The characteristic function of $U$1 is the function$\chi _{U}\colon X\to \{ \mathsf{t},\mathsf{f}\} $2 defined by
\[ \chi _{U}\webleft (x\webright ) \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\begin{cases} \mathsf{true}& \text{if $x\in U$,}\\ \mathsf{false}& \text{if $x\not\in U$} \end{cases} \]
for each $x\in X$.
1Further Terminology: Also called the indicator function of $U$.
2Further Notation: Also written $\chi _{X}\webleft (U,-\webright )$ or $\chi _{X}\webleft (-,U\webright )$.
