Boolean function
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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 nonnegative integer called the arity of the function. In the case where k = 0, the "function" is essentially a constant element of B.
Every kary 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} kary functions for every k.
Boolean functions in applications
A function that can be utilized to evaluate any Boolean output in relation to its Boolean input by logical type of calculations. 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
 Algebra of sets
 Boolean algebra
 Boolean algebra topics
 Boolean domain
 Boolean differential calculus
 Booleanvalued function
 Logical connective
 Truth function
 Truth table
 Symmetric Boolean function
 Decision tree model
 Evasive Boolean function
 Indicator function
 Balanced boolean function
 Readonce function
 PseudoBoolean function
 3ary Boolean functions
References
 Crama, Y; Hammer, P. L. (2011), Boolean Functions : Theory, Algorithms, and Applications, Cambridge University Press, doi:10.1017/CBO9780511852008, ISBN 9780511852008.
 Hazewinkel, Michiel, ed. (2001) [1994], "Boolean function", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 9781556080104
 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). Archived from the original (PDF) on 20160305. Retrieved 20150607.
 Mano, M. M.; Ciletti, M. D. (2013), Digital Design, Pearson.
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