Equivalence Relation

Pronunciation: /ɪˈkwɪv ə ləns rɪˈleɪ ʃən/ ?

An equivalence relation is a relationship on a set that shows equality. An example of an equivalence relation on the set of integers is: 5 = 7 + x.

In the table below, R represents the relationship.

Properties of Equivalence Relations
PropertyGeneral ExampleExample With Real NumbersDescription
Reflexivea R a5 = 5A relationship is reflexive if, for every member a of the set, a R a.
Symmetrica R b implies b R aIf a = b then b = aA relationship is symmetric if, for every relation a R b on the set, b R a holds.
Transitivea R b and b R c implies a R cIf a = b and b = c, then a = cA relationship is transitive if, the relationships a R b and b R c imply a R c.
Table 1


  1. equivalence relation. merriam-webster.com. Encyclopedia Britannica. (Accessed: 2010-01-25). http://www.merriam-webster.com/dictionary/equivalence relation.
  2. Gilbert, Jimmie; and Gilbert Linda. Elements of Modern Algebra, 7th edition, pp 55-56. Brooks Cole, October 20, 2008.
  3. R. Hirsch, I. Hodkinson. Relation Algebras by Games, Volume 147 , pg 27. North Holland, August 29, 2002.
  4. T. S. Blyth, E. F. Robertson. Algebra Through Practice: Volume 1, Sets, Relations and Mappings: A Collection of Problems in Algebra with Solutions, pp 6-14. Cambridge University Press, December 28, 1984.

More Information

  • McAdams, David. Equal. allmathwords.org. All Math Words Encyclopedia. Life is a Story Problem LLC. 2009-03-12. http://www.allmathwords.org/article.aspx?lang=en&id=Equal.

Cite this article as:

Equivalence Relation. 2010-01-26. All Math Words Encyclopedia. Life is a Story Problem LLC. http://www.allmathwords.org/en/e/equivalencerelation.html.


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Revision History

2008-07-26: Added wikipedia to more information (McAdams, David.)
2008-04-25: Clarified wording (McAdams, David.)
2008-03-31: Initial version (McAdams, David.)

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