The characteristic relation on $X$1 is the relation2
\[ \chi _{X}\webleft (-_{1},-_{2}\webright )\colon X\times X\to \{ \mathsf{t},\mathsf{f}\} \]
on $X$ defined by3
\[ \chi _{X}\webleft (x,y\webright ) \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\begin{cases} \mathsf{true}& \text{if $x=y$,}\\ \mathsf{false}& \text{if $x\neq y$} \end{cases} \]
for each $x,y\in X$.
1Further Terminology: Also called the identity relation on $X$.
2Further Notation: Also written $\chi ^{-_{1}}_{-_{2}}$, or $\mathord {\sim }_{\text{id}}$ in the context of relations.
3Under the bijection $\mathsf{Sets}\webleft (X\times X,\{ \mathsf{t},\mathsf{f}\} \webright )\cong \mathcal{P}\webleft (X\times X\webright )$ of Item 2 of Proposition 2.5.1.1.4, the relation $\chi _{X}$ corresponds to the diagonal $\Delta _{X}\subset X\times X$ of $X$.
