Cartesian product: Difference between revisions

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In [[mathematics]], the '''Cartesian product''' of two sets ''X'' and ''Y'' is the set of [[ordered pair]]s from ''X'' and ''Y''.
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In [[mathematics]], the '''Cartesian product''' of two sets ''X'' and ''Y'' is the set of [[ordered pair]]s from ''X'' and ''Y'': it is denoted <math>X \times Y</math> or, less often, <math>X \sqcap Y</math>.
 
There are ''projection maps'' pr<sub>1</sub> and pr<sub>2</sub> 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 <math>f:Z \rightarrow X</math> and <math>g:Z \rightarrow Y</math>, then there is a map <math>h : Z \rightarrow X \times Y</math> such that the compositions <math>h \cdot \mathrm{pr}_1 = f</math> and <math>h \cdot \mathrm{pr}_2 = g</math>.  This map ''h'' is defined by
 
:<math> h(z) = ( f(z), g(z) ) . \, </math>
 
==General products==
The product of any finite number of sets may be defined inductively, as
 
:<math>\prod_{i=1}^n X_i = X_1 \times (X_2 \times (X_3 \times (\cdots X_n)\cdots))) . \, </math>
 
The product of a general family of sets ''X''<sub>λ</sub> 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''<sub>λ</sub> for all λ in Λ.  It may be denoted
 
:<math>\prod_{\lambda \in \Lambda} X_\lambda . \, </math> 
 
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<sub>λ</sub> from the product to each ''X''<sub>λ</sub>.
 
The Cartesian product has a [[universal property]]: if there is a set ''Z'' with maps <math>f_\lambda:Z \rightarrow X_\lambda</math>, then there is a map <math>h : Z \rightarrow \prod_{\lambda \in \Lambda} X_\lambda</math> such that the compositions <math>h \cdot \mathrm{pr}_\lambda = f_\lambda</math>.  This map ''h'' is defined by
 
:<math> h(z) = ( \lambda \mapsto f_\lambda(z) ) . \, </math>
 
===Cartesian power===
The ''n''-th '''Cartesian power''' of a set ''X'' is defined as the Cartesian product of ''n'' copies of ''X''
 
:<math>X^n = X \times X \times \cdots \times X . \,</math>
 
A general Cartesian power over a general index set Λ may be defined as the set of all functions from Λ to ''X''
 
:<math>X^\Lambda = \{ f : \Lambda \rightarrow X \} . \,</math>
 
 


==References==
==References==
* {{cite book | author=Paul Halmos | authorlink=Paul Halmos | title=Naive set theory | series=The University Series in Undergraduate Mathematics | publisher=[[Van Nostrand Reinhold]] | year=1960 | pages=24 }}
* {{cite book | author=Paul Halmos | authorlink=Paul Halmos | title=Naive set theory | series=The University Series in Undergraduate Mathematics | publisher=[[Van Nostrand Reinhold]] | year=1960 | pages=24 }}
* {{cite book | author=Keith J. Devlin | authorlink=Keith Devlin | title=Fundamentals of Contemporary Set Theory | series=Universitext | publisher=[[Springer-Verlag]] | year=1979 | isbn=0-387-90441-7 | pages=12 }}
* {{cite book | author=Keith J. Devlin | authorlink=Keith Devlin | title=Fundamentals of Contemporary Set Theory | series=Universitext | publisher=[[Springer-Verlag]] | year=1979 | isbn=0-387-90441-7 | pages=12 }}

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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