a) Prove using the notion of "without loss of generality" that 5x + 5y + 1 is an even integer when x and y are of opposite parity.
b) Prove that if x and y are real numbers, then max(x,y) + min(x,y) = x+y.
[Hint: Use a proof by cases with the two cases corresponding to x ≥ y and x < y, respectively.]
c) Prove that the following is true for all positive integers n: n^2 is odd if and only if n is odd.