Problem $3.$ MBSTs

One of the basic motivations behind the MST Problem is the goal of designing a spanning network for a set of nodes with minimum total cost. Let us now explore another type of objective: designing a spanning network for which the most expensive edge is as cheap as possible.

Specifically, let $G=(V,E)$ be a connected graph with n vertices, m edges, and positive edges costs that are all distinct. Let $T=(V,E0)$ be a spanning tree of G ; we define the bottleneck edge of T to be the edge of T with the greatest cost. A spanning tree T of G is a minimum$-$bottleneck spanning tree $($MBST$)$ if there is no spanning tree $T0$ of G with a cheaper bottleneck edge.

a$)$ Is every MBST of G an MST of G$?$ Prove or give a counterexample.

b$)$ Is every MST of G an MBST of G$?$ Prove or give a counterexample.