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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.