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Partitions and Equivalence Relations

Definition: The relation induced by a partition D is the set of all pairs of elements of the same block of D:

x ~ D y : <=> exists d in D: x in d /\  y in d.

Proposition: Let D be a partition of S. The relation induced by D is an equivalence relation on S:

forall S, D: D is partition of S =>
    ~ D is equivalence relation on S.

Every partition defines an equivalence relation.

Author: Wolfgang Schreiner
Last Modification: January 12, 2000

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