The Clowder Project

The union of the family $\webleft\{ R_{i}\webright\} _{i\in I}$ is the relation $\bigcup _{i\in I}R_{i}$ from $A$ to $B$ defined as follows:

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

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

  for each $a\in A$.