View source for Equivalence relation

An '''equivalence relation''' is a [[binary relation]] on a [[set]] that groups elements into categories<ref>Not to be confused with [[category]] in [[category theory]].</ref> in which all members are considered "equivalent" under some criterion.

== Definition ==
A [[relation]] <math>\sim</math> on set <math>X</math> is a equivalence relation if it satisfies the following properties:

* '''Reflexivity''': <math>\forall x\in X</math>, <math>x\sim x</math>.
* '''Symmetry''': <math>\forall x,y\in X</math> such that <math>x\sim y</math>, <math>y\sim x</math>.
* '''Transitivity''': <math>\forall x,y,z</math>, if <math>x\sim y</math> and <math>y\sim z</math>, then <math>x\sim z</math>.

When <math>x\sim y</math>, then we say that "<math>x</math> is equivalent to <math>y</math>" under the relation <math>\sim</math>.

== Notes ==
<references group="Note" />