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.

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.

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