Contrapositive and contradiction Consider the following claim: For all integers m and n, if mn + 1 is odd then m is even or n is even. Prove the claim using a proof by contrapositive Prove the claim using a proof by contradiction Equivalence Prove the following for all x elementof R: x is rational x/3 is rational x - 2 is rational. Proof by cases Use a proof by cases to show that: x^2 + 2xy + y^2 = (max(x, y) + min(x, y)) + min(x, y))^2 where x, y elementof R. Rational, irrational Prove or disprove that if x and y are rational numbers, then x^y is also rational. Prove that if x^2 is irrational, then x is irrational. Sets Let A = {a, b, c), B = {x, y}, and C = {5, 10}. Find A times B times C and C times A times B. Let A = {1, 4, 8, 16} and B = {2, 4, 16, 32, 64}. Find A union B, A intersection B, A\B, B\A, and |P(A)|. Prove A intersection B = A union B