Definition:Cartesian Product/Family of Sets/Definition 2

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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:

$\displaystyle \bigcup_{i \mathop \in I} S = S$

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

  • Results about Cartesian products can be found here.

Source of Name

This entry was named for René Descartes.