Cardinality

Pronunciation: /ˌkɑr dnˈæl ɪ ti/ ?

The cardinality of a set is the size of a set. If a set has a finite number of members, then the cardinality of the set is the number of members. The cardinality of the set A={a,b,c} is 3.

The cardinality of an infinite set is related to the infinite set of natural numbers. The cardinality of the set of natural numbers is defined to be ℵ0. If a set has a one-to-one correspondence with the set of natural number, then that set also has a cardinality of ℵ0. A set that is finite, or with a cardinality of ℵ0 is said to be countable or denumerable.

Equivalence of Sets

Two sets are said to be equivalent if they have the same cardinality.

Examples

The set of integers has a one to one correspondence to the set of natural numbers. Since the set of natural numbers is a subset of the set of integers, this is not obvious. The following table shows the one to one correspondence of the set of integers with the set of natural numbers.

Natural
Number
Integer
10
21
3-1
42
5-2
......
Table 1: Correspondence of the natural numbers with integers.

Table 1 shows a one-to-one correspondence of the natural numbers to the integers. Some student, on viewing this correspondence, say, "But, you will run out of natural numbers before you run out of integers." The truth is that there are an infinite number of natural numbers. Since both sets are infinite, you can't run out. And, since one integer can be matched to each and every natural number, the two sets must be the same size. Mathematicians say they have the same cardinality.

An infinite set that does not have a one to one correspondence with the natural numbers is called uncountable or nondenumerable. The set of real numbers is uncountable. Table 2 shows a vain attempt to associate natural numbers with real numbers. Notice that this association will never match any number greater than 2 with a natural number. The set of real numbers has a different cardinality from the set of natural numbers. Recent advances in mathematics has shown that the next leap in cardinality to gd1 is the set of real numbers. The set of real numbers had a cardinality of ℵ1.

Natural
Number
Real
Number
11.0
21.1
31.11
41.111
......
Table 2: Correspondence of the natural numbers with real numbers.

More Information

  • McAdams, David. Set. AllMathWords.org. Life is a Story Problem LLC. 2011-04-29. http://www.allmathwords.org/article.aspx?lang=en&id=Set.

Cite this article as:


Cardinality. 2011-06-02. All Math Words Encyclopedia. Life is a Story Problem LLC. http://www.allmathwords.org/en/c/cardinality.html.

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Revision History


2011-06-02: Added equivalence of sets. (McAdams, David.)
2010-09-30: Changed title from 'Countable' to 'Cardinality'. Expanded article. (McAdams, David.)
2010-01-05: Added "References" (McAdams, David.)
2009-12-15: Added "References" (McAdams, David.)
2009-11-21: Added vocabulary link to one to one correspondence. (McAdams, David.)
2008-12-02: Initial version (McAdams, David.)

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