Cardinality

In mathematics, cardinality describes a relationship between sets which compares their relative size.^[1] For example, the sets ${\displaystyle A=\{1,2,3\}}$ and ${\displaystyle B=\{2,4,6\}}$ are the same size as they each contain 3 elements. 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 notions often used when referring to cardinality: one which compares sets directly using bijections and injections, and another which uses cardinal numbers.^[2] The cardinality of a set may also be called its size, when no confusion with other notions of size is possible.^[a]

When two sets, ⁠${\displaystyle A}$⁠ and ⁠${\displaystyle B}$⁠, have the same cardinality, it is usually written as ${\displaystyle |A|=|B|}$; however, if referring to the cardinal number of an individual set ${\displaystyle A}$, it is simply 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 cardinal number of a set ${\displaystyle A}$ may alternatively be denoted by ${\displaystyle n(A)}$, ${\displaystyle A}$, ${\displaystyle \operatorname {card} (A)}$, or ${\displaystyle \#A}$.

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