Question: Please type the answer or if you can not type, write the answer as clear as possible. Problem 1/ 5 (32 points, 8 + 12

Please type the answer or if you can not type, write the answer as clear as possible.

Problem 1/ 5 (32 points, 8 + 12 + 12) For each problem, provide a strict math proof without any possible confusion or ambiguity. DO NOT just list math equations. Instead, write down convincing and sufcient math reasoning. (1) A rational number is a number that can be in the form 2 where p and q are integers and q is not equal to zero. Prove that for real numbers 1: and y, x 7E 0, if x is rational and y is irrational, then 19y is irrational. (2) Suppose x, y E Z, prove that if (x2 + 2) - (y2 2y) is odd, then both 3: and y are odd. (3) Dene a relation R' on Z by xR'y if x + y is even. Show that R' is an equivalence relation

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