question 4

The NAND operator takes two propositions and evaluates to F when both propositions are T and evaluates to T otherwise; in other words, it is given by the truth table Find a compound proposition whose only operation is NAND and that is logically equivalent to M(x, y, z). Consider the compound proposition (p rightarrow q) rightarrow (q rightarrow p) (a) Rewrite this compound proposition in disjunctive normal form (DNF). (b) Rewrite this compound proposition in conjunctive normal form (CNF). (c) Is this compound proposition a tautology? Why or why not? (c) Is this compound proposition a contradiction? Why or why not? The following puzzle is similar to the "This sentence is false" paradox we talked about in class. In thus puzzle, we think about an imaginary island where each inhabitant either always tells the truth or always lies. Imagine meeting two people from the island, person A and person D. Your goal is to determine whether each one of them is a truth-teller or a liar; or whether there's not enough Information to tell. The only information you have is one statement each person makes. For example, if person A says "B lies' and person B says "I tell the truth"; it's possible that A is telling the truth and B is lying (because A is saying that B's statement is false, and B's statement being false means that B is a liar); and it's also possible that A is lying and B is telling the truth (because A lying would he that the statement "B lies" is false, which means that B is telling the truth, and that's what B is claiming). Therefore, there's not enough information to tell who is who in this scenario. (a) First scenario: Person A says "I tell the truth". Person B says "I tell the truth". (b) Second scenario: Person A says "We both tell the truth." Person B says "A lies