Definition:Cartesian Product of Family/Definition 1

Definition

Let $I$ be an indexing set.

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

The cartesian product of $\family {S_i}_{i \mathop \in I}$ is the set of all families $\family {s_i}_{i \mathop \in I}$ with $s_i \in S_i$ for each $i \in I$.

This can be denoted $\ds \prod_{i \mathop \in I} S_i$ or, if $I$ is understood, $\ds \prod_i S_i$.

Axiom of Choice

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

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

Also see

• Equivalence of Definitions of Cartesian Product of Indexed Family

• Results about Cartesian products can be found here.

Source of Name

This entry was named for René Descartes.

Sources

• 1960: Paul R. Halmos: Naive Set Theory ... (previous) ... (next): $\S 9$: Families
• 2008: Paul Halmos and Steven Givant: Introduction to Boolean Algebras ... (previous) ... (next): Appendix $\text{A}$: Set Theory: Families