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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 , 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
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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