Cartesian product: Difference between revisions

From Citizendium
Jump to navigation Jump to search
imported>Richard Pinch
(added Cartesian power)
imported>Richard Pinch
(universal property for the general product)
Line 18: Line 18:


The [[Axiom of Choice]] is equivalent to stating that a product of any family of non-empty sets is non-empty.
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===
===Cartesian power===
Line 27: Line 33:


:<math>X^\Lambda = \{ f : \Lambda \rightarrow X \} . \,</math>
:<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 }}

Revision as of 12:26, 30 December 2008

This article is a stub and thus not approved.
Main Article
Discussion
Related Articles  [?]
Bibliography  [?]
External Links  [?]
Citable Version  [?]
 
This editable Main Article is under development and subject to a disclaimer.

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