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单词 Cardinality
释义

cardinality


Cardinality

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:

Cardinality.

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

and

Cardinality.

Sets A and B have the same cardinalityif 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.

Results

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.

Proof.

Respectively:

  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.

Example

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|=||.

Titlecardinality
Canonical nameCardinality
Date of creation2013-03-22 12:00:42
Last modified on2013-03-22 12:00:42
Owneryark (2760)
Last modified byyark (2760)
Numerical id25
Authoryark (2760)
Entry typeDefinition
Classificationmsc 03E10
Synonymsize
Related topicOrderGroup
Related topicGeneralizedContinuumHypothesis
Related topicCardinalNumber
Related topicDedekindInfinite
Definesequipotence
Definesequipotent
Definesequicardinality
Definesequipollence
Definesequipollent
Definesequinumerosity
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