Definition:Characteristic Function (Set Theory)

Set
Let $$E \subseteq S$$.

The characteristic function of $$E \ $$ is the function $$\chi_E: S \to \left\{{0, 1}\right\}$$ defined as:
 * $$\chi_E \left({x}\right) = \begin{cases}

1 & : x \in E  \\ 0 & : x \notin E \end{cases}$$

It can be expressed in Iverson bracket notation as:
 * $$\chi_E \left({x}\right) = \left[{x \in E}\right]$$

Relation
Let $$\mathcal{R} \subseteq S \times T$$ be a relation.

The characteristic function of $$\mathcal{R}$$ is the function $$\chi_{\mathcal{R}}: S \times T \to \left\{{0, 1}\right\}$$ defined as:
 * $$\chi_{\mathcal{R}} \left({x, y}\right) = \begin{cases}

1 & : \left({x, y}\right) \in \mathcal{R} \\ 0 & : \left({x, y}\right) \notin \mathcal{R} \end{cases}$$

It can be expressed in Iverson bracket notation as:
 * $$\chi_{\mathcal{R}} \left({x, y}\right) = \left[{\left({x, y}\right) \in \mathcal{R}}\right]$$

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

Let $$\mathcal{R} \subseteq \mathbb{S}$$ be an $n$-ary relation on $$\mathbb{S}$$.

The characteristic function of $$\mathcal{R}$$ is the function $$\chi_{\mathcal{R}}: \mathbb{S} \to \left\{{0, 1}\right\}$$ defined as:
 * $$\chi_{\mathcal{R}} \left({s_1, s_2, \ldots, s_n}\right) = \begin{cases}

1 & : \left({s_1, s_2, \ldots, s_n}\right) \in \mathcal{R} \\ 0 & : \left({s_1, s_2, \ldots, s_n}\right) \notin \mathcal{R} \end{cases}$$

It can be expressed in Iverson bracket notation as:
 * $$\chi_{\mathcal{R}} \left({s_1, s_2, \ldots, s_n}\right) = \left[{\left({s_1, s_2, \ldots, s_n}\right) \in \mathcal{R}}\right]$$

Also known as
It is also known as the indicator function, and $$\chi_E \left({x}\right)$$ denoted $$\mathbf{1}_E \left({x}\right)$$.