• The characteristic function of $x$ is the function[1]
    \[ \chi _{x}\colon X\to \{ \mathsf{t},\mathsf{f}\} \]

    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 $y\in X$.

Footnotes

[1] Further Notation: Also written $\chi ^{x}$, $\chi _{X}\webleft (x,-\webright )$, or $\chi _{X}\webleft (-,x\webright )$.

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