# Indicator function

In the mathematical subfield of set theory, the indicator function, or characteristic function, is a function defined on a set X which is used to indicate membership of an element in a subset A of X.

Remark. The term "characteristic function" has an unrelated meaning in probability theory; see characteristic function.

The indicator function of a subset A of a set X is a function

$1_{A}:X\to \lbrace 0,1\rbrace$ defined as

$1_{A}(x)=\left\{{\begin{matrix}1&{\mbox{if}}\ x\in A\\0&{\mbox{if}}\ x\notin A\end{matrix}}\right.$ The indicator function of A is sometimes denoted

$\ \chi _{A}(x)$ or $\ I_{A}(x)$ or even $\ A(x).$ (The Greek letter χ because it is the initial letter of the Greek etymon of the word characteristic.)

The Iverson bracket allows the notation $[x\in A]$ .

Warning. The notation $1_{A}$ may signify the identity function.

## Basic properties

The mapping which associates a subset A of X to its indicator function 1A is injective; its range is the set of functions f:X →{0,1}.

If A and B are two subsets of X, then

$1_{A\cap B}=\min\{1_{A},1_{B}\}=1_{A}1_{B},\,$ $1_{A\cup B}=\max\{{1_{A},1_{B}}\}=1_{A}+1_{B}-1_{A}1_{B},$ $1_{A\triangle B}=1_{A}+1_{B}-2(1_{A\cap B}),$ and

$1_{A^{\complement }}=1-1_{A}.$ More generally, suppose A1, ..., An is a collection of subsets of X. For any xX,

$\prod _{k\in I}(1-1_{A_{k}}(x))$ is clearly a product of 0s and 1s. This product has the value 1 at precisely those xX which belong to none of the sets Ak and is 0 otherwise. That is

$\prod _{k\in I}(1-1_{A_{k}})=1_{X-\bigcup _{k}A_{k}}=1-1_{\bigcup _{k}A_{k}}$ Expanding the product on the left hand side,

$1_{\bigcup _{k}A_{k}}=1-\sum _{F\subseteq \{1,2,\ldots ,n\}}(-1)^{|F|}1_{\bigcap _{F}A_{k}}=\sum _{\emptyset \neq F\subseteq \{1,2,\ldots ,n\}}(-1)^{|F|+1}1_{\bigcap _{F}A_{k}}$ where |F| is the cardinality of F. This is one form of the principle of inclusion-exclusion.

As suggested by the previous example, the indicator function is a useful notational device in combinatorics. The notation is used in other places as well, for instance in probability theory: if X is a probability space with probability measure P and A is a measurable set, then 1A becomes a random variable whose expected value is equal to the probability of A:

$E(1_{A})=\int _{X}1_{A}(x)\,dP=\int _{A}dP=P(A).\quad$ This identity is used in a simple proof of Markov's inequality.