The Info List - Boolean Function

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In mathematics and logic, a (finitary) Boolean function (or switching function) is a function of the form ƒ : Bk → 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 x1, …, xk, and two propositional formulas are logically equivalent if and only if they express the same Boolean function. There are 22k k-ary functions for every k. Boolean functions in applications[edit] 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[edit]


Algebra of sets Boolean algebra Boolean algebra
Boolean algebra
topics Boolean domain Boolean differential calculus Boolean-valued function Logical connective Truth function Truth table Symmetric Boolean function Decision tree model Evasive Boolean function Indicator function Balanced boolean function Read-once function Pseudo-Boolean function 3-ary Boolean functions


Crama, Y; Hammer, P. L. (2011), Boolean Functions, Cambridge University Press . Hazewinkel, Michiel, ed. (2001) [1994], "Boolean function", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, 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). Archived from the original (PDF) on 2016-03-05. Retrieved 2015-06-07.  Mano, M. M.; Ciletti, M. D. (2013), Digital Design, Pearson .

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Mathematical logic


Formal language Formation rule Formal proof Formal semantics Well-formed formula Set Element Class Classical logic Axiom Rule of inference Relation Theorem Logical consequence Type theory Symbol Syntax Theory


Formal system Deductive system Axiomatic system Hilbert style systems Natural deduction Sequent calculus

Traditional logic

Proposition Inference Argument Validity Cogency Syllogism Square of opposition Venn diagram

Propositional calculus Boolean logic

Boolean functions Propositional calculus Propositional formula Logical connectives Truth tables Many-valued logic

Predicate logic

First-order Quantifiers Predicate Second-order Monadic predicate calculus

Naive set theory

Set Empty set Element Enumeration Extensionality Finite set Infinite set Subset Power set Countable set Uncountable set Recursive set Domain Codomain Image Map Function Relation Ordered pair

Set theory

Foundations of mathematics Zermelo–Fraenkel set theory Axiom
of choice General set theory Kripke–Platek set theory Von Neumann–Bernays–Gödel set theory Morse–Kelley set theory Tarski–Grothendieck set theory

Model theory

Model Interpretation Non-standard model Finite model theory Truth value Validity

Proof theory

Formal proof Deductive system Formal system Theorem Logical consequence Rule of inference Syntax

Computability theory

Recursion Recursive set Recursively enumerable set Decision problem Church–Turing thesis Computable function Primitive recursive function

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