Let $U\subset X$ and let $x\in X$.
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The characteristic function of $U$[1] is the function[2]
\[ \chi _{U}\colon X\to \{ \mathsf{t},\mathsf{f}\} \]
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 $x\in X$.
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The characteristic function of $x$ is the function[3]
\[ \chi _{x}\colon X\to \{ \mathsf{t},\mathsf{f}\} \]
defined by
\[ \chi _{x} \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\chi _{\webleft\{ x\webright\} }, \]i.e. by
\[ \chi _{x}\webleft (y\webright ) \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\begin{cases} \mathsf{true}& \text{if $x=y$,}\\ \mathsf{false}& \text{if $x\neq y$} \end{cases} \]for each $y\in X$.
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The characteristic relation on $X$[4] is the relation[5]
\[ \chi _{X}\webleft (-_{1},-_{2}\webright )\colon X\times X\to \{ \mathsf{t},\mathsf{f}\} \]
on $X$ defined by[6]
\[ \chi _{X}\webleft (x,y\webright ) \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\begin{cases} \mathsf{true}& \text{if $x=y$,}\\ \mathsf{false}& \text{if $x\neq y$} \end{cases} \]for each $x,y\in X$.
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The characteristic embedding[7] of $X$ into $\mathcal{P}\webleft (X\webright )$ is the function
\[ \chi _{\webleft (-\webright )}\colon X \hookrightarrow \mathcal{P}\webleft (X\webright ) \]
defined by
\[ \chi _{\webleft (-\webright )}\webleft (x\webright ) \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\chi _{x} \]for each $x\in X$.
