# 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 a 5 = 5 A relationship is reflexive if, for every member a of the set, a R a.
Symmetric a R b implies b R a If a = b then b = a A relationship is symmetric if, for every relation a R b on the set, b R a is true.
Transitive a R b and b R c implies a R c If a = b and b = c, then a = c A relationship is transitive if, the relationships a R b and b R c imply a R c.
Table 1

1. McAdams, David E.. All Math Words Dictionary, equivalence relation. 2nd Classroom edition 20150108-4799968. pg 71. Life is a Story Problem LLC. January 8, 2015. Buy the book
2. equivalence relation. merriam-webster.com. Encyclopedia Britannica. Merriam-Webster. Last Accessed 7/9/2018. http://www.merriam-webster.com/dictionary/equivalence relation. Buy the book
3. Gilbert, Jimmie; and Gilbert Linda. Elements of Modern Algebra. 7th edition. pp 55-56. Thomson, Brooks/Cole. 2005. Last Accessed 7/9/2018. Buy the book
4. R. Hirsch, I. Hodkinson. Relation Algebras by Games, Volume 147. pg 27. North Holland. August 29, 2002. Last Accessed 7/9/2018. Buy the book
5. 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. Last Accessed 7/9/2018. Buy the book

• 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.

McAdams, David E. Equivalence Relation. 4/20/2019. All Math Words Encyclopedia. Life is a Story Problem LLC. http://www.allmathwords.org/en/e/equivalencerelation.html.

### Revision History

4/20/2019: Updated expressions and equations to match new format. (McAdams, David E.)
12/21/2018: Reviewed and corrected IPA pronunication. (McAdams, David E.)