(Equivalence relations.) (a) For any n 2 2, let A 1,2,...n), and let P(A) denote the power set of A. Define a relation ~ on P(A) as follows: for any two elements B, C P(A), we say that B ~ C if and only if Bn 1,2 Cn1,2) Is ~an equivalence relation? Prove your claim (b) Consider the following "proof" that any relation that is symmetric and transitive must be reflexive: Claim: If~ is a relation on a set S that is both symmetric and transitive, then ~is also reflexive. Proof. Take any x S. Because ~ is transitive, we know that for any z, y, z S, if ~y and y~z then ~z must also hold. In particular, if we let z, this tells us that if r~y andy~x then r~r. We also know that our relation ~ is symmetric; in other words, by the definition of symmetric we know that if r~y theny~ Therefore, if we combine these two observations we get that for any rES that r. In other words, our relation is reflexive, as claimed! 0 This proof has a flaw in its logic. Find the logical mistake and explain why it is a