# Boolean function

In mathematics and logic, a **(finitary) Boolean function** (or switching function) is a function of the form *ƒ* : **B**^{k} → **B**, where **B** = {0, 1} is a *Boolean domain* and *k* is a non-negative integer called the arity of the function. In the case where *k* = 0, the "function" is essentially a constant element of **B**.

Every *k*-ary Boolean function can be expressed as a propositional formula in *k* variables *x*_{1}, …, *x*_{k}, and two propositional formulas are logically equivalent if and only if they express the same Boolean function. There are 2^{2k} *k*-ary functions for every *k*.

## Boolean functions in applications

A Boolean function describes how to determine a Boolean value output based on some logical calculation from Boolean inputs. Such functions play a basic role in questions of complexity theory as well as the design of circuits and chips for digital computers. The properties of Boolean functions play a critical role in cryptography, particularly in the design of symmetric key algorithms (see substitution box).

Boolean functions are often represented by sentences in propositional logic, and sometimes as multivariate polynomials over GF(2), but more efficient representations are binary decision diagrams (BDD), negation normal forms, and propositional directed acyclic graphs (PDAG).

In cooperative game theory, monotone Boolean functions are called **simple games** (voting games); this notion is applied to solve problems in social choice theory.

## See also

## References

- Crama, Y; Hammer, P. L. (2011),
*Boolean Functions*, Cambridge University Press. - Hazewinkel, Michiel, ed. (2001), "Boolean function",
*Encyclopedia of Mathematics*, Springer, ISBN 978-1-55608-010-4 - Janković, Dragan; Stanković, Radomir S.; Moraga, Claudio (November 2003). "Arithmetic expressions optimisation using dual polarity property" (PDF).
*Serbian Journal of Electrical Engineering*.**1**(71-80, number 1). Retrieved 2015-06-07. - Mano, M. M.; Ciletti, M. D. (2013),
*Digital Design*, Pearson.