Let A be a nonempty set and let P be a partition of A. Define a relation R (corresponding to P) on A by

(a) Let A = {1,2,3}. Write down ALL possible partitions of A. For each of the partition P, use () to write down the relation R (as a subset of A ? A) corresponding to P.

(b) Observe that all the relations in (a) are equivalence relations. Prove that this is true in general , that is, prove that if A is a nonempty set , and P is a partition of A, then the relation R corresponding to P defined in (*) must be an equivalence relation.

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Ry if there exists SEP such that x, y € S. Ry if there exists SEP such that x, y € S.

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a All possible partitions of A are 123 1 23 13 2 1 2 3 12... View the full answer