Cartesian product

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

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 ${\displaystyle f:Z\rightarrow X}$ and ${\displaystyle g:Z\rightarrow Y}$, then there is a map ${\displaystyle h:Z\rightarrow X\times Y}$ such that the compositions ${\displaystyle h\cdot \mathrm {pr} _{1}=f}$ and ${\displaystyle h\cdot \mathrm {pr} _{2}=g}$. This map h is defined by

${\displaystyle h(z)=(f(z),g(z)).\,}$

General products[edit]

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

${\displaystyle \prod _{i=1}^{n}X_{i}=X_{1}\times (X_{2}\times (X_{3}\times (\cdots X_{n})\cdots ))).\,}$

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

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

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 ${\displaystyle f_{\lambda }:Z\rightarrow X_{\lambda }}$, then there is a map ${\displaystyle h:Z\rightarrow \prod _{\lambda \in \Lambda }X_{\lambda }}$ such that the compositions ${\displaystyle h\cdot \mathrm {pr} _{\lambda }=f_{\lambda }}$. This map h is defined by

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

Cartesian power[edit]

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

${\displaystyle X^{n}=X\times X\times \cdots \times X.\,}$

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

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

References[edit]

• 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.