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'' | 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>. | ||
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'' | 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. | |||
==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 01:11, 4 November 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.
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.