An Operator Approach to the Principle of Inclusion and Exclusion

Using an operator approach we derive Sylvester-Whitworth formulae for sets A'a. By the same token we treat the problem where both sets of A's and B's are involved. Our result extends the Sylvester-Whitworth inclusion and exclusion formula to the resolution of the number of elements in exactly mi sets of A's and mi sets of B's respectively. The formula are applied to the complete graph and complete bipartite graph. The enumeration of spanning subgraphs with any preassigned number of disconnected cycles is solved, together with the case where any preassigned number of vertices have degree one.