# Definition:Cartesian Product/Family of Sets/Definition 2

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## 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:

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

## 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.

## Sources

- 1964: Steven A. Gaal:
*Point Set Topology*... (previous) ... (next): Introduction to Set Theory: $3$. The Axiom of Choice and Its Equivalents - 1975: T.S. Blyth:
*Set Theory and Abstract Algebra*... (previous) ... (next): $\S 6$. Indexed families; partitions; equivalence relations - 1975: Bert Mendelson:
*Introduction to Topology*(3rd ed.) ... (previous) ... (next): Chapter $1$: Theory of Sets: $\S 10$: Arbitrary Products - 1993: Keith Devlin:
*The Joy of Sets: Fundamentals of Contemporary Set Theory*(2nd ed.) ... (previous) ... (next): Naive Set Theory: $\S 1.6$: Functions