Set 4.1: 32, 61 Set 4.2: 20, 25
Set 4.6: 28 (Prove by Contraposition)
Prove the statements in 24-34. In each case use only the definitions of the terms and the Assumptions listed on page 146, not any previously established properties of odd and even integers. Follow the directions given in this section for writing proofs of universal statements.
32. If a is any odd integer and b is any even integer, then 2a + 3b is even.
61. Suppose that integers m and n are perfect squares. Then m + n + 2*sqrt(m*n) is also a perfect square. Why?
Determine which of the statements in 15-20 are true and which are false. Prove each true statement directly from the definitions, and give a counterexample for each false statement. In case the statement is false, determine whether a small change would make it true. If so, make the change and prove the new statement. Follow the directions for writing proofs on page 154.
20. Given any two rational numbers r and s with r < s, there is another rational number between r and s. (Hint: Use the results from exercise 18 and 19.)
Suppose r and s are two rational numbers where r < s.
Derive the statements in 24-26 as corollaries of Theorems 4.2.1, 4.2.2 and the results of exercises 12, 13, 14, 15 and 17.
25. If r is any rational number, then 3r^2 - 2r + 4 is rational.
Prove each of the statements in 23-29 in two ways: (a) by contraposition and (b) by contradiction.
28. For all integers m and n, if mn is even then m is even or n is even.
(Per the assignment instructions, I will only be solving by contraposition)
