# Equivalence relation

From Encyclopedia of Mathematics

A binary relation on a set with the following properties:

1) for all : (reflexivity);

2) (symmetry);

3) (transitivity).

If maps the set into a set , then the relation is an equivalence.

For any the set consisting of all equivalent to is called the equivalence class of . Any two equivalence classes either are disjoint or coincide, that is, any equivalence defines a partition (decomposition) of , and vice versa.

**How to Cite This Entry:**

Equivalence relation.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Equivalence_relation&oldid=17622

This article was adapted from an original article by V.N. Grishin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article