Homework |
Turn in all problems. But usually only the problems appearing in boldface will get graded.
Explain your work: You should always
explain your work, even if the book doesn't ask you to. How much should you
explain? Pretend you're explaining it to a classmate who doesn't know how to do
the problem.
Theorems whose proofs you may be tested on: Mid 2: p.212 Thm 3; p.234 Thms 2 & 3; p.228 Lemma 1.
Mid 3: Sec 8.5 Thm 1.
Final: Prove when a graph does or doesn't have an Euler path or circuit.
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HW # |
Due | Read | Do |
|---|---|---|---|
| Th 12/16 1:00-4:00 PM. Fowler 301 |
The final exam will be cumulative, with some emphasis on material we
learned after the last midterm. The exam will be roughly twice as long
as one midterm (designed for two hours) but you'll have the full three
hours available. Here's a link to my 2005 final (ignore problem 1). Our midterm will be different (and cover different sections), but it might be helpful for you to try these problems anyway. I don't have solutions written up for them; if you have questions on them, you can ask me in class or my office. Extra office hours: W 12/15: 1:00-4:30; Th 12/16: 9:30-12:30. On Th 12/16 morning please come to Fowler 307; I'll be giving a final exam there, but can come outside to answer your questions. |
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31 |
M 12/04 | This is our LAST homework assignment. Yay! :) Do these problems. Also turn in problem 4 from HW # 30. |
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30 |
F 12/03 | Do these problems. | |
29 |
W 12/01 | Sec 9.4: p. 622 only. Sec 9.5: p. 633-635 (for the time being you can ignore the words "directed", "undirected", "multigraph", "psuedograph"; I'll mention them next time in class). Read the definition of "simple graph" on page 590 (it's different from "simple path"). May be on final: Prove when a graph does or doesn't have an Euler path or circuit. |
Sec 9.4: 1. Sec 9.5: 1, 2, 3, 9, 10. Sec 4.1: 74cd. (Read Example 13 on page 277.) |
28 |
M 11/29 | Sec 7.5 p.499-502. | 7.5: 3, 8, 13, 23, Chapter 2 Supplementary Exercises: 8, 9, 14, 15. Solutions |
27 |
M 11/22 |
Sec 7.1 p. 449-453. | 7.1: 5abde, 17, 18a, 23, 37a, 44. Chapter 5 Supplementary Exercises: 3, 4, 5. Solutions |
| W 11/17 | Covers HWs 17-26 and their corresponding sections. Here's a link to my 2005 midterm 3. Our midterm will be different (and cover different sections), but it might be helpful for you to try these problems anyway. I don't have solutions written up for them; if you have questions on them, you can ask me in class or my office. |
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26 |
F 11/12 |
Sec 4.3 p.294-297. | 4.3: 1bc, 3bd, 5ade, 6abc, 9, 12, 13. Also do the following problem: Define f : N --> N by: for n = 0, 1, 2, f(n) = n; for n > 2, f(n) = f(n-1) + f(n-2) + f(n-3). True or false: for all n, f(n) has the same parity as n? Prove your answer. ("Same parity" means both are odd or both are even.) Solutions |
25 |
W 11/10 |
Sec 2.4 p. 160. | 2.4: 46 (Ignore the book's hint. Use diagonalization.) Also do these problems. Solutions |
24 |
F 11/5 | Sec 8.5 p. 560-562. | 8.5: 15, 16, 26a, 39, 40, 46cde, 47a, 57. 4.1: 42. Also do the following problem: (i) Prove that n^3 - 9n^2 + 20n is divisible by 6 for all positive integers n. Solutions |
23 |
W 11/3 | Sec 8.5 up to and including Theorem 1 (the proof of Theorem 1 may be on exams). | 8.5: 1ab, 2bc, 3abd, 8, 21, 35, 41, 44ab,
45ab. 2.3: 12ac, 13ac. Prove your answers. Solutions |
22 |
M 11/1 | Sec 8.1 p. 519-524. Skip "antisymmetric". | 8.1: 1ae, 3bcd, 4ad, 6ac, 7adef. Ignore all references to "antisymmetric". |
21 |
F 10/29 | Sec 6.2 p. 404-405. | 6.2: 1, 23. 4.1: 22, 43. Also do these additional problems (one or more may be graded). Solutions Solutions to the additional problems |
20 |
W 10/27 | Sec 5.4 p. 363-365. Sec 6.1. |
5.4: 1, 5, 9, 15. 6.1: 5, 23, 29, 33, 34. 4.1: 19. Solutions |
19 |
M 10/25 | Sec 5.2 p. 347-350. |
5.2: 5, 6, 9, 13, 14. 5.3: 3, 4, 19, 27, 30. Solutions |
| F 10/22 | The exam will cover HWs 7- Here's a link to my 2005 midterm 2. Our midterm will be different (and cover different sections), but it might be helpful for you to try these problems anyway. I don't have solutions written up for them; if you have questions on them, you can ask me in class or my office. |
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18 |
W 10/20 | Sec 5.1: As you work on the homework problems, you can just read parts of this section that seem similar to homework. (Normally I don't recommend doing this, but this section is different.) | For all problems, show work and explain your reasoning. Giving only the final answer will not earn any credit. Sec 5.1: 5, 13, 21adegh, 33, 34, 38, 39, 43, 47, 49. For 49, use a calculator or computer. For 21 & 47, use (and do) the following problem:: (i) Prove that a|x and b|x iff lcm(a,b)|x. Solution for extra problem (i) above. Solutions |
17 |
F 10/15 | Sec 4.2. | Sec 4.2: 3, 12, 17, 23, 25. (Problem #12 might be easier if you don't read the hint!). |
16 |
W 10/13 | Sec 4.1 p. 263-271. | Sec 4.1: 5, 6, 10, 23, 41, 49, 53. |
15 |
M 10/11 | Sec 3.6 p.227-229. | Sec 3.6: 23ab, 29. |
14 |
F 10/8 | Sec 2.4 p.158-159. | Sec 2.4: 31, 32abc, For 35, use 40. For 37, use 36 (without proving 36). For 40, assume A, B are disjoint. Solutions |
13 |
W 10/6 | Sec 3.7: p. 231-235. Proof of Theorems 2 & 3 (page 234) may be on
exams. |
Sec 3.7: 1a, 5, 9, 10, 11, 14, 15. (For #11, use #5.) Sec 3.5: 32. Extra problem: (i) Prove these two statements are equivalent: (A) If a prime p divides ab, then it divides a or b. (B) If a prime p divides ab but not a, then it divides b." (ii) Use Solutions |
12 |
M 10/4 | Sec 3.5. Pages 213-214 are optional (but recommended for math majors). |
Sec 3.5: 5, 6, 13, 17, 18, 21abf, 23abf, 27, 31. For #6, prove your answer. Many of the proofs in the back of the book are missing details (e.g., #31); provide all the necessary details and justifications. Extra Problems: (i) Prove the "Linear Combination of Multiples Theorem": Let a, b, m, n, x be integers. Prove that if x|a and x|b, then x|(ma+nb). (ii) True or False: Let a, b, c be integers such that a+b+c=0. If an integer x divides a and b, then it divides c (hint: use (i) above). Solutions |
11 |
F 10/1 | Sec 3.4: p.203-205. | Sec 3.4: 9abe-h, 19c, 20, 21, 22, 23, 24, 33, 34.
(Read the explanation that's before 33 and 34.) Solutions |
10 |
W 9/29 | Sec 3.4: p.200-202. (Note: Solutions for Midterm 1 posted below.) |
Sec 2.1: 23, 26, 28-31. Sec 3.4: 1b, 4, 5-7, 8. Solutions |
9 |
M 9/27 | Sec 2.3: p. 139-146. Sec 2.1: p. 117-118 (Cartesian Products). |
Sec 2.3: 15ab, 18ab, 26a, 28, 29, 30, 31, 36, 37,
39. Solutions |
8 |
F 9/24 | Sec 2.3: p. 133-138, and Definition 12. (We will cover the rest of this section next time.) | 2.3: 1, 9, 10, 11, 16, 17, 19. For problems 16, 17, and 19, if you claim "not 1-1" or "not onto", prove your answer; if you claim "is 1-1" or "is onto", no proof necessary
here (but later we'll learn to prove "1-1" and "onto"). Solutions |
Midterm-1 |
W 9/22 | Will cover HWs 1-6 and their corresponding sections. |
|
7 |
M 9/20 | Sec 2.2. Study Table 1. ("Membership Tables" : optional.) | 2.2: 3, 15a, 19, 23, 26ac, 29, 30ab, 31, 45, 46, 49. Solutions |
6 |
F 9/17 | Sec 1.7: only read Examples 10 and 11. Sec 2.1: skip "Cartesian Products" for now; we'll cover it later. Also skip "Truth Sets". |
1.7: 3, 7, 10. 2.1: 5, 6df, 9, 13-15, 17, 19. Solutions |
5 |
W 9/15 | Sec 1.5: optional: p. 63-65 only. |
1.5: 5. 1.6: 9, 11, 16, 23, 30, 33, 35. Solutions |
4 |
M 9/13 | Sec 1.4. May skip Loops and Example 16. | 1.4: 1, 9, 10cde, 19, 21, 23, 27a-e, 28bef, 33e. Solutions |
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3 |
F 9/10 | Sec 1.3 p.34-43. (May skip "Binding Variables" and computer related topics.) | 1.3: 9, 10cde, 33, 35, 43, 45, 50, 51, 53, 59. For #51, find a different example from that in the back of the book. Solutions |
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2 |
W 9/8 | Sec 1.2; ignore "contingency". | 1.2: 5, 6, 11ce, 12abc, 17. And Sec 1.1:
20. Solutions |
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1 |
F 9/3 | Sec 1.1 pages 1-11. Skip "inverse". | 1.1: 1, 5de, 9defg, 13, 17ab, 18a-e, 23(no inverse),
26ad, 29a, 32c, 33c. Solutions |