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Math 220 Assignment 2 Due September 25th 1. Let P , Q, and R be statements. Prove that (P ∧ Q) → R is logically equivalent to P → (Q → R), using two methods: (a) using a truth table. (b) using logical equivalences, such as ¬(P ∧ Q) ≡ ¬P ∨ ¬Q. 2. Let n be an integer. Prove that n is divisible by 6 if and only if n is even and is divisible by 3. 3. Let n be an integer. Prove that n is disivible by 6 if and only if n2 is divisible by 6. Hint: you are allowed to use the last problem, and any other results covered in class. 4. A real number x is rational if there exists integers p and q such that x = pq . Let x and y be numbers. (a) Prove or disprove: If x and y are rational, then x + y is rational. (b) Prove or disprove: If x is rational, and y is irrational, then x + y is irrational. (c) (Extra credit): If x and y are both irrational, then x + y is irrational. You may assume the following true result: If x + y is an integer, then x and y are integers. 5. Which of the following numbers are rational? Justify your answer. Your response should be ‘x is rational’, or x is irrational, followed by a proof. √ (a) 3. √ (b) 4. √ (c) 6. 1