In the following two sentences, I think that the first one is more precisely correct:
(1) Let G be a graph with n vertices, and S be the set of all subgraphs of G.
(2) Let G be a graph with n vertices, and let S be the set of all subgraphs of G.
In the following two sentences, I think that the first one is more precisely correct:
(1) Let G be a graph with n vertices, and S be the set of all subgraphs of G.
(2) Let G be a graph with n vertices, and let S be the set of all subgraphs of G.