For any set A , let P(A) denote the set of all partitions of A, and

Question:

For any set A ≠ ∅, let P(A) denote the set of all partitions of A, and let E(A) denote the set of all equivalence relations on A. Define the function f: E(A) → P(A) as follows: If R is an equivalence relation on A, then f(R) is the partition of A induced by R. Prove that f is one-to-one and onto, thus establishing Theorem 7.8.