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


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