Let $A$ and $B$ be sets.
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The set of relations from $A$ to $B$ is the set $\mathrm{Rel}\webleft (A,B\webright )$ defined by
\[ \mathrm{Rel}\webleft (A,B\webright ) \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\webleft\{ \text{Relations from $A$ to $B$}\webright\} . \]
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The poset of relations from $A$ to $B$ is the poset
\[ \mathbf{Rel}\webleft (A,B\webright )\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\webleft (\mathrm{Rel}\webleft (A,B\webright ),\subset \webright ) \]
consisting of:
- The Underlying Set. The set $\mathrm{Rel}\webleft (A,B\webright )$ of Item 1.
- The Partial Order. The partial order
\[ \subset \colon \mathrm{Rel}\webleft (A,B\webright )\times \mathrm{Rel}\webleft (A,B\webright )\to \{ \mathsf{true},\mathsf{false}\} \]
on $\mathrm{Rel}\webleft (A,B\webright )$ given by inclusion of relations.
- The category of relations from $A$ to $B$ is the posetal category $\mathbf{Rel}\webleft (A,B\webright )$[1] associated to the poset $\mathbf{Rel}\webleft (A,B\webright )$ of Item 2 via Chapter 8: Categories, Definition 8.1.3.1.1.
