View source for Boolean domain

{{short description|Concept in mathematical logic}}
{{distinguish|Binary Domain}}
{{use dmy dates|date=August 2019|cs1-dates=y}}
{{use list-defined references|date=January 2022}}
{{cs1 config|mode=cs1|name-list-style=none|display-authors=all}}
In [[mathematics]] and [[abstract algebra]], a '''Boolean domain''' is a [[Set (mathematics)|set]] consisting of exactly two elements whose interpretations include ''false'' and ''true''. In [[logic]], mathematics and [[theoretical computer science]], a Boolean domain is usually written as {0, 1},<ref name="vanDalen_2004"/><ref name="Makinson_2008"/><ref name="Boolos-Jeffrey_1980"/><ref name="Mendelson_1997"/><ref name="Hehner_2010"/> or <math>\mathbb{B}.</math><ref name="Parberry_1994"/><ref name="Cortadella-Kishinevsky-Kondratyev-Lavagno-Yakovlev_2002"/>

The [[algebraic structure]] that naturally builds on a Boolean domain is the [[two-element Boolean algebra|Boolean algebra with two elements]]. The [[initial object]] in the [[category (mathematics)|category]] of [[bounded lattice]]s is a Boolean domain.

In [[computer science]], a Boolean variable is a [[Variable (programming)|variable]] that takes values in some Boolean domain. Some [[programming language]]s feature [[reserved word]]s or symbols for the elements of the Boolean domain, for example <code>false</code> and <code>true</code>. However, many programming languages do not have a [[Boolean data type]] in the strict sense. In [[C (programming language)|C]] or [[BASIC]], for example, falsity is represented by the number 0 and truth is represented by the number 1 or −1, and all variables that can take these values can also take any other numerical values.

== Generalizations ==
The Boolean domain {0, 1} can be replaced by the [[unit interval]] {{closed-closed|0,1}}, in which case rather than only taking values 0 or 1, any value between and including 0 and 1 can be assumed. Algebraically, negation (NOT) is replaced with <math>1-x,</math> conjunction (AND) is replaced with multiplication (<math>xy</math>), and disjunction (OR) is defined via [[De Morgan's law]] to be <math>1-(1-x)(1-y)=x+y-xy</math>.

Interpreting these values as logical [[truth value]]s yields a [[multi-valued logic]], which forms the basis for [[fuzzy logic]] and [[probabilistic logic]]. In these interpretations, a value is interpreted as the "degree" of truth – to what extent a proposition is true, or the probability that the proposition is true.

==See also==
* [[Boolean-valued function]]
* [[GF(2)]]

==References==
{{reflist|refs=
<ref name="vanDalen_2004">{{cite book |author-first=Dirk |author-last=van Dalen |author-link=Dirk van Dalen |title=Logic and Structure |publisher=Springer |date=2004 |edition= |isbn= |page=15}}</ref>
<ref name="Makinson_2008">{{cite book |author-first=David |author-last=Makinson |author-link=David Makinson |title=Sets, Logic and Maths for Computing |publisher=Springer |date=2008 |edition= |isbn= |page=13|bibcode=2008slmc.book.....M }}</ref>
<ref name="Boolos-Jeffrey_1980">{{cite book |author-first1=George S. |author-last1=Boolos |author-link1=George S. Boolos |author-first2=Richard C. |author-last2=Jeffrey |author-link2=Richard C. Jeffrey |title=Computability and Logic |publisher=[[Cambridge University Press]] |date=1980 |edition= |isbn= |page=99}}</ref>
<ref name="Mendelson_1997">{{cite book |author-first=Elliott |author-last=Mendelson |author-link=Elliott Mendelson |title=Introduction to Mathematical Logic |edition=4 |publisher=[[Chapman & Hall]]/CRC |date=1997 |page=11}}</ref>
<ref name="Hehner_2010">{{cite book |author-first=Eric C. R. |author-last=Hehner |author-link=Eric C. R. Hehner |title=A Practical Theory of Programming |publisher=Springer |date=2010 |orig-date=1993 |page=3}}</ref>
<ref name="Parberry_1994">{{cite book |author-first=Ian |author-last=Parberry |title=Circuit Complexity and Neural Networks |url=https://archive.org/details/circuitcomplexit00parb |url-access=limited |date=1994 |publisher=[[MIT Press]] |isbn=978-0-262-16148-0 |pages=[https://archive.org/details/circuitcomplexit00parb/page/n86 65]}}</ref>
<ref name="Cortadella-Kishinevsky-Kondratyev-Lavagno-Yakovlev_2002">{{cite book |author-first1=Jordi |author-last1=Cortadella |author-link1=Jordi Cortadella |author-first2=Michael |author-last2=Kishinevsky |author-first3=Alex |author-last3=Kondratyev |author-first4=Luciano |author-last4=Lavagno |author-first5=Alex |author-last5=Yakovlev |title=Logic Synthesis for Asynchronous Controllers and Interfaces |date=2002 |publisher=[[Springer-Verlag Berlin Heidelberg New York]] |isbn=3-540-43152-7 |series=Springer Series in Advanced Microelectronics |volume=8 |issn=1437-0387 |page=[https://archive.org/details/logicsynthesisfo0000unse/page/73 73] |url=https://archive.org/details/logicsynthesisfo0000unse/page/73}}</ref>
}}

==Further reading==
* {{cite book |editor-first=Bernd |editor-last=Steinbach |editor-link=:de:Bernd Steinbach |title=Recent Progress in the Boolean Domain |publisher=[[Cambridge Scholars Publishing]] |publication-place=Newcastle upon Tyne, UK |location=Freiberg, Germany |edition=1 |date=2014-04-01 |orig-date=2013-09-25 |isbn=978-1-4438-5638-6 |url=https://books.google.com/books?id=_pwxBwAAQBAJ |access-date=2019-08-04}} [https://web.archive.org/web/20210228020207/https://www.tau.ac.il/~ilia1/publications/rpbd_book.pdf<!-- https://m.tau.ac.il/~ilia1/publications/rpbd_book.pdf draft version -->] (xxx+428 pages) [https://web.archive.org/web/20200131123408/http://www.informatik.tu-freiberg.de/prof2/ws_bp10/index.html] (NB. Contains extended versions of the best manuscripts from the 10th International Workshop on Boolean Problems held at the [[Technische Universität Bergakademie Freiberg]], Germany on 2012-09-19/21.)
* {{cite book |editor-first=Bernd |editor-last=Steinbach |editor-link=:de:Bernd Steinbach |title=Problems and New Solutions in the Boolean Domain |publisher=[[Cambridge Scholars Publishing]] |publication-place=Newcastle upon Tyne, UK |location=Freiberg, Germany |edition=1 |date=2016-05-01 |isbn=978-1-4438-8947-6 |url=https://books.google.com/books?id=ODX5DAAAQBAJ&pg=PP1 |access-date=2019-08-04}} (xxxv+1+445+1 pages) [https://web.archive.org/web/20200112133809/http://www.informatik.tu-freiberg.de/prof2/ws_bp11/index.html] (NB. Contains extended versions of the best manuscripts from the 11th International Workshop on Boolean Problems held at the Technische Universität Bergakademie Freiberg, Germany on 2014-09-17/19.)
* {{cite book |editor-first=Bernd |editor-last=Steinbach |editor-link=:de:Bernd Steinbach |title=Further Improvements in the Boolean Domain |publisher=[[Cambridge Scholars Publishing]] |publication-place=Newcastle upon Tyne, UK |location=Freiberg, Germany |edition=1 |date=2018-01-01 |isbn=978-1-5275-0371-7 |url=https://books.google.com/books?id=DnSFDwAAQBAJ |access-date=2019-08-04}} [https://www.cambridgescholars.com/download/sample/64246] {{webarchive |url=https://web.archive.org/web/20190804164806/https://www.cambridgescholars.com/download/sample/64246 |date=2019-08-04}} (xli+1+494 pages) [https://web.archive.org/web/20200112133313/http://www.informatik.tu-freiberg.de/prof2/ws_bp12/index.html] (NB. Contains extended versions of the best manuscripts from the 12th International Workshop on Boolean Problems held at the Technische Universität Bergakademie Freiberg, Germany on 2016-09-22/23.)
* {{cite book |editor-first1=Rolf |editor-last1=Drechsler |editor-link1=Rolf Drechsler |editor-first2=Mathias |editor-last2=Soeken |title=Advanced Boolean Techniques - Selected Papers from the 13th International Workshop on Boolean Problems |publisher=[[Springer Nature Switzerland AG]] |publication-place=Cham, Switzerland |location=Bremen, Germany |edition=1 |date=2020 |orig-date=March 2019 |isbn=978-3-030-20322-1 |doi=10.1007/978-3-030-20323-8 |s2cid=240782759}} (vii+265+7 pages) [https://web.archive.org/web/20210515084442/http://www.informatik.uni-bremen.de/iwsbp/2018/] (NB. Contains extended versions of the best manuscripts from the 13th International Workshop on Boolean Problems (IWSBP 2018) held at the [[University of Bremen]], Bremen, Germany on 2018-09-19/21.)
* {{cite book |editor-first1=Rolf |editor-last1=Drechsler |editor-link1=Rolf Drechsler |editor-first2=Daniel |editor-last2=Große |title=Recent Findings in Boolean Techniques - Selected Papers from the 14th International Workshop on Boolean Problems |publisher=[[Springer Nature Switzerland AG]] |publication-place=Cham, Switzerland |edition=1 |date=2021-04-30 |isbn=978-3-030-68070-1 |doi=10.1007/978-3-030-68071-8}} (vii+1+197+5 pages) [https://web.archive.org/web/20210515084207/http://www.informatik.uni-bremen.de/iwsbp/] (NB. Contains extended versions of the best manuscripts from the 14th International Workshop on Boolean Problems (IWSBP 2020) held [[COVID-19|virtually]] on 2020-09-24/25.)
* {{cite book |editor-first=Bernd |editor-last=Steinbach |editor-link=:de:Bernd Steinbach |title=Advances in the Boolean Domain |publisher=[[Cambridge Scholars Publishing]] |publication-place=Newcastle upon Tyne, UK |location=Freiberg, Germany |edition=1 |date=2022-09-29 |isbn=978-1527-58872-1 |url=https://books.google.com/books?id=5K2REAAAQBAJ |access-date=2024-07-15}} (xxii+231+1 pages)
* {{cite book |editor-first1=Rolf |editor-last1=Drechsler |editor-link1=Rolf Drechsler |editor-first2=Sebastian |editor-last2=Huhn |title=Advanced Boolean Techniques - Selected Papers from the 15th International Workshop on Boolean Problems |publisher=[[Springer Nature Switzerland AG]] |publication-place=Cham, Switzerland |location=Bremen, Germany |edition=1 |date=2023-05-30 |isbn=978-3-031-28915-6 |doi=10.1007/978-3-031-28916-3}} (viii+172+6 pages) [https://web.archive.org/web/20240616190220/https://www.informatik.uni-bremen.de/iwsbp/2022/] (NB. Contains extended versions of the best manuscripts from the 15th International Workshop on Boolean Problems (IWSBP 2022) held at the University of Bremen, Bremen, Germany on 2022-09-22/23.)

[[Category:Boolean algebra]]