Difference between revisions of "Cardinality"
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| − | '''Cardinality''' is | + | '''Cardinality''' is a property of [[set]]s. |
==Definition== | ==Definition== | ||
| − | For [[finite]] | + | For [[finite]] sets, the cardinality of is the number of [[element]]s in that set. The cardinality of <math>\{3, 4\}</math> is 2; of <math>\{1, \{2, 3\}, \{1, 2, 3\}\}</math> is 3; and of the [[empty set]] is 0. |
==Notation== | ==Notation== | ||
Revision as of 14:52, 16 April 2008
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Cardinality is a property of sets.
Contents
Definition
For finite sets, the cardinality of is the number of elements in that set. The cardinality of 
is 2; of 
is 3; and 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.
