Editing Set (mathematics) (section)

===Cartesian product===
{{main|Cartesian product|Direct product}}
The Cartesian product of two sets has already be used for defining functions.

Given two sets {{tmath|A_1}} and {{tmath|A_2}}, their ''Cartesian product'', denoted {{tmath|A_1\times A_2}} is the set formed by all ordered pairs {{tmath|(a_1, a_2)}} such that {{tmath|a_1\in A_1}} and {{tmath|a_i\in A_1}}; that is,
<math display=block>A_1\times A_2 = \{(a_1, a_2) \mid a_1\in A_1 \land a_2\in A_2\}.</math>

This definition does not supposes that the two sets are different. In particular,
<math display=block>A\times A = \{(a_1, a_2) \mid a_1\in A \land a_2\in A\}.</math>  

Since this definition involves a pair of indices (1,2), it generalizes straightforwardly to the Cartesian product or [[direct product]] of any indexed family of sets:
<math display=block>\prod_{i\in \mathcal I} A_i= \{(a_i)_{i\in \mathcal I}\mid (\forall i\in \mathcal I) \;a_i\in A_i\}.</math>
That is, the elements of the Cartesian product of a family of sets are all families of elements such that each one belongs to the set of the same index. The fact that, for every indexed family of nonempty sets, the Cartesian product is a nonempty set is insured by the [[axiom of choice]].