Prove or disprove $($by giving a counterexample$)$ each statement below. Assume that the graph G $=($V$,$E$)$ is a connected undirected weighted graph such that all edge weights are positive. Do not assume that edge weights are distinct unless this is specifically stated.

$($a$)$ If G has more than $|$V $|1$ edges, and there is a unique heaviest edge, then this edge cannot be part of any MST$.$

$($b$)$ Let e be any edge of minimum weight in G$.$ Then e must be part of some MST$.$

$($c$)$ If G contains a unique edge e of minimum weight, then e must be part of every MST$.$

$($d$)$ If e is part of some MST of G$,$ then it must be a minimum weight edge across some cut of G$.$

$($e$)$ For every pair of vertices in G$,$ a shortest path between them is necessarily part of some MST$.$