Prove that A → B E (Vx)A → (Vy)B, if y is not free in B.
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- Prove that the following instance of PCP does not have a match [aba, bb], [ab, a], [bab, aa], [ab, ba]Let R=ABCDEGHK and F= {ABK→C, A→DG, B→K, K→ADH, H→GE} . Is it in BCNF? Prove your answer.Show that property (3) of a matroid implies: For any S ∈ I, {x} ∈ I, there exists y ∈ S, such that S \ {y} ∪ {x} ∈ I.
- Prove that ((P Ꚛ Q) →¬R) ↔¬P is a tautology, a contradiction or contingency.(a) Give the definition of an isomorphism from a graph G to a graph H. (b) Consider the graphs G and H below. Are G and H isomorphic?• If yes, give an isomorphism from G to H. You don’t need to prove that it isan isomorphism.• If no, explain why. If you claim that a graph does not have a certain feature,you must demonstrate that concretely. (c) Consider the degree sequence (1, 2, 4, 4, 5). For each of the following, ifthe answer is yes, draw an example. If the answer is no, explain why. (i) Does there exist a graph with this degree sequence?(ii) Does there exist a simple graph with this degree sequence?Simplify the following k-map and compare it to boolean simplification. Answer both item.
- Prove that for any m ∈ N, there exist an NFA with m states such that the equivalentDFA has at least 2^m−1 statesUsing Logically Equivalent Law's, Show that (p -> q) > (r -> s) and (p -> r) -> (q -> s) are logically NOT equivalent?Prove that the following wff is a tautology, a contradiction or neither. Be sure to say which one. (A → B) ∨ (A′→ B)