Definition:Cartesian Product of Family/Definition 1

Definition
Let $I$ be an indexing set.

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

The Cartesian product of $\family {S_i}_{i \mathop \in I}$ is the set of all families $\family {s_i}_{i \mathop \in I}$ 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, $\displaystyle \prod_i S_i$.

Also see

 * Equivalence of Definitions of Cartesian Product of Indexed Family