**Discrete Math

1.

Define the symmetric difference between two sets to be A^B = (A-B) U (B-A). It should be clear that the symmetric difference is commutative since union is commutative. But in order to speak loosely about the symmetric difference among THREE sets, A^B^C, we must first show that the operation is both commutative AND associative. Otherwise, we would have to indicate the order of the sets and where the parentheses went!

Part 1 - Show that the symmetric difference is associative, i.e. (A^B)^C = A^(B^C).

For the next part, let P(S) be the powerset of some set S where |S| = n, so that |P(S)| = 2^n.

Part 2 - What is the probability that the symmetric difference of three randomly chosen elements in P(S) is the empty set? Repetition (choosing the same element multiple times) IS ALLOWED!!!

2. For EACH of the 16 relations over the set S = {a, b}, list whether the relation is reflexive, symmetric, and/or transitive. Please organize your response so that it's easy to read... thanks!!!

3. Let f: A -> A where A is a non-empty set. Prove or disprove: the identity function is the only such function f which is also an equivalence relation.