The product of $R$ and $S$1 is the relation $R\times S$ from $A\times X$ to $B\times Y$ defined as follows:
- Viewing relations from $A\times X$ to $B\times Y$ as subsets of $\webleft (A\times X\webright )\times \webleft (B\times Y\webright )$, we define $R\times S$ as the Cartesian product of $R$ and $S$ as subsets of $A\times X$ and $B\times Y$;2
- Viewing relations from $A\times X$ to $B\times Y$ as functions $A\times X\to \mathcal{P}\webleft (B\times Y\webright )$, we define $R\times S$ as the composition
\[ A\times X \overset {R\times S}{\to } \mathcal{P}\webleft (B\webright )\times \mathcal{P}\webleft (Y\webright ) \overset {\mathcal{P}^{\otimes }_{B,Y}}{\hookrightarrow } \mathcal{P}\webleft (B\times Y\webright ) \]
in $\mathsf{Sets}$, i.e. by
\[ \webleft [R\times S\webright ]\webleft (a,x\webright ) \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}R\webleft (a\webright )\times S\webleft (x\webright ) \]for each $\webleft (a,x\webright )\in A\times X$.
1Further Terminology: Also called the binary product of $R$ and $S$, for emphasis.
2That is, $R\times S$ is the relation given by declaring $\webleft (a,x\webright )\sim _{R\times S}\webleft (b,y\webright )$ iff $a\sim _{R}b$ and $x\sim _{S}y$.
