\begin{definition}[Definition:Characteristic Function (Set Theory)/Set]
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 $S$.
\end{definition}