Fundamental Theorem of Equivalence Relations
A key result links equivalence relations and partitions:
- An equivalence relation ~ on a set X partitions X.
- Conversely, corresponding to any partition of X, there exists an equivalence relation ~ on X.
In both cases, the cells of the partition of X are the equivalence classes of X by ~. Since each element of X belongs to a unique cell of any partition of X, and since each cell of the partition is identical to an equivalence class of X by ~, each element of X belongs to a unique equivalence class of X by ~. Thus there is a natural bijection from the set of all possible equivalence relations on X and the set of all partitions of X.
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