Definition:Cartesian Product/Family of Sets

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Definition

Definition 1

Let $I$ be an indexing set.

Let $\left\langle{S_i}\right\rangle_{i \mathop \in I}$ be an family of sets indexed by $I$.

The Cartesian product of $\left\langle{S_i}\right\rangle_{i \mathop \in I}$ is the set of all families $\left\langle{s_i}\right\rangle_{i \mathop \in I}$ with $s_i \in S_i$ for each $i \in I$.


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


Definition 2

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:

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

where $f$ denotes a mapping.


When $S_i = S$ for all $i \in I$, the expression is written:

$\displaystyle S^I := \set {f: \paren {f: I \to S} \land \paren {\forall i \in I: \paren {f \paren i \in S} } }$

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 denoted as

Some sources use $\def \bigtimes {\mathop {\vcenter {\huge \times}}} \bigtimes_{i \mathop \in I} S_i$ for $\displaystyle \prod_{i \mathop \in I} S_i$.


Also see

  • Results about Cartesian products can be found here.

Source of Name

This entry was named for René Descartes.