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}$

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:
 * $\chi_E \left({x}\right) = \left[{x \in E}\right]$