Cardinality

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Cardinality is a property of sets.

Definition

For finite sets, the cardinality of is the number of elements in that set. The cardinality of $\{3, 4\}$ is 2; of $\{1, \{2, 3\}, \{1, 2, 3\}\}$ is 3; and of the empty set is 0.

Notation

The cardinality of a set $A$ is denoted by $|A|$. In the above example, the cardinality of $\{3, 4\} = |\{3, 4\}| = 2$. (It can also be notated $n(A)$)

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