“Anyone passing his history exams and winning the lottery is happy. But anyone who studies or is lucky can pass all his exams. John did not study but he is lucky. Anyone who is lucky wins the lottery. Is John happy?”(a) [2] Change the sentences to predicate form. (b) [2] Change the predicate statements to clause form.(c) [8] Use binary resolution method with the set of support strategy to prove the conclusion “Is John happy?”.(d) [8] Use binary resolution method with the unit preference strategy to prove the conclusion“Is John happy?”.

3. Formalize the following arguments in L1 providing a suitabledictionary. Make your formalization as detailed as possible. Ifyou find an ambiguity in the argument, explain it to justifyyour choice of formalization. (c) It’s not true that time travel is possible. For if I travelback in time and kill my grandfather, I will be born andunborn. But I will not be born and unborn. And I willtravel back in time and kill my grandfather, provided timetravel is possible.

3. Prove ⊢ (¬A → A) → A in Hilbert deductive system. Note: In addition to the axioms and rule of inference of H, you may use any of the derived rules and/or theorems 3.20-3.30 (as numbered in the textbook). You may not use theorem 3.31, as this is precisely that theorem.Theorem 3.20 ⊢ ¬A → (A → B). Theorem 3.21 ⊢ A → (¬A → B). Theorem 3.22 ⊢ ¬¬A → A. Theorem3.23 ⊢A→¬¬A.Theorem3.25 ⊢(A→B)→(¬B→¬A).Theorem3.28 ⊢(¬A→false)→A.Rule 3.29 (Reductio ad absurdum) Theorem3.30 ⊢(A→¬A)→¬A.   Prove {¬A} ⊢ (¬B → A) → B in H. Note: In addition to the axioms and rule of inference of Hilbert deductive system, you may use any of the derived rules and/or theorems 3.20-3.30 (as numbered in the textbook).

Using predicate logic, prove that the following argument is valid. Provide the translation and the proof sequence. Every computer science student works harder than somebody, and everyone who works harder than any other person gets less sleep than that person.  Riley is a computer science student.  Therefore, Riley gets less sleep than someone else. Use the symbols, C(x), W(x,y), S(x,y), r.

For the following conditional statements, write (i) the contrapositive of the statement (ii) the converse of the statement, and (iii) the negation of the statement.In part (a), your answer should be written in English (not in propositional logic).For  (a), you might have to use DeMorgan’s Laws as part of the process.(a) (6 marks) If the schools are open and the roads are safe, then the game is not cancelled