Definition:Characteristic Function (Set Theory)/Set

Definition
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}$

That is:
 * $\chi_E \left({x}\right) = \begin{cases}

1 & : x \in E \\ 0 & : x \in \complement_S \left({E}\right) \end{cases}$ where $\complement_S \left({E}\right)$ denotes the complement of $E$ relative to $S$.

Also denoted as
The characteristic function of $E$ can be expressed in Iverson bracket notation as:
 * $\chi_E \left({x}\right) = \left[{x \in E}\right]$