Definition:Cartesian Product/Family of Sets

Definition
Let $\left\langle{S_i}\right\rangle_{i \in I}$ be a family of sets.

The Cartesian product of $\left\langle{S_i}\right\rangle_{i \in I}$ is the set of all families $\left\langle{s_i}\right\rangle_{i \in I}$ with $s_i \in S_i$ for each $i \in I$.

This can be denoted $\displaystyle \prod_{i \mathop \in I} S_i$ or, if $I$ is understood, $\displaystyle \prod_i S_i$.