# Boolean function

## Function returning one of only two values / From Wikipedia, the free encyclopedia

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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}).[1][2] Alternative names are **switching function**, used especially in older computer science literature,[3][4] 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.