesis

— Induction
— Recursion
Connections

— Hypercomputability Template:Main

Template:See also

Bibliography: theory and proposed and solved exercises
In English: In Spanish:
  • —¤— Kenneth A. Rosen. Discrete mathematics and its applications. 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, Template:ISBN
Software
In English: In Spanish:
Error creating thumbnail: Multimedia
In English: In Spanish:


To find out more
  1. 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
  2. Continuum hypothesis. Encyclopedia of Mathematics. Disponible en: http://www.encyclopediaofmath.org/index.php?title=Continuum_hypothesis
  3. 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/.
  4. 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/
  5. Portal:Mathematics
  6. And more:
      7. * Category:Set theory

Template:Set theory

Template:Large Template:Navbar
Template:Aligned table
Template:Aye indicates that the column's property is always true for the row's term (at the very left), while Template:N& indicates that the property is not guaranteed in general (it might, or might not, hold). For example, that every equivalence relation is symmetric, but not necessarily antisymmetric, is indicated by Template:Aye in the "Symmetric" column and Template:N& in the "Antisymmetric" column, respectively.

All definitions tacitly require the homogeneous relation R be transitive: for all a,b,c, if aRb and bRc then aRc.
A term's definition may require additional properties that are not listed in this table.

Algebraic structures
File:Associativity of binary operations (without question marks).svg
Key concepts

— Algebraic structures Template:Main Template:Main category

— Magma, semigroup and monoid

— Group Template:Main

See also: Cyclic permutation (Cycle), Symmetric group, Permutation group, Isometry group, Stirling numbers of the first kind and List of small groups

— Ring, integral domain and field

— Homomorphisms

Connections

— Cryptography Template:Main Template:Main category

— Category theory Template:Main

Live article: Baez, John (November 8, 2019). 'Qué es la teoría de categorías y cómo se ha convertido en tendencia'. El País (In: Café y Teoremas) (In Spanish).

— Coding theory Template:Main

See also: Block code, Group code and Hamming code
Bibliography: theory and proposed and solved exercises
In English: In Spanish:
Software
In English: In Spanish:
Error creating thumbnail: Multimedia
In English: In Spanish:

— Algebras

— Groups

— Examples of groups

— Homomorphism of groups

— Rings

— Integral domains

To find out more
  1. Portal:Mathematics
  2. And more:
    1. Multiplicative group of integers modulo n

Theme 2.- Number theory

Number theory
File:Symmetrical 5-set Venn diagram LCM 2 3 4 5 7.svg
Key concepts

Template:Main Template:Main category

— Divisibility and modular arithmetic

Template:See also

— Primes and greatest common divisor
— Solving congruences
— Applications of congruences
— Divisibility rules
— Diophantine equations
— Paradoxes
Bibliography: theory and proposed and solved exercises
In English: In Spanish:
  • —¤— Thomas Koshy. Elementary number theory with applications. Academic Press (an imprint of Elsevier Inc.), New York, United States, 2nd edition, 2007, ISBN: 978-0-12-372487-8
  • —¤— Kenneth A. Rosen. Discrete mathematics and its applications. 7th edition. (Chapter 4 and related exercises). McGraw-Hill, New York, New York, United States, 2012, Template:ISBN
  • Kenneth A. Rosen. Elementary number theory and its applications. Addison-Wesley, Reading, Massachusetts, United States, 1986, ISBN 0-201-06561
  • —¤— Máximo Anzola y José Caruncho. Problemas de Álgebra. Tomo 2. Anillos - Polinomios - Ecuaciones. 3ª edición. Primer Ciclo, Madrid, Spain. (Capítulo 7 «Divisibilidad en ℕ y ℤ», 94 ejercicios resueltos; Capítulo 8 «Ecuaciones diofánticas», 27 ejercicios resueltos; Capítulo 9 «Sistemas de numeración», 25 ejercicios resueltos), 1982. Template:ISBN.
  • —¤— Template:Cite web
  • —¤— Template:Cite web
  • —¤— Template:Cite web
  • —¤— Template:Cite web
  • —¤— Kenneth A. Rosen. Matemática discreta y sus aplicaciones. 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
  • —¤— Template:Cite web
Software
In English: In Spanish:
Error creating thumbnail: Multimedia
In English: In Spanish:

— Divisibility and modular arithmetic

— Primes and GCD

— Solving congruences and their applications

— Diophantine equations

— Cryptography

— Divisibility

— Primes and GCD

— Bézout's lemma

— Modular arithmetic. Euler's φ function (totient function)

— Diophantine equations

— Congruences

Ecuaciones
Sistemas

— Congruences: Casting out nines

— False positives: Casting out elevens

— Power residues

— Divisibility rules

See also
To find out more
  1. Portal:Mathematics
  2. And more:
    1. Divisibility
      1. Division algorithm (Algorithms for division)
    2. Primality
      1. Quadratic residue
      2. Quadratic reciprocity
      3. Primality test
    3. Pseudo-random number generation
      1. List of random number generators
    4. Cryptography
      1. Highly totient number
      2. Highly composite number
      3. Smooth number
      4. Rough number
      5. Semiprime
      6. Elliptic curve cryptography
    5. List of prime numbers
    6. List of numbers

Template:Number theory Template:Divisor classes Template:Classes of natural numbers Template:Number theoretic algorithms Template:Prime number classes Template:Prime number conjectures

Theme 3.- Combinatorics

Combinatorics
File:Proofs-of-Fermats-Little-Theorem-bracelet1.svg
Key concepts

Template:Main Template:Main category Template:See also

— The basics of counting

- Rules of sum, product, substraction and division

- Drawer principle and its generalisation

- Binomial coefficients and identities

- (Ordinary) (i.e., without repetition) variations, permutations and combinations, and with repetition, and circular permutations

- Counting with restrictions

— Combinatorial proofs: 1st, bijective proofs; 2nd, double counting proofs; 3rd, using distinguished element, and 4th, using the inclusion-exclusion principle

Template:Main Template:Main category

— Combinatorial modeling

- I: Sample selection and unit labelling with and without repetition

- II: Grouping units (distribution, storage or placement of objects into recipients)
(Occupancy problems)

- III: Partition of sets

· Catalan and Narayana numbers. Noncrossing partitions

- IV: Additive decompositions of numbers

— Paradoxes
Bibliography: theory and proposed and solved exercises
In English: In Spanish:
Software
In English: In Spanish:
Error creating thumbnail: Multimedia
In English: In Spanish:

— Basic principles

— Variations, permutations and combinations

— Binomial numbers

— Inclusion-exclusion principle

— Partitions

See also
To find out more
  1. Portal:Mathematics
  2. And more:
    1. Generating functions
    2. Examples of generating functions

Theme 4.- Difference equations

Finite difference equations (recurrence relations)
File:ADI-stencil.svg
Key concepts

Template:Main Template:Main category

Some useful previous concepts: Recursive definition, Recursion and Recursion (computer science)
— Linear difference equations

Template:Main

— Linear discrete dynamical systems

- Population dynamics

- Linear discrete dynamical models

- BIDE models

- Markov chains

— Solving equations numerically

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Bibliography: theory and proposed and solved exercises
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To find out more
  1. Integer sequences
  2. List of integer sequences in the OEIS that have their own English Wikipedia entries
  3. Index to OEIS: Section Recurrent Sequencies
  4. Recursion (computer science)
  5. Exponential factorial
  6. Ackermann function
  7. McCarthy 91 function
  8. Tower of Hanoi
  9. Josephus problem

Appendices

Graphs
File:Graph isomorphism.svg
Key concepts
Bibliography: theory and proposed and solved exercises
In English: In Spanish:
  • —¤— Kenneth A. Rosen. Discrete mathematics and its applications. 7th edition. (Chapters 10 and 11 and corresponding exercises). McGraw-Hill, New York, New York, United States, 2012, Template:ISBN
  • —¤— Kenneth A. Rosen. Matemática discreta y sus aplicaciones. 5th edition. (Chapters 8 and 9 and corresponding exercises). McGraw-Hill/Interamericana de España, S.A.U., Aravaca (Madrid), Madrid, Spain, 2004, Template:ISBN
Software
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See also
To find out more
  1. Gallery of named graphs
  2. Portal:Mathematics
  3. Mesh networking
Numerical calculus
File:BilinearInterpolExample2.png
Key concepts
Main articles: Numerical analysis and Numerical linear algebra
Main categories: Numerical analysis and Numerical linear algebra

— Interpolation

Bibliography: theory and proposed and solved exercises
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Software
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— Interpolation

— Newton's divided differences interpolation polynomial

— Lagrange polynomial

— Interpolation

— Newton's divided differences interpolation polynomial

— Lagrange polynomial

To find out more
In English: In Spanish:
  1. Archer, Branden and Weisstein, Eric W. "Lagrange Interpolating Polynomial." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/LagrangeInterpolatingPolynomial.html
  2. Weisstein, Eric W. "Newton's Divided Difference Interpolation Formula." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/NewtonsDividedDifferenceInterpolationFormula.html
  3. Template:Cite web
  4. Portal:Mathematics
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Conjectures

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Open problems

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Paradoxes
Some more problems, either not solved or solved
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Sample exam questions, instrumental and relational[note 1], and some answers

(Illustrative examples, cases, exercises, problems).

'It is axiomatic that the greater the student's individual effort, the more thorough will be his (sic) learning.'
Timothy J. Fitikides: Common mistakes in English, Longmans, 6ª edición / 6th edition, 2000, p. vii.

Very important warning

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Theme 1.- Fundamentals

Logic
Propositional logic

Question L1. (2.5 points).
On the island of truthfuls and deceitfuls — another nomenclature in the literature for the couple have been (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, A and B, standing together in the front yard of a house. You passed by and asked them, 'Are you truthful or deceitful persons?'

  • a) A answered, 'If B is a truthful person then I am a deceitful person.' Can it be determined whether A and B were truthfuls or deceitfuls? (1.25 p.)
  • b) Afterward, B said, 'Don't believe A; he's lying.' With this new information, can it be determined whether A and B were truthfuls or deceitfuls? (1.25 p.)
Solution:
Let us use X for 'X is a truthful person' --- therefore, ¬X is an abbreviation for 'X is a deceitful person' ---.
  • a) A's statement, 'If B is a truthful person then I am a deceitful person', can be expressed as B→¬A and the fact that A says it, as A↔(B→¬A). In view of the truth table: ABA↔(B→¬A)111𝟎1001101𝟏0101010𝟎1110000𝟎0110 the only model for A↔(B→A) is the 2nd interpretation, so it can be determined that A is a truthful person and B a deceitful person.
  • b) B's statement, 'Don't believe A; he's lying', is equivalent to 'A is a deceitful person', which can be expressed as ¬A and the fact that B says it, as B↔¬A, which simply says that neither both A and B 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: AB(A↔(B→¬A))∧(B↔¬A)11101001𝟎100110110101𝟏010101001110𝟎111000000110𝟎0010 we notice that everything remains the way it was, the 2nd interpretation is again a model, but now for (A↔(B→¬A))∧(B↔¬A).

Question L2. (2.5 points)
With the help of propositional logic, prove that the following argument is 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.'
Important: Do not solve it using truth tables.

Solution:
You can check the complete solution through semantic tableaux of several examples in this document (in Spanish, for the time being); (in particular, see exercise 7).

Question L3. (2.5 points).

  • a) Define adequate set of connectives (asc), also called completely expressive or functionally complete set of connectives.
  • 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 {¬,∧,∨,→,↔}.
Solution:
  • 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.
  • b) As it is said in the wording, we assume that we know that the set of the most usual connectives, {¬,∧,∨,→,↔} is an asc. Two two-element asc are the sets {¬,∧} and {¬,∨}. 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: {¬,∧}:p∨q≡¬(¬p∧¬q)p→q≡¬(p∧¬q)p↔q≡¬(p∧¬q)∧¬(¬p∧q){¬,∨}:p∧q≡¬(¬p∨¬q)p→q≡¬p∨qp↔q≡¬(¬(¬p∨q)∨¬(p∨¬q))

Predicate logic

Question L4. (2.5 points).
There are 77 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.'

  • (a) Formalise in predicate logic what the local said.
  • (b) How many of them are two-legged and how many are four-legged?
Solution:
Considering the set of animals on the hill as the universe of discourse, let be:

Dx←'x is two-legged',Cx←'x is four-legged'.

  • (a) ∃xDx∧∀xy(Cx∨Cy);
  • (b)
    𝐒𝐭𝐞𝐩𝐑𝐞𝐚𝐬𝐨𝐧1.∃xDxPremise2.∀xy(Cx∨Cy)Premise3.Ca∨CbUI 2 (universal instantiation)4.∀x(Cx↔¬Dx)Premise5.Ca↔¬DaUI 4 (universal instantiation)6.Ca→¬DaBIE1 5 (biconditional elimination)7.Cb↔¬DbUI 4 (universal instantiation)8.Cb→¬DbBIE1 7 (biconditional elimination)9.¬Da∨¬DbDil3 3, 6, 8 (complex constructive dilemma: {A∨B,A→C,B→D}⊢C∨D )10.¬(Da∧Db)DM1 9 (De Morgan's law, negation of the conjunction)11.∀x∀y¬(Dx∧Dy)UG210 (multiple universal generalisation)12.∀y¬(Da∧Dy)UE 11 (universal instantiation)13.¬∃y(Da∧Dy)EN 12 (existential negation)14.∀x¬∃y(Dx∧Dy)UG 13 (universal generalisation)15.¬∃x∃y(Dx∧Dy)EN 14 (existential negation)
    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.

Question L5. (2.5 points).
Source: Lewis Carroll, Symbolic Logic: Part I. Elementary (Macmillan, 1896), pg. 118. Public Domain.
40.
(1) No kitten, that loves fish, is unteachable;
(2) No kitten without a tail will play with a gorilla;
(3) Kittens with whiskers always love fish;
(4) No teachable kitten has green eyes;
(5) No kittens have tails unless they have whiskers.
Universe = 'kittens'; A = loving fish; B = green-eyed; C = tailed; D = teachable; E = whiskered; H = will play with a gorilla.

You are required to:

  • (a) Formalise all these statements into Predicate Logic.
  • (b) In the universe of kittens and using Predicate Logic, deduce the one conclusion that follows from these statements and makes the argument valid.
  • (c) Translate your symbolic answer into English.
Solution:
Considering the set of kittens as the universe of discourse, let be: Dx←«x is teachable»,Ax←«x loves fish»,Cx←«x has tail»,Hx←«x plays with a gorilla»,Ex←«x has whiskers»,Bx←«x has green eyes».
  • (a) (1) ∀x(¬Dx→¬Ax);
    (2) ∀x(¬Cx→¬Hx);
    (3) ∀x(Ex→Ax);
    (4) ∀x(Dx→¬Bx);
    (5) ∀x(¬Ex→¬Cx).
  • (b)
    𝐒𝐭𝐞𝐩𝐑𝐞𝐚𝐬𝐨𝐧1.∀x(¬Dx→¬Ax)Premise2.∀x(¬Cx→¬Hx)Premise3.∀x(Ex→Ax)Premise4.∀x(Dx→¬Bx)Premise5.∀x(¬Ex→¬Cx)Premise6.¬Da→¬AaUI 1 (universal instantiation)7.¬Ca→¬HaUI 2 (universal instantiation)8.Ea→AaUI 3 (universal instantiation)9.Da→¬BaUI 4 (universal instantiation)10.¬Ea→¬CaUI 5 (universal instantiation)11.¬Aa→¬EaCp1 8 (contraposition [also called transposition] law)12.Ba→¬DaCp2 9 (contraposition [also called transposition] law)13.(Ba→¬Da)∧(¬Da→¬Aa)CI 12, 6 (conjunction introduction)14.(Ba→¬Aa)Syl 13 (hypothetical syllogism)15.(Ba→¬Aa)∧(¬Aa→¬Ea)CI 14, 11 (conjunction introduction)16.(Ba→¬Ea)Syl 15 (hypothetical syllogism)17.(Ba→¬Ea)∧(¬Ea→¬Ca)CI 16, 10 (conjunction introduction)18.(Ba→¬Ca)Syl 17 (hypothetical syllogism)19.(Ba→¬Ca)∧(¬Ca→¬Ha)CI 18, 7 (conjunction introduction)20.(Ba→¬Ha)Syl 19 (hypothetical syllogism)21∀x(Bx→¬Hx)UG 20 (universal generalisation)
  • (c) No green-eyed kitten will play with a gorilla.

Question L6. (2.5 points).
Formalise into Predicate Logic:

  • (a) 'All A that is B, it is also C.'
  • (b) 'If all A is B, then it is also C.'
  • (c) 'There is none which is A or B and is not C.'
Solution:
  • (a) (∀x)((Ax∧Bx)→Cx).
  • (b) It can be rewritten as 'if all A is B, then all A is C too,' therefore, (∀x(Ax→Bx))→(∀x(Ax→Cx)). Warning!, (∀x(Px→Qx))⇒((∀xPx)→(∀xQx)) but ((∀xPx)→(∀xQx))⇏(∀x(Px→Qx)).
  • (c) (¬∃x)((Ax∨Bx)∧¬Cx). Or put it in other words, 'if something is A or B then it is C,' which in the language of Predicate Logic is written as follows: (∀x)((Ax∨Bx)→Cx).

Sets, relations and functions

Question SRF1. (2.5 points)

  • (a) Propose three sets A, B and C, such that A∈B, B∈C and A∉C. (0,5 points).
  • (b) According to a 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 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. (2 points).


Algebraic structures

Question AS1. (2.5 points)
Let x∗y=u be a binary relation defined on the set A={1,3,5,7,9}, for every two elements x and y in A, where u is the figure of the units of the usual product x⋅y between two natural numbers (for example, 3∗9=7).

  • a) Find out theCayley table for the operation ∗ on A.
  • b) Is (A;∗) an abelian group? (You can reason using the Cayley table).
Solution:
  • a) Here is the Cayley table of the binary operation ∗ defined on the set A: ∗13579113579339517555555771593997531
  • b) Let us check if (A,∗) satisfies the five requirements (axioms) to be qualified as an abelian group:
    • a) A is closed under ∗ (it is also said that ∗ is a closed operation or an internal composition law on A) since for all a and b in A, a∗b∈A. It is easy to reason using the table: every number in the table is an element of A.
    • b) ∗ is associative on A — we could check every triad, 1∗(3∗5)=1∗5=5=3∗5=(1∗3)∗5), and so on, however it is easier to reason using the fact that the product (⋅) of natural numbers is associative: simply, ∀x,y,z∈ℕ, x⋅(y⋅z)=(x⋅y)⋅z⇒x∗(y∗z)=(x∗y)∗z is true since when we multiply the unit digits among natural numbers we have to carry no number —;
    • c) ∗ is conmutative (the table is symmetric with respect to the main diagonal);
    • d) the identity for ∗ in A is 1 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;
    • e) not every element is invertible — 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, 7 is the inverse of 3 because 3∗7=7∗3=1 [we search for 1 on the second row (headed by 3), finding it on the fourth column (headed by 7])—: the inverse of 1 is 1, of 3 is 7, of 7 is 3 and of 9 is 9, but 5 has no inverse — when operating any other number with 5 the result is always 5 (such a number, as 5 in this case, is called an absorbing element for ∗ in A [as zero for the product of integers]), so it is impossible to obtain the identity as a result —.

    In short, (A;∗) has not abelian group structure (it has an abelian monoid structure).


Cardinality, induction and recursion

Question C1. (2.5 points).
Proof by definition that ℕ is an infinite set.

Solution:
A set is infinite precisely if there exists a bijection between it and one of its proper subsets (definition by Dedekind). As an example, consider f:ℕ→ℕ∖{0}, defined by n↦f(n)=n+1. Let us prove that it is a bijective mapping. In effect:
  • f is a mapping ⇔(∀x∈ℕ)(∃y∈ℕ∖{0})(f(x)=y)∧(∀x,x′∈ℕ)(x=x′→f(x)=f(x′)), which is trivial, as if x∈ℕ is given and because of the definition of f, there exists yx=x+1∈ℕ∖{0}, this yx being unique for each x, that is, that if x=x′, then, because of the definition of f, f(x)=x+1=yx=yx′=x′+1=f(x′);
  • f is injective ⇔(∀x,x′∈ℕ)(f(x)=f(x′)→x=x′), which is trivial because of the definition of f, as if f(x)=f(x′), that is, if x+1=x′+1, then, x=x′;
  • f is surjective ⇔(∀y∈ℕ∖{0})(∃x∈ℕ)(f(x)=y), which is also trivial due to the definition of f, as if y is given, then x=y−1 satisfies f(x)=f(y−1)=(y−1)+1=y.

Question C2. (2.5 points).
Knowing that ℤ (integers) is a denumerable set and that the denumerable union of denumerable sets is a denumerable set, prove that ℚ (rationals) is a denumerable set.

Solution:
ℚ is a denumerable set as it can be expressed by the denumerable union ℚ=A1∪A2∪…∪An∪…, where every Ai={0,−1i,1i,…,−ki,ki,…} is a denumerable set, since f:ℤ→Ai, defined by f(0)=0 and f(±n)=±ni is a bijection. Note that the set Ai is the set of all the rational numbers that have the same denominator i.

Cuestión C3. (2.5 points).
Let A be a denumerable set and let x∉A. Prove that A∪{x} is a denumerable set.

Solution:

As A is a denumerable set then --- by definition of denumerable set --- there is a bijection f:ℕ→A. Let g:ℕ→A∪{x}, defined by g(n)=x, if n=0 and by g(n)=f(n−1), if n≠0, that is, the correspondence g is defined on two subdomains, on {0} as the constant correspondence x, a bijection, and on ℕ+ as f, also a bijection, and as such subdomains are disjoint and their images, A and {x} are disjoint too, then g is a bijection.


Theme 2.- Number theory

Congruences

Question NT1. (2.5 points).
Use congruence relation theory to respond.

  • a) Prove that, for any n∈ℕ, 21⋅25n+1−4⋅33n+1 is divisible by 5. (1.25 p.)
  • b) Calculate the remainder of 36n+1+32n+1⋅192n−3 (for any n∈ℕ), when it is divided by 28. (1.25 p.)
Solution:
We use congruence relation theory.
  • a) On the one hand:
    25=32≡2(mod5)33=27≡2(mod5)}
    ⇒(i)25≡33(mod5)⇒(ii)25n≡33n(mod5)(1)
    On the other:
    7≡2(mod5)⇒(iii)7⋅2≡2⋅2(mod5)⇒(iv)7⋅2⋅3≡2⋅2⋅3(mod5)(2)
    Substituting (2) in (1):
    7⋅3⋅2⋅25n≡2⋅2⋅3⋅33n(mod5)21⋅25n+1≡4⋅33n+1(mod5)
    which, by definition of congruence relations, means that 21⋅25n+1−4⋅33n+1 is divisible by 5.
  • b) On the one hand:
    33=27≡−1(mod28)⇒(v)36n=(33)2n≡(−1)2n=1(mod28)⇒(iv)36n+1≡3(mod28)(3)
    On the other:
    57=3⋅19≡1(mod28)⇒(v)32n⋅192n≡1(mod28)⇒(iv)32n+1⋅192n≡3(mod28)(4)
    If we add side by side (3) and (4):
    36n+1+32n+1⋅192n≡6(mod28)⇒36n+1+32n+1⋅192n−3≡3(mod28)
    In other words, the requested remainder is 3.

(i) Because congruence relations are symmetric and transitive.
(ii) Rising each side of the congruence relation to the power n.
(iii) Multiplying each side of the congruence relation by 2.
(iv) Multiplying each side of the congruence relation by 3.
(v) Rising each side of the congruence relation to the power 2n.


Power residues and divisibility rules

Question NT2. (2.5 points)
In base-ten (decimal numeral system), find the digits x,y such that the number 12xy567 be divisible by 33.

Solution:
33=3×11.

A number is divisible by 3 precisely if the sum of all its digits is divisible by 3: 3|12xy567⇔3|(7+6+5+y+x+2+1) this is: 3|12xy567⇔3|(21+x+y)⇔21+x+y=3˙⇔x+y=3˙−21 Moreover: x,y are digits in base-ten ⇒0⩽x,y⩽9 then: 0⩽x+y⩽18 We have to find out what differences 3˙−21 satisfy the fact of belonging to [0,18]: 3˙−21={…,0(=21−21),3(=24−21),6(=27−21),9(=30−21),12(=33−21),15(=36−21),18(=39−21),…} so there are 7 possible cases: x+y=0⊻x+y=3⊻x+y=6⊻x+y=9⊻x+y=12⊻x+y=15⊻x+y=18

A number is divisible by 11 precisely if the sum of its digits at even places minus the sum of its digits at odd places is divisible by 11: 11|12xy567⇔11|((7+5+x+1)−(6+y+2)) this is: 11|12xy567⇔11|(5+x−y)⇔5+x−y=11˙⇔x−y=11˙−5 Moreover: x,y are digits in base-ten ⇒0⩽x,y⩽9 then: −9⩽x−y⩽9 We have to find out what differences 11˙−5 satisfy the fact of belonging to [−9,9]: 11˙−5={…,−5(=0−5),6(=11−5),…} so there are 2 possible cases: x−y=−5⊻x−y=6

Therefore, there are 7×2=14 possible cases:

Table of possible cases
Λ x+y=0 x+y=3 x+y=6 x+y=9 x+y=12 x+y=15 x+y=18
x−y=−5 2x=−5
x=−52		 2x=−2
x=−1		 2x=1
x=12		 2x=4
x=2
y=7		 2x=7
x=72		 2x=10
x=5
y=10		 2x=13
x=132
No:Template:Padx is not a digit
in base-ten		 No:Template:Padx is not a digit
in base-ten		 No:Template:Padx is not a digit
in base-ten		 Yes:Template:Padx,y are digits
in base-ten		 No:Template:Padx is not a digit
in base-ten		 No:Template:Pady is not a digit
in base-ten		 No:Template:Padx is not a digit
in base-ten
x−y=6 2x=6
x=3
y=−3		 2x=9
x=92		 2x=12
x=6
y=0		 2x=15
x=152		 2x=18
x=9
y=3		 2x=21
x=212		 2x=24
x=12
No:Template:Pady is not a digit
in base-ten		 No:Template:Padx is not a digit
in base-ten		 Yes:Template:Padx,y are digits
in base-ten		 No:Template:Padx is not a digit
in base-ten		 Yes:Template:Padx,y are digits
in base-ten		 No:Template:Padx is not a digit
in base-ten		 No:Template:Padx is not a digit
in base-ten

So there are three possible solutions: (xy)∈{(27),(60),(93)}.

Thus, the possible numbers are: 1227567,1260567,1293567,

which are divisibles by 33. Their quotients are: 37199,38199,39199.


Diophantine equations

Question NT3. (2.5 points).
One company spent 100000 euros in buying 100 electronic devices, some of them ground breaking and providing maximum performance. Smartphones were 50 euros each, tablets were 1000 euros each and laptops were 5000 euros each. How many of each device did they buy? Solve this question using the theory of:

  • a) diophantine equations;
  • b) congruence equations.
Solution:
Once translated the information from the wording into a system of linear equations and simplifying the latter:

{x+y+z=10050x+1000y+5000z=100000⇒{x+y+z=100x+20y+100z=2000⇒{x+y+z=100x+y+z+19y+99z=2000⇒100+19y+99z=2000⇒99z+19y=1900


Question NT4. (2.5 points).
How could we distribute 100 litres of water in a total of 40 different containers of 1, 4 and 12 litres? Solve this question using the theory of Diophantine equations.

Solution:
Let x, y, and z denote the number of containers of 1, 4 and 12 liters, respectively, used in the solution. The conditions set forth in the statement of the question are taken up by the equations: x+y+z=40 and x+4y+12z=100. By subtracting the first equation from the second equation: 3y+11z=60, that is a Diophantine equation. As gcd⁡(3,11)=1∣60, such Diophantine equation has integer solutions. As 1=4⋅3+(−1)⋅11, the Bézout coefficients are p=4 and q=−1. A particular solution is: y0=4⋅601=240, z0=(−1)⋅601=−60. The general solution is: y=240+11k, z=−60−3k, where k∈ℤ. Assuming that at least one container of each type is involved in the solution, then 0<y,z≤40−(1+1)=38. By replacing y and z with the general solution found, then, on the one side: 0<240+11k≤38, hence −24011<k≤−20211, thus −21,82<k≤−18,36 and therefore, k∈{−21,−20,−19}, and on the other: 0<−60−3k≤38, hence 603<−k≤983, thus −983≤k<−603, that is, −32,67≤k<−20 and therefore, k∈{−32,…,−22,−21}. Thus, k=−21. If k is replaced in the general solution with its value, y=240+11⋅(−21)=9 and z=−60−3⋅(−21)=3, then it is obtained that x=40−(9+3)=28.

Sol.: Assuming that at least one container of each type must be filled, then the 100 liters of water can be distributed using 28 containers of 1 liter, 9 of 4 liters and 3 of 12 liters. (If such a thing is not assumed, then another solution would be possible: 20 containers of 1 liter, 20 of 4 liters and 0 of 12 liters. [Verify this]).


Cryptography

Question NT5. (2.5 points).
Abigail wants to send Balbina the most simple call message: eh. They can only send numbers. Abigail and Balbina use the letters' position in the alphabet to code them (thus, Abigail codes e as 06 and h as 08). They use RSA to encrypt their messages. If Abigail choose p=3 and q=7 as the ground primes for RSA:

  • a) imagine you are Abigail and obtain the encrypted message that you have to send to Balbina;
  • b) imagine you are Balbina and decrypt the encrypted message that Abigail has sent to you.
Solution:
Following the steps of RSA algorithm:
  • 1) p=3, q=7.
  • 2) r=p⋅q=21.
  • 3) ϕ(21)=12 (Euler phi of 21).
  • 4) We have to choose as the secret key (SK) a relatively prime with 12 and less than 12, at the same time; we choose SK=5.
  • 5) The secret key and the public key (PK) are linked by the equation SK⋅PK≡1(modϕ(r)), in this case: 5⋅PK≡1(mod12), so PK=5.
  • 6) If we code the original message, (eh), we have 0608. It can be proven that that if the coded message X satisfies 0≤X≤r−1, then the ciphered message Y, also satisfies 0≤Y≤r−1. 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 r−1=20. Let X1=06 and X2=08 be the coded blocks and let Y1 and Y2 be the ciphered blocks. As RSA method establishes, the Xi encryption is performed by solving the congruence equation XiPK≡Yi(modr) and the Yj decryption by solving the congruence equation YjSK≡Xj(modr).

Let us answer now the two sections of the question.

  • a) Putting ourselves in the shoes of Abigail, let us assess the ciphered message that we have to send to Balbina. Let us cipher X1=06: the solution of 65≡Y1(mod21) is Y1=6. Let us cipher X2=08: the solution of 85≡Y2(mod21) is Y2=8. Thus, the message that we have to send is: 0608.
  • b) Putting ourselves in the shoes of Balbina, let us decipher the ciphered message that Abigail has sent us. Let us decipher Y1=06: the solution of 65≡X1(mod21) is X1=6. Let us decipher Y2=08: the solution of 85≡X2(mod21) is X2=8. Thus, the message we have just deciphered is: 0608.

Theme 3.- Combinatorics

Combinatorics

Question CT1. (2.5 points)
Let D be the set of decimal digits, that is, D={0,1,2,3,4,5,6,7,8,9}. Using combinatorial reasoning, calculate:

  • a) The number of subsets of D which elements are all primes.
  • b) The number of subsets of D having a prime number of elements.
Solution:
  • a) Let P={2,3,5,7} be the set consisting of the the prime numbers in D. What is actually requested is the number of nonempty (nonvoid) subsets of P, in other words, subtracting one (the empty set) from the total number of subsets of P: |P|−1=24−1=15 Sol.: 15 subsets.
  • b) The total number of subsets of k elements of a set of n elements is given by Cn,k=(nk). Thus, running over the prime numbers in D: (102)+(103)+(105)+(107)=10!2!⋅8!⋅10!3!⋅7!⋅10!5!⋅5!⋅10!7!⋅3!=537 Sol.: 537 subsets.

Question CT2. (2.5 points)
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.

Solution:
This involves finding the number of derangements of 12 objects. Instead of calculating for 12, we are going to do for the general case of having n objects. Let 1,2,…,n denote the objects themselves. Let P be the set of all permutations of the objects and let Pk be the set of all derangements that have k fixed elements. Then, the set of all derangements is:

D=P−(⋃i=1nPi) Let us see it:

  • How many permutations fix one specific number? The answer is the number of permutations of the other n−1 (non fixed) numbers, that is to say, (n−1)!, and as there are n numbers, then, the number of permutations that fix any of these n numbers is (n−1)!⋅n.
  • How many permutations fix two specific numbers? The answer is the number of permutations of the other n−2 (non fixed) numbers, that is to say, (n−2)!, and as there are (n2) ways of choosing two different numbers out of n numbers, then, the number of permutations that fix any two of these n numbers is (n2)(n−2)!.

Let us note that in the case of n=2, that is, {1,2}, when we subtract those which fix the 1, then we are subtracting once those which fix both the 1 and the 2 and when we subtract those that fix the 2 we are subtracting again those which fix both the 1 and the 2. 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: |D|=|P|−|(⋃i=1nPi)|=(n0)n!−(n1)(n−1)!+(n2)(n−2)!−⋯+(−1)n+1(nn)0! Thus, if n=12, there are: (120)12!−(121)(12−1)!+(122)(12−2)!−⋯+(−1)12+1(1212)0!=176 214 841 derangements.
Sol.: In 176 214 841 ways.


Question CT3. (2.5 points)
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.

Solution:
There are 7 positions for the figures. At both ends, hypotheses imply even figure. There are three even figures between 1 and 7: 2, 4 and 6. Being guided by the distribution of objects into recipients models, consider these 3 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 5 intermediate positions. The number of these possibilities is given by the permutations of 5 elements (one new abstraction as 5 distinguishable objects [the intermediate positions] being distributed into 5 distinguishable recipients [the figures 1, 3 and 5 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:

V3,2⋅P5=32_⋅5!=(3⋅2)⋅(5!)=6⋅120=720 Sol.: 720 numbers.


Question CT4. (2.5 points)
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.

Solution:
Let us use the occupancy model for the non ordered distribution of balls into boxes; balls and boxes representing ballots and people, respectively. Consider the 17 persons (distinguishable boxes) and the 7 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 7 elements from a set of 17 elements —. In any case, there are C17,7 ways of distributing the assenting ballots into the boxes. For each one of these cases (each one of the combinations), there are 10 empty boxes left. Now, using a similar reasoning, there are C10,5 ways of distributing the dissenting ballots into the boxes, with 5 empty boxes remaining for each one of these cases. Similarly, there are C5,3 ways of distributing the blank ballots into the boxes, with 2 empty boxes remaining for each one of the cases. So, lastly, there are C2,2 ways in which invalid ballots can be placed into the boxes. Applying the rule of product:

C17,7⋅C10,5⋅C5,3⋅C2,2=(177)⋅(105)⋅(53)⋅(22)=17!7!⋅10!⋅10!5!⋅5!⋅5!3!⋅2!⋅2!2!⋅0!=17!⋅10!⋅5!⋅2!7!⋅10!⋅5!⋅5!⋅3!⋅2!⋅2!⋅0!=49 008 960 Sol.: In 49 008 960 ways.


Question CT5. (2.5 points).
Use a combinatorial reasoning to respond.

  • 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.)
  • b) Let us assume a n sided polygon network (n-gon network). Calculate n, the number of nodes (vertices) of the network, knowing that the number of line segments (sides + diagonals) is 253. (1.25 p.)
Solution:
  • a) A seven digit palindrome fits into the model abcdcba, where a≠0. There are 9 possibilities for a, 10 for b, 10 for c and 10 other possibilities for d. Because of the rule of product, there is a total of 9⋅10⋅10⋅10=9000 seven digit palindromes.
  • b) If n 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 n elements (the n nodes), and this number is, by definition of combination, (n2). Then, (n2)=253⇒n(n−1)2=253⇒n=−22∨n=23. Thus, n=23.

Theme 4.- Finite difference equations (recurrence relations)

Question RR1. (2.5 points)
Let the following be the definition of the sum of two natural numbers n and m: S(n,0)=nS(n,m)=S(n,m−1)+1 Prove that the solution of this recurrence is S(n)=n+m.

Solution:
Let us note that n is alien to recursion. Thus, in an easier but equivalent way, denoting S(n,m) by f(m), 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:

f(0)=nf(m)−f(m−1)=1

  • a) General solution of the homogeneous:
    The characteristic polynomial is: P(x)=x−1 thus, 1 is a simple characteristic root.
    The general solution of the homogeneous is: f(m)=c11m;c1∈ℝ
  • b) Particular solution of the non homogeneous:
    As the function on the RHS is constant, let us try out a general constant (real number) as a possible particular solution: k−k=1 but this is a contradiction, so we have to increase the degree of the polynomial. Let us try out with a first degree polynomial, P(m)=Am. Substituting: Am−A(m−1)=1⇒Am−Am+A=1⇒A=1 Thus: f(m)=m is a particular solution of the non homogeneous.
  • c) General solution of the non homogeneus: f(m)=c1+m;c1∈ℝ(1)
  • d) Considering the initial conditions:
    From the initial condition, f(0)=n, we have: n=f(0)=c1+0⇒c1=n Substituting in (1), we get the desired solution: f(m)=n+m or, in other words: S(n,m)=n+m

Question RR2. (2.5 points)
Let x(t) and y(t) be the numbers of malicious software belonging to two malware types, in the day t, that coexist in a certain insecure wide area network (WAN) under malware evolution daily control. Let us assume that the original memberships were of x(0)=3 and y(0)=7 and that the coexistence evolution is as follows:

  • every day, the growth in malware type x is the sum of the triple of the growth in type x on the previous day and the growth in type y also on the previous day plus seven new malware (that were classified as type x),
  • and also every day, the growth in malware type y is the result of subtracting the growth in type x on the previous day from the growth in type y on the previous day, plus three new malware (that were classified as type y).

Find out and solve the system of recurrence equations of the evolution of the malware.

Solution:
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 Xt and Yt denote the growths from time t to time t+1, i.e. Xt=x(t+1)−x(t) and Yt=y(t+1)−y(t). The system of linear recurrence equations that correponds to this situation is:

{Xt+1=3Xt+Yt+7(1)Yt+1=Yt−Xt+3(2)

  • a) Calculating Xt:
    From the first equation we get: Yt=Xt+1−3Xt−7(3)Xt+2=3Xt+1+Yt+1+7(4) Substituting (2) in (4): Xt+2=3Xt+1+Yt−Xt+3+7 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: Xt+2−4Xt+1+4Xt=3
    • a.1) General solution of the homogeneous:
      The characteristic polynomial is: P(X)=X2−4X+4 that is: P(X)=(X−2)2 thus, 2 double characteristic root.
      The general solution of the homogeneous is: Xt=c12t+c2t2t;c1,c2∈ℝ
    • a.2) Particular solution of the non homogeneous:
      As the function on the RHS is constant, let us try out a general constant (real number) as a possible particular solution: k−4k+4k=3 then k=3. Thus: Xt=3 is a particular solution of the non homogeneous.
    • a.3) General solution of the non homogeneous: Xt=c12t+c2t2t+3;c1,c2∈ℝ(5)
  • b) Calculating Yt:
    Substituting (5) in (1), and simplifying, grouping and sorting, we get: Yt=(2c2−c1)2t−c2t2t−13;c1,c2∈ℝ
  • c) Considering the initial conditions:
    From the initial conditions, x0=3 and y0=7 we have: X0=x1−x0=x1−3=c120+c2⋅0⋅20+3=c1+3⇒c1=x1−6(6) On the other side: Y0=y1−y0=y1−7=(2c2−c1)20−c2⋅0⋅20−13=2c2−c1−13(7) and now, substituting (6) in (7): y1−7=2c2−x1+6−13⇒y1=2c2−x1⇒c2=x1+y12
  • d) The solution to the situation under the wording of the question (evolution of these two types of malware on their own growths) is: {Xt=(x1−6)2t+(x1+y1)t2t−1+3Yt=(y1+6)2t−(x1+y1)t2t−1−13 where x1 and y1 are the populations of both types of malware by the end of the first hour (data not provided in the wording).

Apéndices

Graphs

Question G1. (2.5 points)
The accompanying graph shows the connections among four tram stations. You may:

  • a) Write the adjacency matrix G of that graph.
  • b) Interpret the matrices G2 and G3 (reason what situations they represent).
  • c) Reason, using those matrix representations, if it is a strongly connected graph or not.
  • d) Reason, using those matrix representations, what is the length of the shortest path from A to D and how many paths may be considered as the "shortest" ones.
 
+—+       +—+
|D| <———> |C|
+—+       +—+
    \      ⋀
     \     |
      \    |
       \   |
        ╶┘ |
+—+       +—+
|A| <———> |B|
+—+       +—+
Solution:
  • a) The adjacency matrix G of this graph is: G=(0100101000010110) We draw up the adjacency matrix of the graph, by matching the positional subscripts 1, 2, 3 and 4 of its elements with the labels A, B, C and D, so that, for example, g23 corresponds to a possible path from B to C, B⇝C. Thus, the element gij of G is the number of direct connections --- with no intermediate station (paths of length one in the graph) --- between the tram stations corresponding to i and j, in the direction i⇝j. In this way, we interpret g23=1 as the existence of one direct connection from 2 to 3, 2→3, this is, B→C, whilst g32=0 corresponds to the non existence of a direct connection from 3 to 2, 3↛2, this is, C↛B.
  • b) The powers 2 and 3 of G are: G2=(1010010101101011)G3=(0101112010110211) The element gij(2 of G2 is the number of connections with exactly one station in the middle (length two paths in the graph) from the corresponding station to i to the corresponding station to j, under the previous formalization. Similarly, the element gij(3 of G3 represents the number of connections with two intermediate stations (three length paths in the graph) from the corresponding station to i to the corresponding station to j, again according to the previous formalization.
  • 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 G, with n vertices, it is possible to know if there exists a path from the vertex i to the vertex j, regardless of the length but depending on the element pij of the matrix P=G+G2+G3+⋯+Gn−1 as it is the total number of paths from i to j (if there were i,j such that pij=0 then no path would be possible from i to j and the graph would not be strongly connected).
    The graph under our study, G, with 4 vertices, is strongly connected because P=G+G2+G3 has not zero elements: P=G+G2+G3=(1211223111221332) 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, G2+G3 neither has zero elements: G2+G3=(1111122111211222) that is to say, any two stations are connected each other via one or two intermediate stops.
  • d) Denoting by gij(k) the element at position (i,j) of the matrix Gk, we note that g14=g14(2)=0 and that g14(3)=1, 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 g14(3) is one).

Numerical algebra and calculus

Question NAC1. (2.5 points)
Find a possible general formula for computing the nth term, that is, an, of the sequence a1=1,a2=2,a3=4,a4=7 using the Newton's divided differences interpolation polynomial.

Solution:
Because of custom and maybe tradition we begin considering n=0, and so we start dealing this question with b0=1,b1=2,b2=4,b3=7 and then adjust to match what was required. Let f(n) be bn. Then, the table of divided differences is:

ijxif(xi)f(xi,xj)ijklf(xi,xj,xk)f(xi,xj,xk,xl)0010111120121/2122012302241231/2233337 where: f(xi,xj)=f(xj)−f(xi)xj−xif(xi,xj,xk)=f(xj,xk)−f(xi,xj)xk−xi⋮f(x0,x1,…,xn)=f(x1,x2,…,xn)−f(x0,x1,…,xn−1)xn−x0 The interpolating polynomial is: fn(x)=f(x0)+(x−x0)f(x0,x1)+(x−x0)(x−x1)f(x0,x1,x2)+⋯+(x−x0)(x−x1)…(x−xn−1)f(x0,x1,…,xn) as well as in recurrence form: fn(x)=fn−1(x)+(x−x0)(x−x1)…(x−xn−1)f(x0,x1,…,xn) Thus:

  • f0(x)=f(x0)=f(0)=1, that is satisfied by x0, but no longer by x1 since f0(x1)=f0(1)=1≠2=f(1).
  • f1(x)=f0(x)+(x−x0)f(x0,x1)=1+(x−0)⋅1=1+x, that is satisfied by x0 and x1, but no longer by x2 since f1(x2)=f1(2)=3≠4=f(2).
  • f2(x)=f1(x)+(x−x0)(x−x1)f(x0,x1,x2)=1+x+(x−0)(x−1)⋅1/2=1+x2+x22, which is satisfied by all of the points, even by x3 since f2(x3)=f2(3)=7=f(3).
  • The next difference is zero, which confirms that the interpolating polynomial, suggested by this method, is a second degree polynomial:

f2(x)=1+x2+x22

In summary, starting with n=0, the general term is bn=1+n2+n22, and adjusting to match the n=1 beginning as required, the general term is: an=1+n−12+(n−1)22 Sol.: an=1+n−12+(n−1)22.


Sample preparatory exams

Template:Ambox

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
Academic year 2018-2019
Academic year 2019-2020

Past qualifying activities and real exams with some solutions

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 Maria Martinello says).

Academic year 2016-2017

Academic year 2017-2018

Academic year 2018-2019

Academic year 2019-2020

Tentative course outline (chronogram for the 2019-2020 academic year)

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. Template:Ambox

Calendar of activities

File:UN emblem blue.svg

4 January
World Braille Day

24 January
International Day of Education

27 January
International Holocaust Remembrance Day

Further Mathematics

Academic Year 2019-2020

2nd semester

Class (large group) meetings (48 h) and seminar/laboratory meetings (12 h)

Dates Topics Basic texts readings and study Following is a selection of training exercises, essentially instrumental. Do not forget those that are worked in class and seminar/lab meetings. 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.
Rosen 5th ed. Spain/USA Rosen 7th ed. USA Others Rosen 5th ed. Spain/USA Rosen 7th ed. USA Others

Theme 1

Fundamentals

(16 h LG and 5 h S/L)

Wed
29/1
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.
  • § 1.1 (8, 15, 30, 48, 42, 51-55, 59).
  • § 1.1 (12, 21, 38);
  • § 1.2 (12, 16, 19-23, 35).

Thu
30/1

Symbolic logic, I: Propositional logic
  • Propositional equivalences;
  • Formal derivation.
  • § 1.2.
  • § 1.3.
  • § 1.2 (8, 10, 29, 51).
  • § 1.3 (10, 12, 29, 57).
Error creating thumbnail: 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).

Group B
Fri
31/1

Groups A and E
Mon
3/2

Seminar/Laboratory No. 1:
Proofs and refutations, I
  • (Occasionally computer-assisted) hands-on word-problem-solving on issues related to formal and informal arguments, essentially using:
    • truth tables,
    • reductio ad absurdum, or
    • normal forms.
  • § 1.1, 1.2;
  • § 1.5.7;
  • —.
  • § 1.1, 1.2, 1.3;
  • § 1.7.7;
  • —.
  • Rosen 7th Global Edition: § 1.7 (normal forms);
  • WP+ (Logic).

Tue
4/2
File:UN emblem blue.svg
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);
  • WP+ (Logic).
  • Rosen 7th Global Edition: § 1.7 (normal forms) (1, 2, 3, 4, 5, 6);
  • WP+ (Logic).

Wed
5/2

Symbolic logic, II: Predicate logic
  • Reverse translation (from English to predicate logic).
  • § 1.3.
  • § 1.4.
  • § 1.3 (8, 10, 22, 41, 48, 55).
  • § 1.4 (8, 10, 24, 43, 52, 59).

Thu
6/2
File:UN emblem blue.svg
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.
  • § 1.4 (5, 8, 13, 18, 21, 28, 37, 44).
  • § 1.5 (5, 8, 13, 18, 21, 28, 39, 48).

Group B
Fri
7/2

Groups A and E
Mon
10/2
File:UN emblem blue.svg
World Pulses Day

Seminar/Laboratory No. 2:
Proofs and refutations, II
  • (Occasionally computer-assisted) hands-on word-problem-solving on issues related to formal and informal arguments, essentially using:
    • natural deduction calculus (Gentzen-style).
  • § 1.5 (particularly, § 1.5.3, 1.5.6).
  • § 1.6 (particularly, § 1.6.4, 1.6.7).

Tue
11/2
File:UN emblem blue.svg
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.
  • § 1.5 (10, 12).
  • § 1.6 (14, 16).

Wed
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.
  • § 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).

Thu
13/2
File:UN emblem blue.svg
World Radio Day

Sets
  • Sets;
  • Set operations; Boolean algebra; partitions.
  • § 1.6;
  • § 1.7.
  • § 2.1;
  • § 2.2.
  • § 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).

Group B
Fri
14/2

Groups A and E
Mon
17/2

Seminar/Laboratory No. 3:
Proofs and refutations, III
  • (Occasionally computer-assisted) hands-on word-problem-solving on issues related to formal and informal arguments, essentially using:
    • Beth and Hintikka semantic tableaux (Smullyan&Jeffrey-style).
 — —
  • Antón and Casañ, 1987: § 3.2;
  • Manzano and Huertas, 2006: § 4 (for Propositional logic), § 11 (for First-order logic);
  • WP+ (Logic).
 — —

Tue
18/2

Functions
  • Correspondences, functions and mappings; injectivity, surjectivity and bijectivity; inverse function.
  • § 1.8.
  • § 2.3.
  • § 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).

Wed
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.
  • § 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).
Error creating thumbnail: 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 project page and on the contributions page).

Thu
20/2
File:UN emblem blue.svg
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;
  • § —.
  • § 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).

Group B
Fri
21/2
File:UN emblem blue.svg
International Mother Language Day

Groups A and E
Mon
24/2

Seminar/Laboratory No. 4:
Induction and recursion
  • (Occasionally computer-assisted) hands-on word-problem-solving on issues related to:
    • mathematical induction;
    • strong induction and well order;
    • structural induction.
  • (§ 1.5.7, 1.5.8, 1.5.9, 1.5.10; § 3.1);
  • § 3.3;
  • § 3.3;
  • § 3.4.
  • (§ 1.7; § 1.8);
  • § 5.1;
  • § 5.2;
  • § 5.3.
  • G. Polya. How to solve it. Princeton, New Jersey (US-NJ), USA: Princeton University Press;
  • WP+ (Logic).

Tue
25/2

Relations, III
  • Solving questions on relations.

—

—

—

—

Is there something greater than infinity? (Cardinality, I)
  • Countable sets: ℕ, ℤ and ℚ are countable sets.
  • § 3.2.5.
  • § 2.5.1, 2.5.2.
  • § 3.2.5 (31, 32, 34, 38);
  • § 2.5 (1, 4, 16, 28).

Wed
26/2

Is there something greater than infinity? (Cardinality, II)
  • ℝ 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.
  • § 3.2.5 (31, 32, 34, 38);
  • § 2.5 (1, 4, 16, 28).

Thu
27/2

Algebraic structures, I
  • Algebraic structures;
  • Semigroups, monoids and groups;
 — — — —
  • Rosen 7th Global Edition: § 12.1 (1, 2);
  • Rosen 7th Global Edition: § 12.2 (2, 4, 5, 8, 12, 17, 18, 19, 20, 26, 31, 36, 40);
  • WP+ (Algebraic structures).

Group B
Fri
28/2

Groups A and E
Mon
2/3

Seminar/Laboratory No. 5:
Cardinality and Algebraic Structures
  • (Occasionally computer-assisted) hands-on word-problem-solving on issues related to:
    • cardinality;
    • algebraic structures.
 — —
  • Rosen 7th Global Edition: § 12.1;
  • Rosen 7th Global Edition: § 12.2;
  • Rosen 7th Global Edition: § 12.3;
  • Rosen 7th Global Edition: § 12.4;
  • WP+ (Algebraic structures).
 — —

Sun
1/3
File:UN emblem blue.svg
Zero Discrimination Day

Tue
3/3
File:UN emblem blue.svg
World Wildlife Day

Algebraic structures, II
  • Homomorphisms;
  • Rings, integral domains and fields.
 — — — —

Wed
4/3

Algebraic structures, III
  • Solving some questions on algebraic structures.
 — —
  • Rosen 7th Global Edition: § 12.1;
  • Rosen 7th Global Edition: § 12.2;
  • Rosen 7th Global Edition: § 12.3;
  • Rosen 7th Global Edition: § 12.4;
  • WP+ (Algebraic structures).
 — —
  • Rosen 7th Global Edition: § 12.1 (1, 2);
  • Rosen 7th Global Edition: § 12.2 (2, 4, 5, 8, 12, 17, 18, 19, 20, 26, 31, 36, 40);
  • Rosen 7th Global Edition: § 12.3 (4, 5, 7, 8);
  • Rosen 7th Global Edition: § 12.4 (2, 3, 4, 5);
  • WP+ (Algebraic structures).

Theme 2

Number theory

(9 h LG and 3 h S/L)

(1h LG Solving the mid-course preparatory exam)

Thu
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.
  • § 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).

Group B
Fri
6/3

Groups A and E
Mon
9/3

Seminar/Laboratory No. 6:
Divisibility, modular arithmetic, primes, GCD and congruences
  • (Occasionally computer-assisted) hands-on word-problem-solving on issues related to:
    • divisibility;
    • modular arithmetic;
    • primes;
    • GCD;
    • congruences.
  • (Everything we have studied on the subject).
  • (Everything we have studied on the subject).
  • (Everything we have studied on the subject).
  • (Everything we have studied on the subject).

Sun
8/3
File:UN emblem blue.svg
International Women's Day

Tue
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.
  • § 2.4 (8, 12, 14, 15, 20, 24, 25, 26, 27).
  • § 4.3 (2, 4, 6, 11, 18, 20, 21, 22, 23).

Wed
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.
  • § 2.4 (17, 28);
  • § 2.5 (21, 22);
  • § 2.5 (2, 50).
  • § 4.3 (15, 24);
  • § 4.3 (33, 32);
  • § 4.3 (40, 44).

Thu
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.
  • § 2.6 (4, 5, 6, 7, 8, ...);
  • § 2.6 (... ... ...);
  • § 2.6 (... ... ...).
  • § 4.4 (2, 5a, 6a, 5b, 6c, ...);
  • § 4.4 (... ... ...);
  • § 4.4 (... ... ...).

Group B
Fri
13/3

Groups A and E
Mon
16/3

Seminar/Laboratory No. 7:
Diophantine and congruence equations, I
  • (Occasionally computer-assisted) hands-on word-problem-solving on issues related to:
    • diophantine equations;
    • congruence equations.
  • —;
  • § 2.6.8, 2.6.9, 2.6.10.
  • —;
  • § 4.6.4, 4.6.5, 4.6.6, 4.6.7, 4.6.8.
  • —;
  • § 2.6 (46, 47, 45*).
  • —;
  • § 4.6 (24, 27, 23*).

Tue
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.
  • § 2.6 (17, 28, 32, 34, 43, 44, 52, 56).
  • § 4.4 (19, 38, 46, 48,41, 42, 58, 62).

Wed
18/3

Divisibility rules
  • Power residues and divisibility rules.
 — — — —

Thu
19/3

Diophantine equations
  • Diophantine equations.
 — — — —

Group B
Fri
20/3
File:UN emblem blue.svg
International Francophonie Day
International Day of Happiness

Groups A and E
Mon
23/3
File:UN emblem blue.svg
World Meteorological Day

Seminar/Laboratory No. 8:
Diophantine and congruence equations, II
  • (Occasionally computer-assisted) hands-on word-problem-solving on issues related to:
    • diophantine equations;
    • congruence equations;
    • applications of congruences.
  • (Everything we have studied on the subject).
  • (Everything we have studied on the subject).
  • (Everything we have studied on the subject).
  • (Everything we have studied on the subject).
  • (Everything we have studied on the subject).

Sat
21/3
File:UN emblem blue.svg
World Poetry Day
International Day for the Elimination of Racial Discrimination
International Nowruz Day (es)
World Down Syndrome Day
International Day of Forests

Sun
22/3
File:UN emblem blue.svg
World Water Day

Tue
24/3
File:UN emblem blue.svg
International Day for the Right to the Truth Concerning Gross Human Rights Violations and for the Dignity of Victims (es)
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.
  • § 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).

Wed
25/3
Annunciation
File:UN emblem blue.svg
International Day of Solidarity with Detained and Missing Staff Members
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.
  • § 2.6 (45, 46, 47).
  • § 4.6 (23, 24, 27).

Thu
26/3

File:Group full edit.svg 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).

Theme 3

Combinatorics

(8 h LG and 3 h S/L)

Group B
Fri
27/3

Groups A and E
Mon
30/3

Seminar/Laboratory No. 9:
Combinatorics, I
  • (Occasionally computer-assisted) hands-on word-problem-solving on issues related to:
    • combinatorics.
  • (Everything we have studied on the subject).
  • (Everything we have studied on the subject).
  • (Everything we have studied on the subject).
  • (Everything we have studied on the subject).

Tue
31/3

Combinatorics
  • The basics of counting.
  • § 4.1.
  • § 6.1.
  • § 4.1 (6, 16, 22, 28, 33, 37, 41, 51, 52).
  • § 6.1 (8, 16, 26, 32, 37, 41, 49, 67, 68).

Wed
1/4

Combinatorics
  • The pigeonhole principle.
  • § 4.2.
  • § 6.2.
  • § 4.2 (9, 10, 16, 17, 18, 20, 24, 25, 26, 36).
  • § 6.2 (9, 10, 16, 17, 18, 20, 26, 27, 28, 40).

Thu
2/4
File:UN emblem blue.svg
World Autism Awareness Day

Combinatorics
  • Permutations and combinations;
  • Binomial coefficients and identities.
  • § 4.3;
  • § 4.4
  • § 6.3;
  • § 6.4.
  • § 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).
Error creating thumbnail: 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 contributions page of the project. Furthermore, you must publish, also on an ongoing basis, in your logbook (sandbox), the part of your self-report that deals with what you have developed so far.

Group B
Fri
3/4
Lent
Friday of Sorrows

Groups A and E
Mon
20/4

Seminar/Laboratory No. 10:
Combinatorics, II
  • (Occasionally computer-assisted) hands-on word-problem-solving on issues related to:
    • combinatorics.
  • (Everything we have studied on the subject).
  • (Everything we have studied on the subject).
  • (Everything we have studied on the subject).
  • (Everything we have studied on the subject).

Sat
4/4
Lazarus Saturday
File:UN emblem blue.svg
International Day for Mine Awareness and Assistance in Mine Action (es)

Mon
6/4
Holy Week
Holy Monday
File:UN emblem blue.svg
International Day of Sport for Development and Peace

Tue
7/4
Holy Week
Holy Tuesday
File:UN emblem blue.svg
World Health Day

Wed
8/4
Holy Week
Holy Wednesday

Thu
9/4
Holy Week
Maundy Thursday

Fri
10/4
Holy Week
Good Friday

Sun
12/4
Holy Week
Resurrection Sunday
File:UN emblem blue.svg
International Day of Human Space Flight

Mon
13/4
Eastertide
Easter Monday

Tue
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.
  • § 4.5 (10, 15, 16, 34, 56).
  • § 6.5 (10, 15, 16, 34, 66).

Wed
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.
  • § 4.5.5 (22, 47, 50).
  • § 6.5.5 (22, 47, 50).

Thu
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.
  • § 4.5.5 (22, 47, 50).
  • § 6.5.5 (22, 47, 50).

Group B
Fri
17/4

Groups A and E
Mon
4/5

Seminar/Laboratory No. 11:
Combinatorics, III
  • (Occasionally computer-assisted) hands-on word-problem-solving on issues related to:
    • combinatorics.
  • (Everything we have studied on the subject).
  • (Everything we have studied on the subject).
  • (Everything we have studied on the subject).
  • (Everything we have studied on the subject).

Sun
19/4
File:UN emblem blue.svg
UN Chinese Language Day

Tue
21/4
File:UN emblem blue.svg
World Creativity and Innovation Day

Combinatorics
  • Partitions of a set.

Wed
22/4
File:UN emblem blue.svg
Earth Day

Combinatorics
  • Additive decompositions of numbers.

Thu
23/4
Saint George's Day
File:UN emblem blue.svg
World Book and Copyright Day
UN English Language Day
UN Spanish Language Day
International Girls in ICT Day (es)

Theme 4

Difference equations

(8 h LG and 2 h S/L)

(1h LG Solving the end-course preparatory exam)

Group B
Fri
24/4
File:UN emblem blue.svg
International Day of Multilateralism and Diplomacy for Peace (ref)

Groups A and E
Mon
27/4

Seminar/Laboratory No. 12:
Difference equations, I
  • (Occasionally computer-assisted) hands-on word-problem-solving on issues related to:
    • linear finite difference equations.
  • (Everything we have studied on the subject);
  • § 6.3.
  • (Everything we have studied on the subject);
  • § 8.3.
  • (Everything we have studied on the subject);
  • § 6.2 (45, 46, 47);
  • § 6.3 (10, 11, 14, 15, 16).
  • (Everything we have studied on the subject);
  • § 8.2 (45, 46, 47);
  • § 8.3 (10, 11, 14, 15, 16).

Sat
25/4
File:UN emblem blue.svg
World Malaria Day
International Delegate's Day (ref)

Sun
26/4
File:UN emblem blue.svg
World Intellectual Property Day
International Chernobyl Disaster Remembrance Day (ref)

Tue
28/4
File:UN emblem blue.svg
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.
  • § 6.1 (17, 22, 23, 25, 27, 36, 37, 42, 46).
  • § 8.1 (1, 6, 7, 9, 11, 20, 21, 26, 30).

Wed
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.
  • § 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).

Thu
30/4
File:UN emblem blue.svg
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.
  • § 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).

Fri
1/5
International Workers' Day

Sat
2/5
File:UN emblem blue.svg
World Tuna Day (es)

Sun
3/5
File:UN emblem blue.svg
World Press Freedom Day

Tue
5/5
File:UN emblem blue.svg
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.
  • § 6.2 (23, 24, 26, 31).
  • § 8.2 (23, 24, 26, 31).

Wed
6/5

Finite difference equations (recurrence relations)
  • Non-homogeneous linear finite difference equation with constant coefficients, II.
  • § 6.2.3.
  • § 8.2.3.
  • § 6.2 (23, 24, 26, 31).
  • § 8.2 (23, 24, 26, 31).

Thu
7/5
File:UN emblem blue.svg
Vesak Day
(Held on the/a full moon day of May each year)

Finite difference equations (recurrence relations)
  • Systems of linear finite difference equations, I.
 — — — —
Error creating thumbnail: 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 contributions page of the project. Furthermore, you must publish, also on an ongoing basis, in your logbook (sandbox), the part of your 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.

Fri
8/5
Academic celebration, Cáceres School of Technology (es)
File:UN emblem blue.svg
Time of Remembrance and Reconciliation for Those Who Lost Their Lives during the Second World War

Sat
9/5
File:UN emblem blue.svg
Time of Remembrance and Reconciliation for Those Who Lost Their Lives during the Second World War
World Migratory Bird Day (es)

Group B
Mon
11/5
(please come and attend as far as possible)

Groups A and E
Mon
11/5

Seminario/Laboratorio N.º 13:
Difference equations, II
  • (Occasionally computer-assisted) hands-on word-problem-solving on issues related to:
    • finite difference equations.
  • (Everything we have studied on the subject).
  • (Everything we have studied on the subject).
  • (Everything we have studied on the subject).
  • (Everything we have studied on the subject).

Tue
12/5

Finite difference equations (recurrence relations)
  • Systems of linear finite difference equations, II.
 — — — —

Wed
13/5

Finite difference equations (recurrence relations)
  • Systems of linear finite difference equations, III.
 — — — —

Thu
14/5
End of classes

File:Group full edit.svg 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).
Error creating thumbnail: University project Discrete and numerical mathematics (optional out-of-class activity): Ending date of the academic component in the 2nd semester of the academic year 2019-2020.

File:UN emblem blue.svg

15 May
International Day of Families

16 May
International Day of Living Together in Peace (es)
International Day of Light (es)

17 May
World Telecommunication and Information Society Day

Mon
20/5
Start of June exam period

File:UN emblem blue.svg

20 May
World Bee Day

21 May
World Day for Cultural Diversity for Dialogue and Development
International Tea Day (UN) (ref)

22 May
International Day for Biological Diversity

23 May
International Day to End Obstetric Fistula (es)

29 May
International Day of United Nations Peacekeepers

31 May
World No Tobacco Day

___ __/_ Error creating thumbnail: 2019-2020 Final exam.

File:UN emblem blue.svg

...

Sat
6/7
End of June exam period

File:UN emblem blue.svg

...

Mon
22/6
Start of July exam period

File:UN emblem blue.svg

...

___ __/_ Error creating thumbnail: 2019-2020 Resit final exam.

File:UN emblem blue.svg

...

Fri
10/7
End of July exam period

File:UN emblem blue.svg

...

Mon
20/7
End of term

File:UN emblem blue.svg

...

(See: International days currently observed by the United Nations).

Coda

Ex post I: Arts & Humanities Classroom 'Juanelo Turriano'

(For the time being, please see Ex post I on the Spanish Wikipedia).

Ex post II: Humour, entertainment and curiosities

Template:Humorous essays

Notes

Template:Reflist

References

Template:Reflist

See also

Inner links
Interwiki links

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.

Declaration of conformity

Juan Miguel León Rojas declares under his own responsibility that the learning plan specified here meets all the essential requirements of the academic program (ficha12a) corresponding to the course Further Mathematics taught at the School of Technology, University of Extremadura.

About this page on the English Wikipedia


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Transverse information 1.- Some points about Free/Libre & Open Knowledge (FLOK)

'True poems of cante jondo are attributable to no one at all
but float on the wind like golden thistledown
and each generation clothes them in its own distinctive color,
in releasing them to the future.'
Federico García Lorca (1898-1936): Importancia histórica y artística del primitivo canto andaluz llamado «cante jondo» (Historical and artistic importance of the primitive Andalusian song, that which is called deep song, cante jondo.). (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 Poetry in Translation, vid. http://www.poetryintranslation.com/PITBR/Spanish/DeepSong.htm.

Some referencies about free/libre & open licenses and free knowledge and culture

Practising lawyers who are specialised in intellectual property and computer law (IP & IT lawyers)

Some open repositories

Take into account the right of quotation

And the possible plagiarism

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Libraries (texts, courses)

Transverse information 2.- About more topics of interest

Accessibility and usability

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Library

Fair trade

— Technology

Hacker ethic

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Philosophy

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Periodical library

Tools

LaTeX

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Spanish language

(Spanish language)
— U.S. Spanish
— Collocations dictionaries
(Collocation)
— Spelling and grammar checkers
— Plain Spanish
(Plain language)

English language

(English language)
— Academic English
— American English
— British English
— Collocations dictionaries
— Dictionaries and thesauri
— English and mathematics
— Plain English

Mathematics

— Web sites
— Android apps
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Programming

— Languages
— Online interpreters

Digital preservation

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Knowledge representation

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Computer security

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Free and open-source software

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— & —

Snoozing

Bubbles

Miscellanea

Wikimedia, in English

— Wikimedia
— Wikipedia
— Wikibooks

Wikimedia, in Spanish

— Wikipedia
— Wikilibros
  1. ↑ Template:Cite journal


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