Problem $3.(10$ points$)$ Let $G=(V,E)$ be a directed graph with $n$ nodes, and let $s(G)$ be the number of strongly connected components it has. Let hat$(G)$ be a new graph obtained by adding one or more directed edges that are not in $G,$ and let $s(hat(G))$ be the number of strongly connected components of hat$(G).$ Prove that $s(hat(G))s(G).$ Prove whether or not it is ever possible for $s(hat(G))=s(G).$