1) Is the following argument valid? Zack is either a pianist or a writer. If he is a pianist, then he has a keen sense of hearing. Zack does not have a keen sense of hearing, so he is a writer. The following are the variables: p: Zack is a pianist. w: Zack is a writer. h: Zack has a keen sense of hearing. Cite the rules of inference used in every step of your proof. 2) The following premises are given: 1. A tourist in this tour group has not visited Gardens by the Bay. 2. Every tourist in this tour group has visited Sentosa. Can you conclude that \\Someone who has visited Sentosa has not visited Gardens by the Bay"? a. Define appropriate predicates and write out the argument using these predicates with quantifiers. b. Prove your argument, justifying every step. 3) The logician Raymond Smullyan describes an island containing two types of people: knights who always tell the truth and knaves who always lie. You visit the island and have the following encounters with two natives A and B. A says: I am a knave or B is a knight. B says nothing. What are A and B? Begin your proof by assuming that A is a knave. 4) Prove that for all integers n, if n > 2 then there is a prime number p such that < < !. 5) Recall the Quotient-Remainder Theorem: For any integer n, and any positive integer d, there exist integers q; r such that = + , where 0 r < d. Use Mathematical Induction to prove the Quotient-Remainder Theorem for all . Q1 () () ~ ~ () ~ ~ ~ ~ ~ ( Q2 () = () = , ~() () , () () Prove , () ^ ~() -- Am I correct? Not too sure how to prove here Q3) Assume A is a knave A says \"I am a knave or B is a knight\" ( or is inclusive or ) Hence the statement is false contradicts because knaves always lie. A cannot be a knave Hence A must be a knight. If A is a knight, B is a knight because knights tell the truth and the second part of the statement has to be true. Q4) Q5)