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imported>Richard Pinch (Added anchor superset) |
imported>Richard Pinch (define and anchor inclusion map) |
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In [[set theory]], a '''subset''' of a [[set (mathematics)|set]] ''X'' is a set ''A'' whose elements are all elements of ''X'': that is, <math>x \in A \Rightarrow x \in X</math>, denoted <math>A \subseteq X</math>. The [[empty set]] Ø and ''X'' itself are always subsets of ''X''. The containing set ''X'' is a '''superset''' of ''A''. The relation between the subset and the superset is '''inclusion'''. | In [[set theory]], a '''subset''' of a [[set (mathematics)|set]] ''X'' is a set ''A'' whose elements are all elements of ''X'': that is, <math>x \in A \Rightarrow x \in X</math>, denoted <math>A \subseteq X</math>. The [[empty set]] Ø and ''X'' itself are always subsets of ''X''. The containing set ''X'' is a '''superset''' of ''A''. The relation between the subset and the superset is '''inclusion''', and the '''inclusion map''' is the map from ''A'' → ''X'' which is the [[identity map|identity]] on ''A''. | ||
The [[power set]] of ''X'' is the set of all subsets of ''X''. | The [[power set]] of ''X'' is the set of all subsets of ''X''. | ||
Revision as of 11:42, 30 December 2008
In set theory, a subset of a set X is a set A whose elements are all elements of X: that is, , denoted . The empty set Ø and X itself are always subsets of X. The containing set X is a superset of A. The relation between the subset and the superset is inclusion, and the inclusion map is the map from A → X which is the identity on A.
The power set of X is the set of all subsets of X.