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Boolean function - Wikipedia

Boolean function

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In mathematics, a Boolean function is a function whose arguments and result assume values from a two-element set (usually {true, false}, {0,1} or {−1,1}).¹² Alternative names are switching function, used especially in older computer science literature,³⁴ and truth function (or logical function), used in logic. Boolean functions are the subject of Boolean algebra and switching theory.⁵

A Boolean function takes the form $f:\{0,1{\}}^k\to \{0,1\}$, where $\{0,1\}$ is known as the Boolean domain and $k$ is a non-negative integer called the arity of the function. In the case where $k=0$, the function is a constant element of $\{0,1\}$. A Boolean function with multiple outputs, $f:\{0,1{\}}^k\to \{0,1{\}}^m$ with $m>1$ is a vectorial or vector-valued Boolean function (an S-box in symmetric cryptography).⁶

There are $2^{2^k}$ different Boolean functions with $k$ arguments; equal to the number of different truth tables with $2^k$ entries.

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.

Printed 2026-06-28.

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Footnotes

1. “Boolean function - Encyclopedia of Mathematics”. encyclopediaofmath.org. Retrieved 2021-05-03. ↩

2. Weisstein, Eric W. “Boolean Function”. mathworld.wolfram.com. Retrieved 2021-05-03. ↩

3. “switching function”. TheFreeDictionary.com. Retrieved 2021-05-03. ↩

4. Davies, D. W. (December 1957). “Switching Functions of Three Variables”. IRE Transactions on Electronic Computers. EC-6 (4): 265–275. doi:10.1109/TEC.1957.5222038. ISSN 0367-9950. ↩

5. McCluskey, Edward J. (2003-01-01), “Switching theory”, Encyclopedia of Computer Science, GBR: John Wiley and Sons Ltd., pp. 1727–1731, ISBN 978-0-470-86412-8, retrieved 2021-05-03 ↩

6. Carlet, Claude. “Vectorial Boolean Functions for Cryptography” (PDF). University of Paris. Archived (PDF) from the original on 2016-01-17. ↩

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