The product of the family $\left\{ R_{i}\right\} _{i\in I}$ is the relation $\smash {\prod _{i\in I}R_{i}}$ from $\smash {\prod _{i\in I}A_{i}}$ to $\smash {\prod _{i\in I}B_{i}}$ defined as follows:
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Viewing relations as subsets, we define $\smash {\prod _{i\in I}R_{i}}$ as its product as a family of sets, i.e. we have
\[ \prod _{i\in I}R_{i} \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\left\{ (a_{i},b_{i})_{i\in I}\in \prod _{i\in I}(A_{i}\times B_{i})\ \middle |\ \begin{aligned} & \text{for each $i\in I$,}\\ & \text{we have $a_{i}\sim _{R_{i}}b_{i}$}\end{aligned} \right\} . \] -
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Viewing relations as functions to powersets, we define
\[ \left[\prod _{i\in I}R_{i}\right]((a_{i})_{i\in I}) \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\prod _{i\in I}R_{i}(a_{i}) \]for each $(a_{i})_{i\in I}\in \prod _{i\in I}R_{i}$.
