Let H = (V,E) be a connected, undirected graph. Let A be a subset of E.Let C =  (W , F) be a connected component (tree) in the forest G = (V,A). Let (u,v) be an edge connecting C to some other component in G.Now doesnt it imply that (u,v) must belong to A since it is in forest G and it doesnot matter which component it connects? Is this proof wrong:For purpose of contradiction lets assume that it does not belong to A. As (u,v) connects two components of graph it must be an edge. As Set of edge is given by A in graph G it must belong to A which contradict our assumption.
What do we mean when we say an edge (u,v) connects some component to other component in forest G = (V,A)
formal languages;graph theory;graphs;reductions;proof techniques