Write the Negation

Write the negation of the following statements in English using the logical equivalence of ~∀x P(x) = ∃x~P(x) and ~∃x P(x) = ∀x~P(x). No credit will be given if you didn’t use these logical equivalences. (a) ∀x∀y (P(x,y) → ~Q(x,y) (b)∃x∀y ( P(x,y) V ~Q(x,y) ). Note: When you are done with simplification of the quantifiers then also use the equivalences of P→ Q = ~P V Q and Demorgan law to simplify further your answer. I will deduct marks if you ignore this. Q1) Use the principle of resolution to show that the hypothesis “Chohan works hard”, “If Chohan works hard then he is a dull boy”, “if Chohan is a dull boy, he will not get a job” imply the conclusion “Chohan will not get the job”. P1: C1: P2: C2: P3: C3: C: C4: C5: From ___ and ____ C6: From ___ and ____ C7: From ___ and ____. Q4: The following question relates to the inhabitants of the island of knights and knaves created by Smullyan. The knights only speak the truth when they are happy. The knaves always speak lie regardless they are happy or sad. You encounter two people A and B. Determine if possible what A and B are if they address you in the way. If you cannot determine what these two people are, can you draw any conclusions? (Marks 20) A says “The two of us are both knights” and B says “A is a knave” Solution: P = A is a knight ~P = A is a knave Q = B is a knight ~Q = B is a knave. Scenario 1: Assume all knights to be happy .CASE 1: _________, _____________ 1) 2) CASE 2: _________, _____________ 1) 2) CASE 3: _________, _____________ 1) 2) CASE 4: _________, _____________ 1) 2) Conclusion: A is _______________, B is _____________ Scenario 2: Assume all knights to be sad (means the knights will speak lies now) CASE 1: _________, _____________ 1) 2) CASE 2: _________, _____________ 1) 2) CASE 3: _________, _____________ 1) 2) CASE 4: _________, _____________ 1) 2) Conclusion: A is _______________, B is _____________ Q5: Assume that the statement “if it is sunny day then I will not go to beach” is in contrapositive form. Make the following forms of this statement using English sentences . Converse: Contrapositive: