|
Written assignments ask for a small number of proofs written up carefully, in full sentences, the way you would want them to appear in a book. They are graded on the writing as well as on the mathematics: an argument that is correct but unreadable will not receive full credit. Each assignment will be posted here at least a week before it is due. You may collaborate on these, but you must write your final solutions in your own words and list your collaborators at the top of the page. See the course policies for details. |
| Assignment | Due | Files |
|---|---|---|
| Written Assignment 1 (Chapters 2–4) | Wednesday, September 9 | Assignment (PDF) |
| Written Assignment 2 (Chapters 5–6) | Wednesday, September 23 | Assignment (PDF) |
|
A word about the two kinds of problems in the book. Gorkin and Daepp distinguish Exercises, which appear inside the body of each chapter and have complete solutions at the end of that chapter, from Problems, which appear at the end of the chapter and do not. Work the Exercises as you read - that is what they are for, and checking your answer immediately is the point. The practice problems listed below are all Problems. These are not collected. Nobody will check whether you did them. They are also the single best predictor of how you will do on the quizzes and exams, which between them are 70% of your grade. Quiz problems will frequently be one of these problems, or a close relative of one. How to use the list. Do not read a problem, decide you know how it would go, and move on. Write the proof out in full sentences. The gap between "I see why this is true" and "I can write an argument that convinces a skeptical reader" is exactly what this course is about, and it only becomes visible when you put a pen down on paper. If you get stuck for more than fifteen minutes or so on a problem, that is a good problem to bring to office hours. A problem marked ◆ is one I particularly recommend; these tend to be the ones that either come up again later or catch a common misunderstanding. |
| Date | Chapter | Problems |
|---|---|---|
| Week 1 | ||
| 8/24 | 1. The How, When, and Why of Mathematics | 1.2, 1.3, 1.4, 1.5, 1.6, 1.9 ◆, 1.13 |
| 8/26 | 2. Logically Speaking | 2.1, 2.2 ◆, 2.3 ◆, 2.5, 2.10, 2.12, 2.14, 2.20, 2.22, 2.24 |
| Week 2 | ||
| 8/31 | 3. Contrapositive and Converse | 3.2 ◆, 3.3, 3.5, 3.6, 3.7, 3.9, 3.19, 3.20, 3.21 ◆ |
| 9/2 | 4. Set Notation and Quantifiers | 4.1, 4.2, 4.4, 4.6 ◆, 4.13, 4.14, 4.18, 4.20, 4.22, 4.27 |
| Week 3 | ||
| 9/7 | No Class - Labor Day | |
| 9/9 | 5. Proof Techniques (direct proof, contrapositive) | 5.4, 5.6 ◆, 5.7, 5.14, 5.16 ◆, 5.18, 5.31 |
| Week 4 | ||
| 9/14 | 5. Proof Techniques (contradiction, cases, counterexamples) | 5.8 ◆, 5.9, 5.11, 5.12, 5.19, 5.25, 5.33 ◆, 5.36 |
| 9/16 | 6. Sets | 6.1, 6.3 ◆, 6.4, 6.5, 6.11, 6.12, 6.13, 6.19, 6.22, 6.25 |
| Week 5 | ||
| 9/21 | 7. Operations on Sets | 7.2, 7.3, 7.4, 7.8 ◆, 7.11, 7.17, 7.19 |
| 9/23 | 8. Mathematical Induction | 8.1, 8.2, 8.3, 8.4, 8.5 ◆, 8.9, 8.10 |
| Week 6 | ||
| 9/28 | 8. Complete Induction, Recursion, Well Ordering | 8.7, 8.8, 8.16, 8.17 ◆, 8.18, 8.19 ◆, 8.23, 8.24 |
| 9/30 | 9. More on Operations on Sets (indexed families) | 9.1, 9.2, 9.3, 9.5, 9.8, 9.15 ◆ |
| Week 7 | ||
| 10/5 | 10. The Power Set and the Cartesian Product; Review for Midterm Exam 1 (~20 min) | 10.1, 10.2, 10.3, 10.5, 10.6 ◆, 10.9 ◆, 10.10, 10.13, 10.17, 10.22 Review: rework any problem above that you could not finish the first time, without looking at your earlier attempt. |
| 10/7 | Midterm Exam 1 (Chapters 1-10, 75 minutes) | |
| Week 8 | ||
| 10/12 | 11. Relations | 11.1, 11.2, 11.5, 11.6, 11.7, 11.8, 11.12 ◆, 11.15, 11.21 ◆ |
| 10/14 | 12. Partitions | 12.1, 12.2, 12.3 ◆, 12.5, 12.12, 12.13, 12.14, 12.23 |
| Week 9 | ||
| 10/19 | 13. Order in the Reals | 13.1, 13.2 ◆, 13.10, 13.11 ◆, 13.13, 13.15, 13.19, 13.21, 14.2, 14.7 |
| 10/21 | 15. Functions, Domain, and Range | 15.1 ◆, 15.2, 15.3, 15.6, 15.12, 15.13, 15.16, 15.17, 15.22 |
| Week 10 | ||
| 10/26 | 16. Injective, Surjective and Bijective Functions | 16.1, 16.2, 16.3, 16.6 ◆, 16.7, 16.8, 16.14 |
| 10/28 | 17. Inverses and Composition | 17.1, 17.2, 17.5, 17.8, 17.11 ◆, 17.13 |
| Week 11 | ||
| 11/2 | 18. Images and Inverse Images | 18.1, 18.2, 18.3, 18.5, 18.9, 18.11 ◆, 18.14, 18.19 ◆, 18.21 |
| 11/4 | 19. Sequences | 19.1, 19.2, 19.3, 19.5 ◆, 19.7, 19.9, 19.11, 19.21 |
| Week 12 | ||
| 11/9 | 20. Convergence of Sequences of Real Numbers | 20.1, 20.2, 20.3, 20.4 ◆, 20.5 ◆, 20.7, 20.11 |
| 11/11 | Catch-up day - no new problems | |
| Week 13 | ||
| 11/16 | 20. Cauchy Sequences and Completeness | 20.10, 20.13, 20.14 ◆, 20.24, 20.25 ◆, 20.26 |
| 11/18 | Midterm Exam 2 (Chapters 11-13, 15-20) | |
| Week 14 | ||
| 11/23 | 21. Equivalent Sets | 21.1, 21.2, 21.4 ◆, 21.6, 21.7, 21.15, 21.20, 21.21 ◆ |
| 11/25 | No Class - Thanksgiving Holiday | |
| Week 15 | ||
| 11/30 | 22-24. Finite Sets, Countability, Cantor-Schröder-Bernstein | 22.2, 22.3, 22.11 ◆, 22.13, 22.17, 22.23 23.1, 23.2, 23.3, 23.5, 23.14 ◆ 24.2, 24.6, 24.7, 24.11 ◆ |
| 12/2 | 28. Congruence Modulo m | 28.1 ◆, 28.2, 28.8 ◆, 28.11, 28.12, 28.13, 28.16 |
| Week 16 | ||
| 12/7 | 28. The GCD, the Euclidean Algorithm, Modular Inverses | 28.6, 28.7, 28.9 ◆, 28.10, 28.15, 28.17, 28.20 ◆, 28.21 |
| 12/11 | Final Exam - Friday, December 11, 12:30-2:30 p.m. (cumulative) | |