Inmathematics andabstract algebra, aBoolean domain is aset consisting of exactly two elements whose interpretations includefalse andtrue. Inlogic, mathematics andtheoretical computer science, a Boolean domain is usually written as {0, 1},^{[1][2][3][4][5]} or${\displaystyle \mathbb{B}.}$^[6][7]

Thealgebraic structure that naturally builds on a Boolean domain is theBoolean algebra with two elements. Theinitial object in thecategory ofbounded lattices is a Boolean domain.

Incomputer science, a Boolean variable is avariable that takes values in some Boolean domain. Someprogramming languages featurereserved words or symbols for the elements of the Boolean domain, for examplefalse andtrue. However, many programming languages do not have aBoolean data type in the strict sense. InC orBASIC, 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.

The Boolean domain {0, 1} can be replaced by theunit 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 viaDe Morgan's law to be${\displaystyle 1-(1-x)(1-y)=x+y-xy}$.

Interpreting these values as logicaltruth values yields amulti-valued logic, which forms the basis forfuzzy logic andprobabilistic 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.

• Boolean-valued function
• GF(2)

• 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. Retrieved2019-08-04.[1] (xxx+428 pages)[2] (NB. Contains extended versions of the best manuscripts from the 10th International Workshop on Boolean Problems held at theTechnische 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. Retrieved2019-08-04. (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. Retrieved2019-08-04.[4]Archived 2019-08-04 at theWayback 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 theUniversity 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) heldvirtually 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. Retrieved2024-07-15. (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.)

• Boolean algebra

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