Indicator function

Definition

Let $X$ be a set and let $A$ be a subset of $X$. The indicator function of $A$ is the function $\mathbf{1}_A \colon X \to \{0, 1\}$ with $$\mathbf{1}_A(x) = \begin{cases} 1 & \text{if } x \in A, \\ 0 & \text{if } x \notin A. \end{cases}$$

Remarks

Other names are characteristic function and the notation $\chi_A$. For subsets $A$ and $B$ of $X$, $\mathbf{1}_{A \cap B} = \mathbf{1}_A \mathbf{1}_B$ and $\mathbf{1}_{A^c} = 1 - \mathbf{1}_A$. The map $A \mapsto \mathbf{1}_A$ is a bijection from the power set of $X$ to the set of functions from $X$ to $\{0, 1\}$.

Explanations

L1 human

The indicator function answers "is $x$ in $A$?" with $1$ for yes and $0$ for no. For $A$ the even integers, $\mathbf{1}_A(4) = 1$ and $\mathbf{1}_A(5) = 0$.