• The characteristic relation on $X$[1] is the relation[2]
    \[ \chi _{X}\webleft (-_{1},-_{2}\webright )\colon X\times X\to \{ \mathsf{t},\mathsf{f}\} \]

    on $X$ defined by[3]

    \[ \chi _{X}\webleft (x,y\webright ) \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\begin{cases} \mathsf{true}& \text{if $x=y$,}\\ \mathsf{false}& \text{if $x\neq y$} \end{cases} \]

    for each $x,y\in X$.

Footnotes

[1] Further Terminology: Also called the identity relation on $X$.
[2] Further Notation: Also written $\chi ^{-_{1}}_{-_{2}}$, or $\mathord {\sim }_{\text{id}}$ in the context of relations.
[3] As a subset of $X\times X$, the relation $\chi _{X}$ corresponds to the diagonal $\Delta _{X}\subset X\times X$ of $X$.

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