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Let  be a set with an equivalence relation  . An equivalence class of  under  is a subset
 such that
- If
 and  , then  if and only if 
- If
 is nonempty, then  is nonempty
For  , the equivalence class containing  is often denoted by ![$ [x]$](http://images.planetmath.org:8080/cache/objects/468/l2h/img14.png "$ [x]$") , so that
The set of all equivalence classes of  under  is defined to be the set of all subsets of  which are equivalence classes of  under  , and is denoted by  . The map
![$ x\mapsto [x]$](http://images.planetmath.org:8080/cache/objects/468/l2h/img22.png "$ x\mapsto [x]$") is sometimes referred to as the canonical projection.
For any equivalence relation  , the set of all equivalence classes of  under  is a partition of  , and this correspondence is a bijection between the set of equivalence relations on  and the set of partitions of  (consisting of nonempty sets).
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![$\displaystyle [x] := \{ y \in S \mid x \sim y \}. $](http://images.planetmath.org:8080/cache/objects/468/l2h/img15.png "$\displaystyle [x] := \{ y \in S \mid x \sim y \}. $"){kind=small}