Definition:Cartesian Product of Family/Definition 2

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

The Cartesian product of $\family {S_i}_{i \mathop \in I}$ is the set:
 * $\ds \prod_{i \mathop \in I} S_i := \set {f: \paren {f: I \to \bigcup_{i \mathop \in I} S_i} \land \paren {\forall i \in I: \paren {\map f i \in S_i} } }$

where $f$ denotes a mapping.

Also see

 * Equivalence of Definitions of Cartesian Product of Indexed Family