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 ${\displaystyle \mathbf {1} _{A}(x)=1}$ if ${\displaystyle x\in A,}$ and ${\displaystyle \mathbf {1} _{A}(x)=0}$ otherwise, where ${\displaystyle \mathbf {1} _{A}}$ is a common notation for the indicator function. Other common notations are ${\displaystyle I_{A},}$ and ${\displaystyle \chi _{A}.}$

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

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

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