Which can be converted to inferred equivalent CNF sentence? A. Every sentence of propositional logic B. Every sentence of inference C. Every sentence of first-order logic D. All of the mentioned

Consider the following statements Anyone passing his math exams and winning the lottery is happy. Anyone who studies or is lucky can pass all his exams. Ravi did not study but he is lucky. Anyone who is lucky wins the lottery. Translate these sentences into predicate logic form. Convert them to Conjunctive Normal Form(CNF) if required. Prove by resolution that Ravi is happy

2. Use a tree to test whether the following two propositions are equivalent. (Present the tree, and say whether they’re equivalent or not.) If they’re not equivalent, read off from your tree a model on which the propositions have different truth values (and indicate the path from which you are reading it off). ∀x(Ax → Gx) ∃x(Ax → Gx)

Given Similar as Valid InferenceIf p, Then q   If ~q, Then ~pOnly If p, Then q If q, Then p If ~p, Then ~qUnless p, Then q If ~p, Then q If ~q, Then pEither p or q    If ~p, Then q   If ~q, Then p

Full Statement Clauses in semantic

Exercise b) Recall that a formula ϕ is a tautology if and only if ¬ϕ is not satisfiable, cf. Exercise 1.2.61in Rosen that we have covered in class, that is, to check whether ϕ is a tautology, you can check whether¬ϕ is satisfiable, and reverse the answer. Note also, that since the tableau rules only deal with negation andconjunction, you have to rewrite ϕ such that it only contains these two connectives. Now, make tableaus, asthey are presented in Priest’s book, Section 1.4, showing that the formulas p → (p ∨ q) and ¬p → (p → q)are tautologies, cf. exercises 1.2.11b and 1.2.11c in Rosen’s book