The inverse of $f$ is the relation $f^{-1}\colon B\mathrel {\rightarrow \kern -9.5pt\mathrlap {|}\kern 6pt}A$ defined as follows:
- Viewing relations from $B$ to $A$ as subsets of $B\times A$, we define
\[ f^{-1}\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\webleft\{ \webleft (b,f^{-1}\webleft (b\webright )\webright )\in B\times A\ \middle |\ a\in A\webright\} , \]
where
\[ f^{-1}\webleft (b\webright )\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\webleft\{ a\in A\ \middle |\ f\webleft (a\webright )=b\webright\} \]for each $b\in B$.
- Viewing relations from $B$ to $A$ as functions $B\times A\to \{ \mathsf{true},\mathsf{false}\} $, we define
\[ f^{-1}\webleft (b,a\webright )\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\begin{cases} \mathsf{true}& \text{if there exists $a\in A$ with $f\webleft (a\webright )=b$,}\\ \mathsf{false}& \text{otherwise} \end{cases} \]
for each $\webleft (b,a\webright )\in B\times A$.
- Viewing relations from $B$ to $A$ as functions $B\to \mathcal{P}\webleft (A\webright )$, we define
\[ f^{-1}\webleft (b\webright )\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\webleft\{ a\in A\ \middle |\ f\webleft (a\webright )=b\webright\} \]
for each $b\in B$.
