Difference between revisions of "Cardinality"
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− | + | '''Cardinality''' is one property of [[set]]s | |
− | For [[finite]] [[set]]s, the | + | ==Definition== |
+ | 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. | ||
+ | ==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\} = |\{3, 4\}| = 2</math>. | The cardinality of a set <math>A</math> is denoted by <math>|A|</math>. In the above example, the cardinality of <math>\{3, 4\} = |\{3, 4\}| = 2</math>. | ||
+ | (It can also be notated <math>n(A)</math>) | ||
+ | ==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]] which can be put in [[bijection]] with S. | 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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* [[Set]] | * [[Set]] | ||
* [[Element]] | * [[Element]] | ||
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+ | {{stub}} |
Revision as of 00:07, 15 November 2007
Cardinality is one property of sets
Contents
Definition
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.
Notation
The cardinality of a set is denoted by
. In the above example, the cardinality of
.
(It can also be notated
)
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 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.
See Also
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