Editing Equivalence class (section)

==Properties==

For a set <math>X</math> with an [[equivalence relation]] ~, 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. (The proof is shown below.) 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 <math>[x]</math>, which may be the equivalence classes for other elements of <math>X</math>.<ref>{{harvnb|Maddox|2002|loc=p. 74, Thm. 2.5.15}}</ref> (I.e., all elements in <math>X</math> are grouped into non-empty sets, that are here equivalence classes of <math>X</math>.)

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> 
* <math>[x] \cap [y] \ne \emptyset.</math>

=== Proof ===

* Proof of "<math>x \sim y</math> if and only if <math>[x] = [y]</math>".
** Proof of "If <math>x \sim y</math> then <math>[x] = [y]</math>".
**# For <math>c \in [x]</math>, <math>x \sim c</math>. By symmetry <math>y \sim x</math> from <math>x \sim y</math>, and by transitivity <math>y \sim c</math> or <math>c \in [y]</math>, Thus, <math>[x] \subseteq [y]</math>. 
**# For <math>c' \in [y]</math>, <math>y \sim c'</math>. By transitivity <math>x \sim c'</math> or <math>c' \in [x]</math>, Thus, <math>[y] \subseteq [x]</math>.
**# Thus <math>[x] = [y]</math>.
** Proof of "If <math>[x] = [y]</math> then <math>x \sim y</math>".
*** For <math>c \in [x]</math>, <math>x \sim c</math>, and <math>y \sim c</math> by <math>[x] = [y]</math>. By symmetry and transitivity, <math>x \sim y</math>.

* Proof of "If <math>[x]\cap[y] \neq \emptyset </math> then <math>[x] = [y]</math>".
** If <math>[x]\cap[y] \neq \emptyset </math>, then there is <math>c</math> such that <math>x \sim c</math> and <math>y \sim c</math>. By symmetry and transitivity <math>x \sim y</math>, and by the above theorem, <math>[x] = [y]</math>.