1. Prove the following distributive law: AU(BOC) = (AUB) n (AUC) 2. Suppose A C B and C C D. Show that A X C C B X D. 3. Prove that A - (BOC) = (A - B) U(A - C). 4. An ordered pair (a, b) can be defined as the set {{a}, {a, b} }. Show that (a, b) = (c, d) if and only if a = c and b = d. 5. Given an arbitrary relation R, suppose we compute two new relations: . R1, the reflexive closure of the transitive closure of R . R2, the transitive closure of the reflexive closure of R Prove or disprove: R1 = R2 for all R. 6. Let A = {cat, dog, bird, rat } and R be a relation on A defined by {(x, y) : x and y have at least one letter in common}. (a) Draw R as a directed graph. (b) Is R reflexive, symmetric, and/or transitive? 7. Given a relation R on a set A, prove that if R is transitive, then so is R-1. 8. Suppose R and S are symmetric relations on a set A. Prove that Ro S is symmetric iff ROS = SoR. 9. Prove or disprove: for any set A, there exists a relation R on A such that R is both symmetric and antisymmetric. 10. Find all equivalence relations on {1, 2, 3}