Cardinalities

In mathematics, the cardinality of a set is a measure of the "number of elements of the set". For example, the set A = {2, 4, 6} contains 3 elements, and therefore A has a cardinality of 3. There are two approaches to cardinality – one which compares sets directly using bijections and injections, and another which uses cardinal numbers. The cardinality of a set is also called its size, when no confusion with other notions of size is possible.

The cardinality of a set A is usually denoted | A |, with a vertical bar on each side; this is the same notation as absolute value and the meaning depends on context. Alternatively, the cardinality of a set A may be denoted by n(A), A, card(A), or # A.

While the cardinality of a finite set is just the number of its elements, extending the notion to infinite sets usually starts with defining the notion of comparison of arbitrary (in particular infinite) sets.

If | A | ≤ | B | and | B | ≤ | A | then | A | = | B | (Cantor–Bernstein–Schroeder theorem). The axiom of choice is equivalent to the statement that | A | ≤ | B | or | B | ≤ | A | for every A,B.

Above, "cardinality" was defined functionally. That is, the "cardinality" of a set was not defined as a specific object itself. However, such an object can be defined as follows.

The relation of having the same cardinality is called equinumerosity, and this is an equivalence relation on the class of all sets. The equivalence class of a set A under this relation then consists of all those sets which have the same cardinality as A. There are two ways to define the "cardinality of a set":

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