Problem 1. Exercises 8 and 10 from Section 2.5
Problem 2. What are “DeMorgan’s Laws?”
Problem 3. Exercises 9 and 10 from Section 2.6
Problem 4. Exercises 4, 5, and 8 from Section 2.7
Problem 5. Exercises 4, 10, and 11 from Section 2.10
Problem 6. Describe at least two facts that our textbook explicitly claims we will accept without proof or justification.
Problem 7. Let. Prove that . (Be sure you are only using the definition of least common multiple.)
Problem 8. Here is a proof of some statement. Read it, and then write down the claim that was proved.
proof. Let be arbitrary. If so that , then is even. Suppose, then, that is not even. This means that for all integers . By the Division Algorithm, there exists so that where . This implies that either or . However, if , then we have , which is impossible since we have assumed is not even. Therefore and we have that . This implies that is odd, completing the proof. .
Problem 9. Prove that if and if then .
Problem 10. Prove that if , then .
Problem 11. Exercises 4, 8, 14, and 26 from Chapter 4 exercises.