Indicator function

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In mathematics, an indicator function or a characteristic function of a subset of a set is a function that maps elements of the subset to one, and all other elements to zero. That is, if A is a subset of some set X, then ${\displaystyle \ \mathbf {1} _{A}\!(x)\equiv 1\ }$ if ${\displaystyle \ x\in A\ ,}$ and ${\displaystyle \ \mathbf {1} _{A}\!(x)\equiv 0\ }$ otherwise, where ${\displaystyle \ \mathbf {1} _{A}\ }$ is one common notation for the indicator function; other common notations are ${\displaystyle \ I_{A}\!(x)\ ,}$ ${\displaystyle \ \chi _{A}\!(x)\ ,}$^[a] and ${\displaystyle \ \operatorname {\mathbb {I} } \!{\bigl (}\ x\in A\ {\bigr )}~.}$

The indicator function of A is the Iverson bracket of the property of belonging to A; that is,

$${\displaystyle \ \mathbf {1} _{A}(x)=\left[\ x\in A\ \right]~.}$$

For example, the Dirichlet function is the indicator function of the rational numbers as a subset of the real numbers.
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