Consider two connected components of an undirected graph \(G\), and suppose each has no cycles. Let \(G^{\prime}\) be a new graph whose vertex set is the union of the vertex sets of the two components and whose edge set is the union of the two edge sets, together with a single edge \(\{u, v\}\), where \(u\) is in one component and \(v\) is in the other. Show that \(G^{\prime}\) has no cycles.