I'm working on some questions to bone up on my knowledge of DFA's for a computing class and I've run across the following problem that is giving me some issues. If we have some DFA M = (Q, Σ, δ, q0, F) and some other DFA M' = (Q, Σ, δ, q0, F') where F is a proper subset of F', are the following relations possible or not between the two produced languages?
1) L(M) ⊂ L(M')
2) L(M) ⊃ L(M')
My current theory is that the first one is not possible, due to the fact that the first machine has fewer finish states than the other, thus the language must be larger and cannot be subset to M'. This would of course mean that the second relationship is possible, since M must contain M'. Am I on the right track here and if so, how could I prove this?