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
Property | General Example | Example With Real Numbers | Description |
Reflexive | a 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 |
References
- 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
- 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
- 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
- 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
- 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
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. 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.)
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.)