The union of $R$ and $S$1 is the relation $R\cup S$ from $A$ to $B$ defined as follows:
- Viewing relations from $A$ to $B$ as subsets of $A\times B$, we define2
\[ R\cup S\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\webleft\{ \webleft (a,b\webright )\in B\times A\ \middle |\ \text{we have $a\sim _{R}b$ or $a\sim _{S}b$}\webright\} . \]
- Viewing relations from $A$ to $B$ as functions $A\to \mathcal{P}\webleft (B\webright )$, we define
\[ \webleft [R\cup S\webright ]\webleft (a\webright )\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}R\webleft (a\webright )\cup S\webleft (a\webright ) \]
for each $a\in A$.
1Further Terminology: Also called the binary union of $R$ and $S$, for emphasis.
2This is the same as the union of $R$ and $S$ as subsets of $A\times B$.
