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:
 * $\displaystyle \prod_{i \mathop \in I} S_i := \set {f: \paren {f: I \to \bigcup_{i \mathop \in I} S_i} \land \paren {\forall i \in I: \paren {\map f i \in S_i} } }$

where $f$ denotes a mapping.

When $S_i = S$ for all $i \in I$, the expression is written:
 * $\displaystyle S^I := \set {f: \paren {f: I \to S} \land \paren {\forall i \in I: \paren {\map f i \in S} } }$

which follows from Union is Idempotent:
 * $\displaystyle \bigcup_{i \mathop \in I} S = S$

Also see

 * Equivalence of Definitions of Cartesian Product of Indexed Family