Boolean ___domain

Not to be confused with Binary Domain.

In mathematics and abstract algebra, a Boolean ___domain is a 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},^{[1][2][3][4][5]} or ${\displaystyle \mathbb {B} .}$^[6][7]

The algebraic structure that naturally builds on a Boolean ___domain is the Boolean algebra with two elements. The initial object in the category of bounded lattices is a Boolean ___domain.

In computer science, a Boolean variable is a variable that takes values in some Boolean ___domain. Some programming languages feature reserved words or symbols for the elements of the Boolean ___domain, for example false and true. However, many programming languages do not have a Boolean data type in the strict sense. In 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 [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 ${\displaystyle 1-x,}$  conjunction (AND) is replaced with multiplication (${\displaystyle xy}$ ), and disjunction (OR) is defined via De Morgan's law to be ${\displaystyle 1-(1-x)(1-y)=x+y-xy}$ .

Interpreting these values as logical truth values 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

1. van Dalen, Dirk (2004). Logic and Structure. Springer. p. 15.
2. Makinson, David (2008). Sets, Logic and Maths for Computing. Springer. p. 13. Bibcode:2008slmc.book.....M.
3. Boolos, George S.; Jeffrey, Richard C. (1980). Computability and Logic. Cambridge University Press. p. 99.
4. Mendelson, Elliott (1997). Introduction to Mathematical Logic (4 ed.). Chapman & Hall/CRC. p. 11.
5. Hehner, Eric C. R. (2010) [1993]. A Practical Theory of Programming. Springer. p. 3.
6. Parberry, Ian (1994). Circuit Complexity and Neural Networks. MIT Press. pp. 65. ISBN 978-0-262-16148-0.
7. Cortadella, Jordi; Kishinevsky, Michael; Kondratyev, Alex; Lavagno, Luciano; Yakovlev, Alex (2002). Logic Synthesis for Asynchronous Controllers and Interfaces. Springer Series in Advanced Microelectronics. Vol. 8. Springer-Verlag Berlin Heidelberg New York. p. 73. ISBN 3-540-43152-7. ISSN 1437-0387.

Further reading

• Steinbach, Bernd [in German], ed. (2014-04-01) [2013-09-25]. Written at Freiberg, Germany. Recent Progress in the Boolean Domain (1 ed.). Newcastle upon Tyne, UK: Cambridge Scholars Publishing. ISBN 978-1-4438-5638-6. Retrieved 2019-08-04. [1] (xxx+428 pages) [2] (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.)
• Steinbach, Bernd [in German], ed. (2016-05-01). Written at Freiberg, Germany. Problems and New Solutions in the Boolean Domain (1 ed.). Newcastle upon Tyne, UK: Cambridge Scholars Publishing. ISBN 978-1-4438-8947-6. Retrieved 2019-08-04. (xxxv+1+445+1 pages) [3] (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.)
• Steinbach, Bernd [in German], ed. (2018-01-01). Written at Freiberg, Germany. Further Improvements in the Boolean Domain (1 ed.). Newcastle upon Tyne, UK: Cambridge Scholars Publishing. ISBN 978-1-5275-0371-7. Retrieved 2019-08-04. [4] Archived 2019-08-04 at the Wayback Machine (xli+1+494 pages) [5] (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.)
• Drechsler, Rolf; Soeken, Mathias, eds. (2020) [March 2019]. Written at Bremen, Germany. Advanced Boolean Techniques - Selected Papers from the 13th International Workshop on Boolean Problems (1 ed.). Cham, Switzerland: Springer Nature Switzerland AG. doi:10.1007/978-3-030-20323-8. ISBN 978-3-030-20322-1. S2CID 240782759. (vii+265+7 pages) [6] (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.)
• Drechsler, Rolf; Große, Daniel, eds. (2021-04-30). Recent Findings in Boolean Techniques - Selected Papers from the 14th International Workshop on Boolean Problems (1 ed.). Cham, Switzerland: Springer Nature Switzerland AG. doi:10.1007/978-3-030-68071-8. ISBN 978-3-030-68070-1. (vii+1+197+5 pages) [7] (NB. Contains extended versions of the best manuscripts from the 14th International Workshop on Boolean Problems (IWSBP 2020) held virtually on 2020-09-24/25.)
• Steinbach, Bernd [in German], ed. (2022-09-29). Written at Freiberg, Germany. Advances in the Boolean Domain (1 ed.). Newcastle upon Tyne, UK: Cambridge Scholars Publishing. ISBN 978-1527-58872-1. Retrieved 2024-07-15. (xxii+231+1 pages)
• Drechsler, Rolf; Huhn, Sebastian, eds. (2023-05-30). Written at Bremen, Germany. Advanced Boolean Techniques - Selected Papers from the 15th International Workshop on Boolean Problems (1 ed.). Cham, Switzerland: Springer Nature Switzerland AG. doi:10.1007/978-3-031-28916-3. ISBN 978-3-031-28915-6. (viii+172+6 pages) [8] (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.)