Write down all proofs in acceptable mathematical lanquage: make sure you mark the beginning and end of the proof, define all variables, use complete, grammatically correct sentences, and give a justification for each assertion (e.g. by definition of...). Definitions: I. An integer n is even if and only if there exists an integer k such that n = 2k 2. An integer n is odd if and only if there exists an integer k such that n = 2k + 1. 3. Two integers have the same parity when they are both even or when they are both odd 4. Two integers have opposite parity when one is even and the other one is odd 5. An integer n is divisible by an integer d with d 0, denoted d I n, if and only if 6. A real number r is rational if and only if there exist integers a and b with b 0 7. For any real number x, the absolute value of X, denoted X1, is defined as follows there exists an integer k such that n - dk such that r = a/b x ifx 20 1. Prove each of the following statements using a direct proof, a proof by contrapositive, a proof by contradiction, or a proof by cases. In addition to the proof, you must answer the following questions for each statement: Which proof method did you use? What are the assumptions of the proof (what you suppose)? What is the conclusion of the proof (what you need to show)? . . . a. Any two consecutive integers have opposite parity. b. If a group of 8 kids have won a total of 65 trophies, then at least one of the 8 kids has won at least 9 trophies. c. The difference of any rational number and any irrational number is irrational d. For any three integers x, y, and z, if y is divisible by x and z is divisible by y, then z is divisible by x. e. For all real numbers x and y, max (x, y) = f. For any positive integer n, n is even if and only if 7n + 4 is even and min (x, y) = 2 2 Hint: To prove that a bi-conditional statement of the form p q is true, you must show that p q and q p are both true. For example, to prove that for any integer n, n is odd if and only if n is odd, you must show that (1) if n is odd, then n2 is odd, and (2) if n2 is odd, then n is odod