Cardinality is a notion of the size of a set which does not rely on numbers. It is a relative notion. For instance, two sets may each have an infiniteMathworldPlanetmath number of elements, but one may have a greater cardinality. That is, in a sense, one may have a “more infinite” number of elements. See Cantor diagonalization for an example of how the reals have a greater cardinality than the natural numbersMathworldPlanetmath.

The formal definition of cardinality rests upon the notion mappings between sets:


The cardinality of a set A is greater than or equal to the cardinality of a set B if there is a one-to-one function (an injectionMathworldPlanetmath) from B to A. Symbolically, we write |A||B|.



Sets A and B have the same cardinality if there is a one-to-one and onto functionMathworldPlanetmath (a bijection) from A to B. Symbolically, we write |A|=|B|.

It can be shown that if |A||B| and |B||A| then |A|=|B|. This is the Schröder-Bernstein Theorem.

Equality of cardinality is variously called equipotence, equipollence, equinumerosity, or equicardinality. For |A|=|B|, we would say that “A is equipotent to B”, “A is equipollent to B”, or “A is equinumerous to B”.

An equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmath definition of cardinality is

Cardinality (alt. def.).

The cardinality of a set A is the unique cardinal numberMathworldPlanetmath κ such that A is equinumerous with κ. The cardinality of A is written |A|.

This definition of cardinality makes use of a special class of numbers, called the cardinal numbers. This highlights the fact that, while cardinality can be understood and defined without appealing to numbers, it is often convenient and useful to treat cardinality in a “numeric” manner.


Some results on cardinality:

  1. 1.

    A is equipotent to A.

  2. 2.

    If A is equipotent to B, then B is equipotent to A.

  3. 3.

    If A is equipotent to B and B is equipotent to C, then A is equipotent to C.



  1. 1.

    The identity functionMathworldPlanetmath on A is a bijection from A to A.

  2. 2.

    If f is a bijection from A to B, then f-1 exists and is a bijection from B to A.

  3. 3.

    If f is a bijection from A to B and g is a bijection from B to C, then fg is a bijection from A to C.


The set of even integers 2 has the same cardinality as the set of integers : if we define f:2 such that f(x)=x2, then f is a bijection, and therefore |2|=||.

Title cardinality
Canonical name Cardinality
Date of creation 2013-03-22 12:00:42
Last modified on 2013-03-22 12:00:42
Owner yark (2760)
Last modified by yark (2760)
Numerical id 25
Author yark (2760)
Entry type Definition
Classification msc 03E10
Synonym size
Related topic OrderGroup
Related topic GeneralizedContinuumHypothesis
Related topic CardinalNumber
Related topic DedekindInfinite
Defines equipotence
Defines equipotent
Defines equicardinality
Defines equipollence
Defines equipollent
Defines equinumerosity