The apartness composition of $R$ and $S$ is the relation $S\mathbin {\square }R$ defined as follows:
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Viewing relations from $A$ to $C$ as subsets of $A\times C$, we define
\[ S\mathbin {\square }R \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\left\{ \webleft (a,c\webright )\in A\times C\ \middle |\ \begin{aligned} & \text{for each $b\in B$, we have}\\ & \text{$a\sim _{R}b$ or $b\sim _{S}c$}\end{aligned} \right\} . \] -
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Viewing relations as functions $A\times B\to \{ \mathsf{true},\mathsf{false}\} $, we define
\begin{align*} \webleft (S\mathbin {\square }R\webright )^{-_{1}}_{-_{2}} & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\int _{b\in B}S^{-_{1}}_{b}\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}R^{b}_{-_{2}}\\ & = \bigwedge _{b\in B}S^{-_{1}}_{b}\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}R^{b}_{-_{2}},\end{align*}where the meet $\bigwedge $ is taken in the poset $\webleft (\{ \mathsf{true},\mathsf{false}\} ,\preceq \webright )$ of Chapter 3: Sets, Definition 3.2.2.1.3.
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Viewing relations as functions $A\to \mathcal{P}\webleft (B\webright )$, we define
\[ \webleft [S\mathbin {\square }R\webright ]\webleft (a\webright )\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\bigcap _{b\in B\setminus R\webleft (a\webright )}S\webleft (b\webright ) \]
