Definition:Characteristic Function (Set Theory)/Set

Definition
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 $S$.

Also denoted as
The characteristic function of $E$ can be expressed in Iverson bracket notation as:
 * $\map {\chi_E} x = \sqbrk {x \in E}$