Answers and Hints

These pages contain answers to every exercise appearing in the book. Before getting overly excited, though, keep in mind that a thorough understanding of an exercise encompasses much more than just the correct answer. It will include a mastery of notation and vocabulary, a working knowledge of the material on which the exercise is based, discussions with classmates that reveal alternate methods of solution, and so on. That being said, having an answer key at one's disposal does have its advantages. At the very least an answer serves as a blaze on a trail; it provides an indication that one is (or isn't, as the case may be) headed in the right direction. Hints to all of the writing problems are also included below. They should be used sparingly, of course, and certainly never consulted before a sincere effort has been made to first solve a problem independently. Each hint will hopefully guide the solver in a fruitful direction, perhaps even past an obstacle or two. But good math problems will typically yield to several different approaches, only one of which is indicated by a hint. So don't become worried if you have found a solution that seems to strike out in a different direction than the one suggested by the hint. Just be sure that your solution addresses any issues that may be raised by the hint. Good luck with the problems, and enjoy the math! Chapter 1: Logical Foundations 1.1 Statements and Open Sentences Exercises 1. a) False b) True c) True d) True e) False 2. The open sentence is true for n = 2 and false for n = 11 3. The set of even numbers 4. x = 40 is the only solution 5. n is an integer, k is a positive integer, and x is a real number 6. The equation is satisfied if x + 3 = 0 or x 2 − 9 = 0 7. a) True b) False c) True d) False e) False f) True g) True 8. n = 9 and n = 15 also serve as counterexamples 205