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MyWikiBiz, Author Your Legacy — Wednesday April 09, 2025
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More generally, a function of the form ''f'' : ''X'' → '''B''', where ''X'' is an arbitrary set, is a ''[[boolean-valued function]]''. If ''X'' = '''M''' = {1, 2, 3, …}, then ''f'' is a ''binary sequence'', that is, an infinite [[sequence]] of 0's and 1's. If ''X'' = [''k''] = {1, 2, 3, …, ''k''}, then ''f'' is ''binary sequence'' of length ''k''.
* [[Boolean domain]]
−* [[Boolean logic]]
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* [[Logical connective]]
−* [[Truth function]]
In [[mathematics]], a '''finitary boolean function''' is a [[function (mathematics)|function]] of the form <math>f : \mathbb{B}^k \to \mathbb{B},</math> where <math>\mathbb{B} = \{ 0, 1 \}</math> is a [[boolean domain]] and where <math>k\!</math> is a nonnegative integer. In the case where <math>k = 0,\!</math> the function is simply a constant element of <math>\mathbb{B}.</math>
In [[mathematics]], a '''finitary boolean function''' is a [[function (mathematics)|function]] of the form <math>f : \mathbb{B}^k \to \mathbb{B},</math> where <math>\mathbb{B} = \{ 0, 1 \}</math> is a [[boolean domain]] and where <math>k\!</math> is a nonnegative integer. In the case where <math>k = 0,\!</math> the function is simply a constant element of <math>\mathbb{B}.</math>
−There are <math>2^{2^k}</math> such functions. These play a basic role in questions of [[complexity theory]] as well as the design of circuits and chips for [[digital computer]]s. The properties of boolean functions play a critical role in [[cryptography]], particularly in the design of [[symmetric key algorithm]]s (see [[S-box]]).
There are <math>2^{2^k}</math> such functions. These play a basic role in questions of [[complexity theory]] as well as the design of circuits and chips for [[digital computer]]s. The properties of boolean functions play a critical role in [[cryptography]], particularly in the design of [[symmetric key algorithm]]s (see [[S-box]]).
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==See also==
==See also==
+
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* [[Algebra of sets]]
* [[Algebra of sets]]
* [[Boolean algebra]]
* [[Boolean algebra]]
−* [[List of Boolean algebra topics|Boolean algebra topics]]
+* [[Boolean domain]]
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−* [[Boolean value]]
* [[Boolean value]]
−* [[Boolean-valued function]]
* [[Boolean-valued function]]
−* [[Zeroth order logic]]
* [[Zeroth order logic]]
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