The graph of $f$ is the relation $\text{Gr}\webleft (f\webright )\colon A\mathrel {\rightarrow \kern -9.5pt\mathrlap {|}\kern 6pt}B$ defined as follows:1
- Viewing relations from $A$ to $B$ as subsets of $A\times B$, we define
\[ \text{Gr}\webleft (f\webright )\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\webleft\{ \webleft (a,f\webleft (a\webright )\webright )\in A\times B\ \middle |\ a\in A\webright\} ; \]
- Viewing relations from $A$ to $B$ as functions $A\times B\to \{ \mathsf{true},\mathsf{false}\} $, we define
\[ \webleft [\text{Gr}\webleft (f\webright )\webright ]\webleft (a,b\webright )\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\begin{cases} \mathsf{true}& \text{if $b=f\webleft (a\webright )$,}\\ \mathsf{false}& \text{otherwise} \end{cases} \]
for each $\webleft (a,b\webright )\in A\times B$;
- Viewing relations from $A$ to $B$ as functions $A\to \mathcal{P}\webleft (B\webright )$, we define
\[ \webleft [\text{Gr}\webleft (f\webright )\webright ]\webleft (a\webright )\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\webleft\{ f\webleft (a\webright )\webright\} \]
for each $a\in A$, i.e. we define $\text{Gr}\webleft (f\webright )$ as the composition
\[ A \overset {f}{\to } B \overset {\chi _{B}}{\to } \mathcal{P}\webleft (B\webright ). \]
1Further Notation: We write $\text{Gr}\webleft (A\webright )$ for $\text{Gr}\webleft (\text{id}_{A}\webright )$, and call it the graph of $A$.
