Cartesian product: Difference between revisions
imported>Richard Pinch (added projections, universal property) |
imported>Richard Pinch (universal property for the general product) |
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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=== | |||
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 }} |
Revision as of 12:26, 30 December 2008
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
- Paul Halmos (1960). Naive set theory. Van Nostrand Reinhold, 24.
- Keith J. Devlin (1979). Fundamentals of Contemporary Set Theory. Springer-Verlag, 12. ISBN 0-387-90441-7.