Equivalence Relation

Pronunciation: /ɪˈkwɪv ə ləns rɪˈleɪ ʃən/ Explain

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. Merriam-Webster. Last Accessed 8/6/2018. http://www.merriam-webster.com/dictionary/equivalence relation.
  2. Gilbert, Jimmie; and Gilbert Linda. Elements of Modern Algebra. 7th edition. pp 55-56. Thomson, Brooks/Cole. 2005. Buy the book
  3. R. Hirsch, I. Hodkinson. Relation Algebras by Games, Volume 147. pg 27. North Holland. August 29, 2002. Buy the book
  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. Buy the book

More Information

  • McAdams, David E.. Equal. allmathwords.org. All Math Words Encyclopedia. Life is a Story Problem LLC. 3/12/2009. http://www.allmathwords.org/en/e/equal.html.

Cite this article as:

McAdams, David E. Equivalence Relation. 7/10/2018. All Math Words Encyclopedia. Life is a Story Problem LLC. http://www.allmathwords.org/en/e/equivalencerelation.html.

Revision History

7/5/2018: Removed broken links, updated license, implemented new markup, implemented new Geogebra protocol. (McAdams, David E.)
7/26/2008: Added wikipedia to more information. (McAdams, David E.)
4/25/2008: Clarified wording. (McAdams, David E.)
3/31/2008: Initial version. (McAdams, David E.)

All Math Words Encyclopedia is a service of Life is a Story Problem LLC.
Copyright © 2018 Life is a Story Problem LLC. All rights reserved.
This work is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License