Subsection 7.3.7 (00PY): Binary Intersections of Relations

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The intersection of $R$ and $S$¹ is the relation $R\cap S$ from $A$ to $B$ defined as follows:

Viewing relations from $A$ to $B$ as subsets of $A\times B$, we define²
\[ R\cap S\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\webleft\{ \webleft (a,b\webright )\in B\times A\ \middle |\ \text{we have $a\sim _{R}b$ and $a\sim _{S}b$}\webright\} . \]
Viewing relations from $A$ to $B$ as functions $A\to \mathcal{P}\webleft (B\webright )$, we define

\[ \webleft [R\cap S\webright ]\webleft (a\webright )\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}R\webleft (a\webright )\cap S\webleft (a\webright ) \]

for each $a\in A$.

¹Further Terminology: Also called the binary intersection of $R$ and $S$, for emphasis.

²This is the same as the intersection of $R$ and $S$ as subsets of $A\times B$.

Let $R$, $S$, $R_{1}$, and $R_{2}$ be relations from $A$ to $B$, and let $S_{1}$ and $S_{2}$ be relations from $B$ to $C$.

Interaction With Inverses. We have
\[ \webleft (R\cap S\webright )^{\dagger } = R^{\dagger }\cap S^{\dagger }. \]
Interaction With Composition. We have
\[ \webleft (S_{1}\mathbin {\diamond }R_{1}\webright ) \cap \webleft (S_{2}\mathbin {\diamond }R_{2}\webright ) = \webleft (S_{1}\cap S_{2}\webright ) \mathbin {\diamond }\webleft (R_{1}\cap R_{2}\webright ). \]

Item 1: Interaction With Inverses

Clear.

Item 2: Interaction With Composition

Unwinding the definitions, we see that:

The condition for $\webleft (S_{1}\mathbin {\diamond }R_{1}\webright )\cap \webleft (S_{2}\mathbin {\diamond }R_{2}\webright )$ is:
1. There exists some $b\in B$ such that:
  1. $\require{color}{\color[rgb]{0.835294117647059,0.368627450980392,0.000000000000000}{a\sim _{R_{1}}b}}$ and $\require{color}{\color[rgb]{0.000000000000000,0.447058823529412,0.698039215686274}{b\sim _{S_{1}}c}}$;
  and
  1. $\require{color}{\color[rgb]{0.835294117647059,0.368627450980392,0.000000000000000}{a\sim _{R_{2}}b}}$ and $\require{color}{\color[rgb]{0.000000000000000,0.447058823529412,0.698039215686274}{b\sim _{S_{2}}c}}$;
The condition for $\webleft (S_{1}\cap S_{2}\webright )\mathbin {\diamond }\webleft (R_{1}\cap R_{2}\webright )$ is:
1. There exists some $b\in B$ such that:
  1. $\require{color}{\color[rgb]{0.835294117647059,0.368627450980392,0.000000000000000}{a\sim _{R_{1}}b}}$ and $\require{color}{\color[rgb]{0.835294117647059,0.368627450980392,0.000000000000000}{a\sim _{R_{2}}b}}$;
  and
  1. $\require{color}{\color[rgb]{0.000000000000000,0.447058823529412,0.698039215686274}{b\sim _{S_{1}}c}}$ and $\require{color}{\color[rgb]{0.000000000000000,0.447058823529412,0.698039215686274}{b\sim _{S_{2}}c}}$.

These two conditions agree, and thus so do the two resulting relations on $A\times C$.

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