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Boolean domain

## Summary

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 datatype 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 ${\displaystyle 1-(1-x)(1-y)}$ .

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.

## References

1. ^ Dirk van Dalen, Logic and Structure. Springer (2004), page 15.
2. ^ David Makinson, Sets, Logic and Maths for Computing. Springer (2008), page 13.
3. ^ George S. Boolos and Richard C. Jeffrey, Computability and Logic. Cambridge University Press (1980), page 99.
4. ^ Elliott Mendelson, Introduction to Mathematical Logic (4th. ed.). Chapman & Hall/CRC (1997), page 11.
5. ^ Eric C. R. Hehner, A Practical Theory of Programming. Springer (1993, 2010), page 3.
6. ^ Parberry, Ian (1994). Circuit Complexity and Neural Networks. MIT Press. pp. 65. ISBN 978-0-262-16148-0.
7. ^ Cortadella, Jordi; et al. (2002). Logic Synthesis for Asynchronous Controllers and Interfaces. Springer Science & Business Media. p. 73. ISBN 978-3-540-43152-7.