2. (Equivalence relations (a) For any n 2 2, let A ,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: Ifis a relation on a set S that is both symmetric and transitive, then is also reflexive. Proof. Take any E S. Because~is transitive, we know that for any r, y, zES, ifx ~ y and y ~ z then x ~ z must also hold. In particular, if we let x = z, this tells us that if y and y~ then We also know that our relation~is symmetric; in other words, by the definition of symmetric we know that if z~y then y~. Therefore, if we combine these two observations we get that for any xES that r. In other words, our relation is reflexive, as claimed! This proof has a flaw in its logic. Find the logical mistake and explain why it is a mistake