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 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) ) . \,

[edit] 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) ) . \,

[edit] 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 \} . \,


[edit] References

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