Definition:Cartesian Product of Relations

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Let $\family {S_i}_{i \mathop \in I}$ and $\family {T_i}_{i \mathop \in I}$ be families of sets indexed by $I$.

For each $i \in I$, let $\RR_i \subseteq S_i \times T_i$ be a relation from $S_i$ to $T_i$.

Let $S$ and $T$ be the Cartesian product of $\family {S_i}_{i \mathop \in I}$ and $\family {T_i}_{i \mathop \in I}$ respectively:

$\ds S = \prod_{i \mathop \in I} S_i$
$\ds T = \prod_{i \mathop \in I} T_i$

Then the product of the relations $\RR_i$ is defined as the relation $\RR \subseteq S \times T$ such that:

$x \mathrel \RR y \iff \forall i \in I: x_i \mathrel {\RR_i} y_i$

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