The Clowder Project

The intersection of the family $\left\{ R_{i}\right\} _{i\in I}$ 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

  \[ \bigcup _{i\in I}R_{i}\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\left\{ (a,b)\in (A\times B)^{\times I}\ \middle |\ \begin{aligned} & \text{for each $i\in I$,}\\ & \text{we have $a\sim _{R_{i}}b$}\end{aligned} \right\} . \]
• •

  Viewing relations from $A$ to $B$ as functions $A\to \mathcal{P}(B)$, we define

  \[ \left[\bigcap _{i\in I}R_{i}\right](a)\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\bigcap _{i\in I}R_{i}(a) \]

  for each $a\in A$.