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Relationship between Constructions

Proposition: Let R be an equivalence relation on S. Then R is the relation induced by the quotient set of S with respect to R:

forall S, R: R is equivalence relation on S =>
   R = ~ S/R.

Let D be a partition of S. Then D is the quotient set of the relation induced by D:

forall S, D: D is partition of S =>
   D = S/ ~ D.

Each construction is the inverse of the other.

Author: Wolfgang Schreiner
Last Modification: January 12, 2000

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