Difference between revisions of "Cardinality"
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| − | For [[finite]] [[set]]s, the '''cardinality''' of a set is the number of | + | For [[finite]] [[set]]s, the '''cardinality''' of a set is the number of [[element]]s in that set, so the cardinality of <math>\{3, 4\}</math> is 2, the cardinality of <math>\{1, \{2, 3\}, \{1, 2, 3\}\}</math> is 3, and the cardinality of the [[empty set]] is 0. |
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| + | For [[infinite]] sets, cardinality also measures (in some sense) the "size" of the set, but an explicit formulation is more complicated: the cardinality of a set S is the least [[cardinal]] which can be put in [[bijection]] with S. | ||
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| + | The notion of cardinalities for infinite sets is due to [[Georg Cantor]] and is one aspect of the field of [[set theory]]. Most significantly, Cantor showed that there are multiple infinite cardinalities. In other words, not all infinite sets are the same size. | ||
Revision as of 12:52, 1 November 2006
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For finite sets, the cardinality of a set is the number of elements in that set, so the cardinality of 
is 2, the cardinality of 
is 3, and the cardinality of the empty set is 0.
For infinite sets, cardinality also measures (in some sense) the "size" of the set, but an explicit formulation is more complicated: the cardinality of a set S is the least cardinal which can be put in bijection with S.
The notion of cardinalities for infinite sets is due to Georg Cantor and is one aspect of the field of set theory. Most significantly, Cantor showed that there are multiple infinite cardinalities. In other words, not all infinite sets are the same size.
