1. Use a direct proof to show that the sum of two even integers is even.
3. Show that the additive inverse, or negative, of an even number is an even number using a direct proof.
4. Use a direct proof to show that the product of two odd numbers is odd.
5. Prove that if n is a perfect square, then n + 2 is not a perfect square.
6. Use a direct proof to show that the product of two rational numbers is rational.
7. Prove that if m and n are integers and mn is even, then m is even or n is even.
8. Prove that if n is an integer and 3n + 2 is even, then n is even using
a) a proof by contraposition
b) a proof by contradiction
9. Prove the proposition P(1), where P(n) is the proposition “If n is a positive integer, then n2 ≥ n”. What kind of proof did you use?
11. Show that if you pick three socks from a drawer containing just blue socks and black socks, you must get either a pair of blue socks or a pair of black socks.
12. Prove that if n is a positive integer, then n is even if and only if 7n + 4 is even.
13. Find a counterexample to the statement that every positive integer can be written as the sum of the squares of three integers.