\begin{definition}[Definition:Cartesian Product of Family/Definition 1]
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 $\ds \prod_{i \mathop \in I} S_i$ or, if $I$ is understood, $\ds \prod_i S_i$.
\end{definition}