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''. The product of any finite number of sets may be defined inductively. | ||
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'' on Λ such that ''x''(λ) is in ''X''<sub>λ</sub> for all λ in Λ. The [[Axiom of Choice]] is equivalent to stating that an element of such a product may always be taken. | |||
==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 02:30, 3 November 2008
In mathematics, the Cartesian product of two sets X and Y is the set of ordered pairs from X and Y. The product of any finite number of sets may be defined inductively.
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 on Λ such that x(λ) is in Xλ for all λ in Λ. The Axiom of Choice is equivalent to stating that an element of such a product may always be taken.
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.