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}$$

Alternatively, and equivalently, it can be written as:
 * $$\chi_E \left({x}\right) = \begin{cases}

1 & : x \in E  \\ 0 & : x \in \complement_S \left({E}\right) \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 $$\mathbf 1_E \left({x}\right)$$.