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 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

$\prod _{\lambda \in \Lambda }X_{\lambda }.\,$

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 pr_λ from the product to each X_λ.

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

• 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.