Let G be a bipartite graph, with two sets of vertices A and B, each of size n, and some edges, each with one endpoint in A and one in B. A perfect matching in this graph is a subset of the edges such that every vertex in A and B is an endpoint of exactly one edge. A student recalls that the Gale-Shapley algorithm finds perfect matchings, and proposes the following. Give each vertex a preference list on the other set of vertices that puts all of its neighbors ahead of its non-neighbors. Use Gale-Shapley to find a stable matching M for this list. If all matched pairs in M correspond to graph edges, this is a perfect matching in G. Otherwise, the student claims, G has no perfect matching, because if there were one, but M matches some nodes to non-neighbors, then M would be unstable and not the result of Gale-Shapley. Is the student correct? Argue why or why not.