The Clowder Project

The intersection of the family $\webleft\{ R_{i}\webright\} _{i\in I}$ is the relation $\smash {\bigcup _{i\in I}R_{i}}$ defined as follows:

• Viewing relations from $A$ to $B$ as subsets of $A\times B$, we define¹
  \[ \bigcup _{i\in I}R_{i}\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\webleft\{ \webleft (a,b\webright )\in \webleft (A\times B\webright )^{\times I}\ \middle |\ \begin{aligned} & \text{for each $i\in I$,}\\ & \text{we have $a\sim _{R_{i}}b$}\end{aligned} \webright\} . \]

• Viewing relations from $A$ to $B$ as functions $A\to \mathcal{P}\webleft (B\webright )$, we define
  \[ \webleft[\bigcap _{i\in I}R_{i}\webright]\webleft (a\webright )\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\bigcap _{i\in I}R_{i}\webleft (a\webright ) \]

  for each $a\in A$.