# Cartesian product

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

There are projection maps pr1 and pr2 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 $f:Z \rightarrow X$ and $g:Z \rightarrow Y$, then there is a map $h : Z \rightarrow X \times Y$ such that the compositions $h \cdot \mathrm{pr}_1 = f$ and $h \cdot \mathrm{pr}_2 = g$. This map h is defined by

$h(z) = ( f(z), g(z) ) . \,$

## General products

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

$\prod_{i=1}^n X_i = X_1 \times (X_2 \times (X_3 \times (\cdots X_n)\cdots))) . \,$

The product of a general family of sets Xλ 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 Xλ for all λ in Λ. It may be denoted

$\prod_{\lambda \in \Lambda} X_\lambda . \,$

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 prλ from the product to each Xλ.

The Cartesian product has a universal property: if there is a set Z with maps $f_\lambda:Z \rightarrow X_\lambda$, then there is a map $h : Z \rightarrow \prod_{\lambda \in \Lambda} X_\lambda$ such that the compositions $h \cdot \mathrm{pr}_\lambda = f_\lambda$. This map h is defined by

$h(z) = ( \lambda \mapsto f_\lambda(z) ) . \,$

### Cartesian power

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

$X^n = X \times X \times \cdots \times X . \,$

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

$X^\Lambda = \{ f : \Lambda \rightarrow X \} . \,$

## References

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