State whether or not each of the following is an equivalence relation on the set of vertices of an arbitrary graph. In each case give a proof or a counterexample. i. uvif there is an edge between u and v. ii. uv if there is no edge between u and v. iii. uv if there is a path between u and v.

2. Let H ≤ G and define = on G by a = b iff a-¹b € H. Show that is an equivalence relation.

This question refers to unions and intersections of relations. Since relations are subsets of Cartesian products, their unions and intersections can be calculated as for any subsets. Given two relations R and S from a set A to a set B, RUS = {(x, y) E AXB I (x, y) ER or (x, y) E S} RNS = {(x, y)E AXB| (x, y) ER and (x, y) Es}. Let A = {-2, 2, 3, 5} and B = {2, 3} and define relations R and S from A to B as follows: For every (x, y) E Ax B, x R y - |x| = lyl and xsyex - y is even. Using set-roster notation, state explicitly which ordered pairs are in A x B, R, S, RU S, and RN s. (Enter your answers as comma-separated lists of ordered pairs.) AXB = R = S = RUS = ROS =

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