1. For each of the following statements, write the contrapositive statement, and prove the original statement by proving its contrapositive:
(a) If m^2 + n^2 ≠ 0, then m ≠ 0 or n ≠ 0.
2. What is wrong with the following proof of the conjecture "If n^2 is positive, then n is positive.":
Proof: Suppose that n^2 is positive. Because the conditional statement "If n is positive, then n^2 is positive" is true, we can conclude that n is positive.
3. A rational number is a number that can be expressed as the ratio of two integers p and q such that q ≠ 0.
(a) Prove that the product of any two rational numbers x and y is a rational number.
4. The term modulus (denoted mod or %) is used to describe the remainder when one integer is divided by another. Thus, we write a mod b = r to mean that r is the remainder when a is divided by b. Provide a counterexample to each of the following statements about integers that is false:
(a) If (a mod b) = (b mod c), then a = b.
(b) If (a mod b) = c, then ((a + 1) mod b) = c + 1