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Class Properties

Proposition: Let R be an equivalence relation on S. [x]_R contains x, for every x in S:

forall S, R: R is equivalence relation on S => forall x in S: x in [x]R.

Let x and y be elements of S. The equivalence classes of x and y with respect to R are either identical or disjoint:

forall S, R: R is equivalence relation on S =>

forall x in S, y in S:

[x]R = [y]R \/ [x]R intersection [y]R = 0.