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 or, less often, .

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 and , then there is a map such that the compositions and . This map h is defined by

General products

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

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

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 , then there is a map such that the compositions . This map h is defined by

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


References