1. Give an example of a set A and two nonempty relations R1 and R2 on A such that a) R1 is symmetric and transitive but not reflexive; b) R2 is reflexive and symmetric but not transitive. 2. A relation R is defined on Q by ny if xy > 0. Prove or disprove the following: a) R is reflexive; b) R is symmetric; c) R is transitive; d) R is an equivalence relation. 3. A relation R is defined on R by ny if X y E Z. Prove that R is an equivalence relation. 4. Let R be the relation on Z defined by ny if x2 E y2 (mod 5). Prove that R is an equivalence relation and determine the distinct equivalence classes. 5. Let A be a nonempty set and R be a relation on A such that domain(R) = A. Prove that if R is symmetric and transitive then R is an equivalence. 6. Give an example of an equivalence relation R on the set A = {a, b, c, d, e, f, g, h} such that there are exactly three distinct equivalence classes. List the distinct equivalence classes of R