Cardinality

In mathematics, the cardinality of a set is a measure of the number of elements of the set. For example, the set ${\displaystyle A=\{2,4,6\}}$ contains 3 elements, and therefore ${\displaystyle A}$ has a cardinality of 3. Beginning in the late 19th century, this concept was generalized to infinite sets, which allows one to distinguish between different types of infinity, and to perform arithmetic on them. There are two approaches to cardinality: one which compares sets directly using bijections and injections, and another which uses cardinal numbers.^[1] The cardinality of a set may also be called its size, when no confusion with other notions of size^[2] is possible.

The cardinality of a set ${\displaystyle A}$ is usually denoted ${\displaystyle |A|}$, with a vertical bar on each side;^[3] this is the same notation as absolute value, and the meaning depends on context. The cardinality of a set ${\displaystyle A}$ may alternatively be denoted by ${\displaystyle n(A)}$, ${\displaystyle A}$, ${\displaystyle \operatorname {card} (A)}$, or ${\displaystyle \#A}$.