Discrete Structures, Logic, and Computability, 4th Edition

Answers to Selected Exercises

Chapter 1

Section 1.1

1. Consider the four statements “if 1 = 1 then 2 = 2,” “if 1 = 1 then 2 = 3,” “if 1 = 0 then 2 = 2,” and “if 1 = 0 then 2 = 3.” The second statement is the only one that is false.

3. A and B are both true or they are both false.

4. a. If there are no clouds, then it is not raining. c. If x ≤ 0, then x ≤ 1. e. If L₁ and L₂ are parallel, then L₁ and L₂ don’t divide the plane into four parts. g. If x + y ≤ 0, then (x ≤ 0 or y ≤ 0).

5. a. It is not raining or there are clouds. c. x ≤ 1 or x > 0. e. L₁ and L₂ don’t divide the plane into four parts or L₁ and L₂ are not parallel. g. (x ≤ 0 or y ≤ 0) or (x + y > 0).

6. a. For example, let x = 2 and y = − 1. c. For example, let x = 3 and y = 2.

7.