Indicator function

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 the indicator function of A is the function ${\displaystyle \mathbf {1} _{A}}$ defined by ${\displaystyle \mathbf {1} _{A}\!(x)=1}$ if ${\displaystyle x\in A,}$ and ${\displaystyle \mathbf {1} _{A}\!(x)=0}$ otherwise. Other common notations are 𝟙_A and ${\displaystyle \chi _{A}.}$

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

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

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