# Cartesian product

*14 January 2011*.

In mathematics, the **Cartesian product** of two sets *X* and *Y* is the set of ordered pairs from *X* and *Y*: it is denoted or, less often, .

There are *projection maps* pr_{1} and pr_{2} from the product to *X* and *Y* taking the first and second component of each ordered pair respectively.

The Cartesian product has a universal property: if there is a set *Z* with maps and , then there is a map such that the compositions and . This map *h* is defined by

## [edit] General products

The product of any finite number of sets may be defined inductively, as

The product of a general family of sets *X*_{λ} as λ ranges over a general index set Λ may be defined as the set of all functions *x* with domain Λ such that *x*(λ) is in *X*_{λ} for all λ in Λ. It may be denoted

The Axiom of Choice is equivalent to stating that a product of any family of non-empty sets is non-empty.

There are *projection maps* pr_{λ} from the product to each *X*_{λ}.

The Cartesian product has a universal property: if there is a set *Z* with maps , then there is a map such that the compositions . This map *h* is defined by

### [edit] Cartesian power

The *n*-th **Cartesian power** of a set *X* is defined as the Cartesian product of *n* copies of *X*

A general Cartesian power over a general index set Λ may be defined as the set of all functions from Λ to *X*

## [edit] References

- Paul Halmos (1960).
*Naive set theory*. Van Nostrand Reinhold, 24. - Keith J. Devlin (1979).
*Fundamentals of Contemporary Set Theory*. Springer-Verlag, 12. ISBN 0-387-90441-7.

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