Definition:Cartesian Product of Family/Definition 2

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:
 * $\displaystyle \prod_{i \mathop \in I} S_i := \left\{{f: \left({f: I \to \bigcup_{i \mathop \in I} S_i}\right) \land \left({ \forall i \in I: \left({f \left({i}\right) \in S_i}\right)}\right)}\right\}$

where $f$ denotes a mapping.

When $S_i = S$ for all $i \in I$, the expression is written:
 * $\displaystyle S^I := \left\{{f: \left({f: I \to S}\right) \land \left({ \forall i \in I: \left({f \left({i}\right) \in S}\right)}\right)}\right\}$

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

Axiom of Choice
It is of the utmost importance that one be aware that for many $I$, establishing non-emptiness of $\displaystyle \prod_{i \mathop \in I} S_i$ requires a suitable version of the Axiom of Choice.

Details on this correspondence can be found on Equivalence of Versions of Axiom of Choice.

Also see

 * Equivalence of Definitions of Cartesian Product