Editing Equivalence class (section)

==Properties==

For a set <math>X</math> with an [[equivalence relation]] <math>\sim</math>, every element <math>x</math> of <math>X</math> is a member of the equivalence class <math>[x]</math> by [[Reflexive relation|reflexivity]] (<math>a \sim a</math> for all <math>a \in X</math>). Every two equivalence classes <math>[x]</math> and <math>[y]</math> are either equal if <math>x \sim y</math>, or [[disjoint sets|disjoint]] otherwise.  Therefore, the set of all equivalence classes of <math>X</math> forms a [[partition of a set|partition]] of <math>X</math>: every element <math>x</math> of <math>X</math> belongs to one and only one equivalence class.<ref>{{harvnb|Maddox|2002|loc=p. 74, Thm. 2.5.15}}</ref>

Conversely, for a set <math>X</math>, every partition comes from an equivalence relation in this way, and different relations give different partitions. Thus <math>x \sim y</math> if and only if <math>x</math> and <math>y</math> belong to the same set of the partition.<ref>{{harvnb|Avelsgaard|1989|loc=p. 132, Thm. 3.16}}</ref>

It follows from the properties in the previous section that if <math>\,\sim\,</math> is an equivalence relation on a set <math>X,</math> and <math>x</math> and <math>y</math> are two elements of <math>X,</math> the following statements are equivalent:
* <math>x \sim y</math>,
* <math>[x] = [y]</math>, and 
* <math>[x] \cap [y] \ne \emptyset.</math>