The converse of $R$1 is the relation $\smash {R^{\dagger }}$ defined as follows:
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Viewing relations as subsets, we define
\[ R^{\dagger } \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\left\{ \webleft (b,a\webright )\in B\times A\ \middle |\ \text{we have $b\sim _{R}a$}\right\} . \] -
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Viewing relations as functions $A\times B\to \{ \mathsf{true},\mathsf{false}\} $, we define
\[ {\webleft [R^{\dagger }\webright ]}{}^{a}_{b} \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}R^{b}_{a} \]for each $\webleft (b,a\webright )\in B\times A$.
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Viewing relations as functions $A\to \mathcal{P}\webleft (B\webright )$, we define
\begin{align*} \webleft [R^{\dagger }\webright ]\webleft (b\webright ) & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}R^{\dagger }\webleft (\left\{ b\right\} \webright )\\ & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\left\{ a\in A\ \middle |\ b\in R\webleft (a\webright )\right\} \end{align*}for each $b\in B$, where $R^{\dagger }\webleft (\left\{ b\right\} \webright )$ is the fibre of $R$ over $\left\{ b\right\} $.
- 1Further Terminology: Also called the opposite of $R$ or the transpose of $R$.
