Difference between revisions of "Cardinality"
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| − | '''Cardinality''' is a property of [[set]]s. | + | '''Cardinality''' is a property of [[set]]s. For [[finite]] sets, the cardinality of is the number of [[element]]s in that set, i.e. the size of the set. 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 [[finite]] sets, the cardinality of is the number of [[element]]s in that set. The cardinality of <math>\{3, 4\}</math> is 2 | ||
==Notation== | ==Notation== | ||
| − | The cardinality of a set <math>A</math> is denoted by <math>|A|</math>. In the above example, the cardinality of <math>\{3, 4\} | + | The cardinality of a set <math>A</math> is denoted by <math>|A|</math>. In the above example, the cardinality of <math>\{3, 4\}</math> is <math>|\{3, 4\}| = 2</math>. Sometimes the notations <math>n(A)</math> and <math>\# (A)</math> are also used. |
==Infinite== | ==Infinite== | ||
| − | 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]] | + | 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 <math>S</math> is the least [[cardinal]] that can be put in [[bijection]] with <math>S</math>. |
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. | 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 11:06, 17 April 2008
Cardinality is a property of sets. For finite sets, the cardinality of is the number of elements in that set, i.e. the size of the set. The cardinality of 
is 2, the cardinality of 
is 3, and the cardinality of the empty set is 0.
Notation
The cardinality of a set 
is denoted by 
. In the above example, the cardinality of is 
. Sometimes the notations 
and 
are also used.
Infinite
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 
is the least cardinal that can be put in bijection with .
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.
See Also
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