# Definition:Characteristic Function (Set Theory)

This page is about Characteristic Function in the context of Set Theory. For other uses, see Characteristic Function.

## Definition

### Set

Let $E \subseteq S$.

The characteristic function of $E$ is the function $\chi_E: S \to \set {0, 1}$ defined as:

$\map {\chi_E} x = \begin {cases} 1 & : x \in E \\ 0 & : x \notin E \end {cases}$

That is:

$\map {\chi_E} x = \begin {cases} 1 & : x \in E \\ 0 & : x \in \relcomp S E \end {cases}$

where $\relcomp S E$ denotes the complement of $E$ relative to $S$.

### Relation

The concept of a characteristic function of a subset carries over directly to relations.

Let $\RR \subseteq S \times T$ be a relation.

The characteristic function of $\RR$ is the function $\chi_\RR: S \times T \to \set {0, 1}$ defined as:

$\map {\chi_\RR} {x, y} = \begin {cases} 1 & : \tuple {x, y} \in \RR \\ 0 & : \tuple {x, y} \notin \RR \end{cases}$

It can be expressed in Iverson bracket notation as:

$\map {\chi_\RR} {x, y} = \sqbrk {\tuple {x, y} \in \RR}$

More generally, let $\displaystyle \mathbb S = \prod_{i \mathop = 1}^n S_i = S_1 \times S_2 \times \ldots \times S_n$ be the cartesian product of $n$ sets $S_1, S_2, \ldots, S_n$.

Let $\RR \subseteq \mathbb S$ be an $n$-ary relation on $\mathbb S$.

The characteristic function of $\RR$ is the function $\chi_\RR: \mathbb S \to \set {0, 1}$ defined as:

$\map {\chi_\RR} {s_1, s_2, \ldots, s_n} = \begin {cases} 1 & : \tuple {s_1, s_2, \ldots, s_n} \in \RR \\ 0 & : \tuple {s_1, s_2, \ldots, s_n} \notin \RR \end {cases}$

It can be expressed in Iverson bracket notation as:

$\map {\chi_\RR} {s_1, s_2, \ldots, s_n} = \sqbrk {\tuple {s_1, s_2, \ldots, s_n} \in \RR}$

## Also known as

It is also known as the indicator function, and $\map {\chi_E} x$ denoted $\map {\mathbf 1_E} x$.

Some sources, in an attempt to apply consistency to the terminology, refer to this concept as a characteristic mapping, but this term appears to be rare.

## Also see

• Results about characteristic functions can be found here.