Some problems about cuts. Doesn't need a formal proof, just want some ideas about how to proof these statement.

Cuts are sometimes dened as subsets of the edges of the graph, instead of as partitions of its vertices. We will sometimes go back and forth between these two denitions without making much of a distinction, so in this problem you will prove that these two denitions are almost equivalent. Let G = (V, E) be a directed graph, and let 3,15 6 V. We say that a set of edges X Q E separates s and t if every directed path from s to 75 contains at least one edge in X. For any subset S of vertices, let 6(3) denote the set of edges leaving 5', i.e., let 6(3) = {(u, v) E E : u E 5', 'U 915 S}. (b) Let (S, 3') be an (3,13) cut (i.e., .3 E S and t E 5', where 5' = V\\S). Prove that 6(3) separates .5' and t. (c) Let X be an arbitrary subset of edges that separates s and t. Prove that there is an (s, t)-cut (S, 5') such that 6(8) Q X. (d) Let X be a minimal subset of edges that separates s and t (so for all X'r Q X , we know that X ' does not separate .9 and t). Prove that there is an (s, t)-cut (S, 3') such that 6(8) = X