Indicator Function¶

Definition¶

The indicator function is a function that identifies whether an element of a set satisfies a given property. It is denoted as $\indicator_{A}(x)$ and defined as:

\indicator_{A}(x)= \begin{cases} \displaystyle \,1, & \text{if } x \in A, \\[2pt] \displaystyle \,0, & \text{if } x \notin A. \end{cases}

Interpretation¶

An indicator function encodes set membership as a number: it answers the question "is $x$ in $A$ ?" with $1$ for yes and $0$ for no.

The indicator function of an interval A, equal to 1 on A, drawn as a raised segment, and 0 elsewhere on the line.

Here $A=[1,3]$ $[1, 3]$ . The graph is at height $1$ above $A$ and at height $0$ everywhere else, so it jumps at the two endpoints. The filled dots at $(1,1)$ and $(3,1)$ show that the endpoints belong to $A$ and receive the value $1$ .

Set operations become arithmetic on the values. For all sets $A$ and $B$ ,

\indicator_{A\cap B}=\indicator_A\indicator_B,\quad \indicator_{A^{\complement}}=1-\indicator_A,

and $\indicator_{A\cup B}=\indicator_A+\indicator_B-\indicator_A\indicator_B$ $𝟙 1_{A} 𝟙 1_{B}$ ; each identity is checked by going through the four cases of whether $x$ lies in $A$ and in $B$ . In probability the same idea returns as the indicator random variable, which is $1$ when an event occurs and $0$ otherwise.

Key Results¶

Indicator functions take only the values $0$ and $1\text{.}$
Set operations can be translated into algebraic relations between indicator functions.
Indicator functions are widely used in integration and probability.

Examples¶

$\indicator_{[0,1]}(x)=1$ for $x\in[0,1]$ and $0$ otherwise.
$\indicator_{\mathbb{Q}}(x)$ distinguishes rational from irrational numbers.

Prerequisites¶

Function

Sources¶

Math I: Calculus · Lecture 4

Aliases: characteristic function of a set Symbols: element of, intersection, rational numbers, union

Last updated: 20 October 2026 03:02