In the problem below, a graph is a pair G = (V, E), where V is a finite set and E V 2 is a set of 2-element subsets of V . (None of the questions is about multigraphs.) A path in a graph G = (V, E) is a sequence of distinct vertices p = (v1, v2, . . . , vk), for some k N, such that there is an edge between any two consecutive vertices in p.

In each case, either prove that R is an equivalence relation for every graph G = (V, E) or find a graph G = (V, E) for which R is not an equivalence relation. (a) Let R be the relation on V consisting pairs of vertices (a, b) V V such that a = b or there is a path (a = v1, v2, . . . , vk = b) for some odd integer k. (b) Let R be the relation on V consisting pairs of vertices (a, b) V V such that a = b or there is a path (a = v1, v2, . . . , vk = b) for some even integer k.