Cardinality

Pronunciation: /ˌkɑr.dnˈæl.ɪ.ti/ Explain

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 1 0 2 1 3 -1 4 2 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 ℵ1 is the set of real numbers. The set of real numbers had a cardinality of ℵ1.

Natural Number Real Number 1 1.0 2 1.1 3 1.11 4 1.111 … … Table 2: Correspondence of the natural numbers with real numbers.

References

1. McAdams, David E.. All Math Words Dictionary, cardinality. 2nd Classroom edition 20150108-4799968. pg 29. Life is a Story Problem LLC. January 8, 2015. Buy the book

More Information

• McAdams, David E.. Set. AllMathWords.org. Life is a Story Problem LLC. 6/27/2018. http://www.allmathwords.org/en/s/set.html.

Cite this article as:

McAdams, David E. Cardinality. 4/14/2019. All Math Words Encyclopedia. Life is a Story Problem LLC. http://www.allmathwords.org/en/c/cardinality.html.

Revision History

4/14/2019: Updated equations and expressions to new format. (McAdams, David E.)
12/21/2018: Reviewed and corrected IPA pronunication. (McAdams, David E.)
6/22/2018: Removed broken links, updated license, implemented new markup, updated GeoGebra apps. (McAdams, David E.)
5/5/2011: Initial version. (McAdams, David E.)