Minimum spanning tree

(a) Assume that all edge costs in a graph G are distinct. Let C be a cycle in the graph, and let edge e = (v,w) E C be the most expensive edge of C. Prove that this edge cannot belong to any MST of G. (b) Assume that all edge costs are distinct. Let S be any subset of nodes that is neither empty nor equal to all of V, and let edge e = (u,w) be the minimum cost edge with one end in S and the other in V \ S. Prove that this edge belongs to every MST of G. (c) Use the previous two part to prove the following statement. Edge e = (u,w) does not belong to a minimum spanning tree of graph G if and only if v and w can be joined by a path consisting entirely of edges that are cheaper than e. Make sure you prove both directions