Cardinality

In mathematics, the cardinality of a set is the number of its elements. The cardinality of a set may also be called its size, when no confusion with other notions of size is possible.^[a] Beginning in the late 19th century, this concept of size was generalized to infinite sets, allowing one to distinguish between different types of infinity and to perform arithmetic on them. Nowadays, infinite sets are encountered in almost all parts of mathematics, even those that may seem to be unrelated. Familiar examples are provided by most number systems and algebraic structures (natural numbers, rational numbers, real numbers, vector spaces, etc.), as well as in geometry, by lines, line segments and curves, which are considered as the sets of their points.

There are two approaches to describing cardinality: one which uses cardinal numbers and another which compares sets directly using functions between them, either bijections or injections. The former states the size as a number; the latter compares their relative size and led to the discovery of different sizes of infinity.^[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 (the first approach) and there is a bijection between them (the second approach).

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