Math 135 - Spring 2011
First Steps into Advanced Mathematics

Professor: Erika L.C. King
Email: eking@hws.edu
Office: Lansing 304
Phone: (315) 781-3355
Home Page

Office Hours: and by appointment
Class Schedule: held TTh 1:30-2:55 in Napier 202
Course Syllabus
Proof Writing and Presentation Tips
Course Grade Scale

Colloquium Website

READING/EXAM WEEK

Review Session: Thursday, May 5th 1:00pm-2:30pm in Napier 202. If you would like to present during the review session, please let me know by Wednesday at 1:30pm.

Office Hours:

• Wednesday, May 4: 1:00pm-3:00pm
• Thursday, May 5: 11:00am-NOON and 2:30pm-4:00pm
• Friday, May 6: 1:00pm-2:30pm
• Monday, May 9: NOON-1:30pm
• By appointment

In-Class Final Exam: Sunday, May 8th 1:30pm-4:30pm in Napier 202. Bring your journal! Here is a list of what I will be looking for in your journal.

Take-Home Final Exam Due: Tuesday, May 10th at NOON.

Thanks for an awesome class! Have a great summer! Keep in touch!

WEEK 15

Journal Homework for class Tuesday, May 3:

• Bowen will start class with a proof of Theorem 4.2.26. Work on finalizing your own version of the proof.
• Finish reading Section 4.3 (pages 81-83 in the text).
• Write proofs of Theorems 4.3.15, 4.3.16, Lemma 4.3.20 and Theorem 4.3.21 in your journal. I will be looking for volunteers to present all of these. There are two separate statements to show to prove Theorem 4.3.21, so I will look for two separate volunteers - one for each part. Note it is likely that we will discuss all of these Theorems but not prove all of them on Tuesday. Any remaining proofs will be available to be presented at the review session.
• Complete Problem 13 at the end of Chapter 4 (page 99).

There will be no collected homework due on Tuesday. Use your time to prepare the proofs for class on Tuesday - this is important material and the more you have prepared the more we can get to your real questions during class! Note that in class we will also be discussing exam 2 as part of our review, talking about end of the semester details, and doing evaluations.

WEEK 14

Journal Homework for class Tuesday, April 26:

• Gen did a proof of Theorem 4.2.18 for immediate successors; put a proof of the Theorem for immediate predecessors in your journal.
• Reread Section 4.2 (especially pages 77-78 in the text). Read pages 78-80 of Section 4.3. We will be pushing to finish Sections 4.2 and 4.3 in the next three classes. Please continue working to prepare this material for class. We will start class Tuesday with Brezeck's proof of 4.2.22.
• Write proofs of Lemma 4.2.25 and Theorem 4.2.26 in your journal. I will be looking for volunteers to present these.

COLLABORATIVE Collected Homework (Due Tuesday, April 26 at 5:00pm):

• Remember that although you may discuss this assignment with others, your write up should be your own.
• Complete problem 10 parts (c)-(f) at the end of chapter 4 on pages 98-9. Be sure to give some explanation for your answers in (d)-(f). Note for part (e) you need to address the parenthetical remark as well. (Note that you did parts (a) and (b) on your exam!)

Journal Homework for class Thursday, April 28:

• Work on the proof of Lemma 4.2.25. We will start class with Josh's presentation of this. Make sure you have processed the parts of the proof that Josh already presented, that is, either arrive at a point where it is clear to you, or bring specific questions.
• Read/reread pages 78-81 of Section 4.3.
• Complete the worksheet we started in class. Once you get the hang of it, it should go pretty quickly. Although it will not be collected, you should treat it as if it were. This should help us understand the reading more clearly.
• Complete Problems 4.3.11-4.3.14 in your journal. Write proofs of Theorems 4.3.8 and 4.3.15 in your journal. I will be looking for volunteers to present these as well as Theorem 4.2.26.
• Complete Problems 3 and 7 at the end of Chapter 4 (pages 96-98).

INDIVIDUAL Collected Homework (Due Friday, April 29 at 3:00pm):

• Remember that this assignment should be done on your own. (Of course, you are welcome to ask me questions!)
• Recall that your final exam will include a take-home portion. You will be working in pairs on this. Thus we need to set up partners. On a separate piece of paper than the rest of this assignment, list your top three choices of whom you would like to work with for the take-home final exam. If there is anyone in the class with whom you do not think you could work well, make a note of that too. Do not staple this to the rest of your collected assignment.
• Complete problem 12 at the end of chapter 4 on page 99. For part (a), you will probably want to do the proof in three parts, one for each inequality.

WEEK 13

Journal Homework for class Tuesday, April 19:

• Read pages 75-77 of Section 4.2 in the text.
• You should already have a proof of Theorem 4.2.14 in your journal. If not, put one there! Also write proofs of Theorems 4.2.15 and 4.2.18 in your journal. I will be looking for volunteers to present these three proofs.
• Complete Problems 9 and 11 at the end of Chapter 4 (pages 98-99).
• There will be no collected problem set due today. Work thoroughly on pages 75-77, Problems 9 and 11, and be prepared to present at least one of the three proofs.

No collected homework due on Tuesday! :-)

Field Assignment (Due Thursday, April 21 at 1:30):

1. Go to the HWS library website. Figure out how to find MathSciNet. Hint: It is a database. Write detailed directions explaining to someone how to enter the database.
2. Under author, type in one of the mathematicians we have mentioned this semester, or someone you have heard about and want to know more about. Describe who you chose and how you entered the name. Can you type first and last name? For example, try finding me (probably not as easy as the name you chose!). There are some links in the database to help you figure this out. Can you find an article I have written?
3. What were the results of your first (not of me!) search (how many matches, what sorts of articles, etc.). You have several links available to you for each article or book. What do they allow you to do and why are some highlighted (i.e. able to be clicked on) and some not?
4. Now try searching for a topic - perhaps one we have been spending a lot of time on or one that you read about in Letters to a Young Mathematician. What topic did you choose and what were your results? Choose an article from your results that looks interesting and print an abstract for it. Can you find out what other articles the author (or authors) of your article have written without doing a whole new search? How?
5. Can you figure out how many entries there are for 2010? If so, explain how and reveal your findings!
6. What are author citations? Can you find out how many author citations my graduate advisor, Michael D. Plummer, has? What is the list MathSciNet gives you (i.e. what does it mean)?
7. Is there anything particularly interesting, surprising or exciting you discover? Perhaps something about the way the database is structured as compared to other databases you have used, or some information you find in the database, or something else?

Journal Homework for class Thursday, April 21:

• Finish reading Section 4.2 (pages 76-78 in the text).
• In reference to the remark at the top of page 76, write down in your journal the theorem about least and minimal elements that is analogous to Theorem 4.2.14. Then prove it. Then do the same for Theorem 4.2.15.
• Write proofs of Theorem 4.2.22 and of Lemma 4.2.25 in your journal. (A proof of Theorem 4.2.18 should already be there.)
• Review your work on Problems 9 and 11 at the end of Chapter 4 (pages 98-99). I may be calling on people to discuss the answers to these.
• I will be looking for volunteers to present 4.2.18, 4.2.22 and 4.2.25.

INDIVIDUAL Collected Homework (Due Friday, April 22 at 3:00pm):

• Remember that this assignment should be done on your own. (Of course, you are welcome to ask me questions!)
• Complete problem 1 at the end of chapter 4 on page 96. You can just state your conjecture and prove it, since we have already talked about what this should be in class (i.e. you don't have to show me any examples).

WEEK 12

Remember our second exam is Thursday (April 14th) 7PM-9PM in Gulick 206B. Remember to bring your journal to be turned in!

Due to committee meetings, I will need to start my office hours this Wednesday a little later. They will be 2:30-4:00. It is possible that I will be done earlier and able to start my office hours earlier, but I will not know until then. PLEASE let me know if you have questions and cannot make those hours.

Journal Homework for class Tuesday, April 12:

• Read pages 69-75 of Section 4.2 in the text. Also be sure to work through Examples 4.2.2 and 4.2.10 in your journal. They are really exercises!
• Write proofs of Theorems 4.2.6 and 4.2.14 in your journal. I will be looking for a volunteer to present a proof part 4 of 4.1.10 and of 4.2.6.
• Catch up on journal work in preparation for your exam. On Tuesday I will post a list of the problems that I expect to see in your journal. Review the journal guidelines in the syllabus to check that you have followed them.

COLLABORATIVE Collected Homework (Due Tuesday, April 12 at 5:00pm):

• Remember that although you may discuss this assignment with others, your write up should be your own.
• Complete proofs for Problem 4.1.10 part 6 (pages 68-69). Use the hint, and make it clear where in your proofs you are using it. The hint essentially gives you two Lemmas you may use.

Journal Homework for class Thursday, April 14:

• Read/reread pages 75-76 of Section 4.2 in the text.
• Write/rewrite proofs of Theorems 4.2.6, 4.2.14 and 4.2.15 in your journal. I will be looking for volunteers to present proofs of these.
• Prepare for your exam! Catch up on journal work. (Check out this list for a condensed version of what should be in your journal.) Bring questions to class.
• Depending on how quickly we work through the newer material and how many general questions people have, we may end class early and have office hours for people to ask questions individually on the induction questions for the last bit.

Since we have an exam on Thursday night, there will be NO collected assignment due on Friday.

WEEK 11

Due to a committee meeting, I will need to start my office hours this Wednesday a little later. They will be 3:00-4:30. It is possible that I will be done earlier and able to start my office hours earlier, but I will not know until then. Please let me know if you have questions and cannot make those hours.

Journal Homework for class Tuesday, April 5:

• Work a proof to problem 3.3.2 in your journal. Also figure out what is going on in Example 3.3.4 and write why it is a "proof" and not a proof. :-)
• Reread Section 4.1 (pages 65-69). As has been said on previous journal assignments, if you have not finished the worksheet from class on March 22nd, finish it now. If you have trouble with any of the questions, try to write out what confuses you in words.
• Read pages 69-73 of Section 4.2 in the text. Be sure to work through example 4.2.2 in your journal. It is really an exercise!
• Write proofs of parts 2, 3 and 4 of Problem 4.1.10 in your journal (note that each of these requires essentially four smaller proofs.), as well as a proof of 4.2.6.
• I will be looking for a volunteer to present a proof of one the parts of 4.1.10.

COLLABORATIVE Collected Homework (Due Tuesday, April 5 at 5:00pm):

• Remember that although you may discuss this assignment with others, your write up should be your own.
• Using induction, prove that the sum of the cubes of any three consecutive natural numbers is a multiple of 9. (Hint: You can represent three consecutive numbers by n, n+1 and n+2.)

Journal Homework for class Thursday, April 7:

• Review the question with the rabbits (Example 3.3.4). Does it make sense? If you are confused, try to carefully articulate a question or questions. We will address these at the beginning of the next class.
• Reread Section 4.1 (pages 65-69). As has been said on previous journal assignments, if you have not finished the worksheet from class on March 22nd, finish it now. If you have trouble with any of the questions, try to write out what confuses you in words.
• Read pages 69-73 of Section 4.2 in the text. Be sure to work through example 4.2.2 in your journal. It is really an exercise!
• Write proofs of parts 2, 3 and 4 of Problem 4.1.10 in your journal (note that each of these requires essentially four smaller proofs.), as well as a proof of 4.2.6.
• I will be looking for a volunteer to present a proof of one the parts of 4.1.10 and of 4.2.6.

INDIVIDUAL Collected Homework (Due Friday, April 8 at 3:00pm):

• Remember that this assignment should be done on your own. (Of course, you are welcome to ask me questions!)
• Prove Theorem 3.3.3 on page 63.

WEEK 10

Due to various committee meetings and the colloquium, I will need to change my office hours this Friday. They will be NOON-1:30. Please let me know if you have questions and cannot make those hours.

Journal Homework for class Tuesday, March 29:

• Prepare your proof with your group. It is a good idea to review the Proof Writing and Presentation Tips website. Also investigate the other groups' problems so that you are at least familiar with the questions and have some ideas about how their proofs will work. The problems that will be presented are 3.2.3-3.2.6 on page 62.
• Reread Section 4.1 (pages 65-69). If you did not finish the worksheet in class on Thursday, try to do so now.

COLLABORATIVE Collected Homework (Due Tuesday, March 29 at 5:00pm):

• Remember that although you may discuss this assignment with others, your write up should be your own.
• Complete Problem 8(c) at the end of Chapter 2, page 55. Be careful! Dealing with complements and subsets is different than dealing with complements and elements. If you are confused by the notation, you can add parentheses around everything before the subset symbol. Use this handout to do your work.

Field Assignment (Due Thursday, March 31 at 1:30pm):

• Read the fourth chapter of Letters to a Young Mathematician, Hasn't It All Been Done? (pages 33-44).
• Write a short essay on this reading (2-3 pages). Your essay should include your reactions to this chapter. This chapter continues some threads that we have been discussing in class. Is there anything that surprises you? Why do we do mathematics? What is the importance of an answer or theorem? How much is out there? How long does it take to solve a conjecture? Do a little research on the internet to find out some more details about Fermat's Last Theorem and the journey to discovering a publishable proof.
• Be prepared to discuss this chapter in class on Thursday, March 31st.

Journal Homework for class Thursday, March 31:

• Think about the proof of Problem 3.2.6 whether you will be presenting it on Thursday or not. If you did present in class on Tuesday, jot down thoughts about your group members so that you are able to fill out your group evaluation form.
• Think carefully about induction and why it works. Do you have questions? Bring them to class.
• Read Section 3.3 in the text (pages 62-63).
• Before doing problem 3.3.2 try this one.
• Work a proof to problem 3.3.2 in your journal. Also figure out what is going on in Example 3.3.4 and write why it is a "proof" and not a proof. :-)
• If you did not finish the worksheet from class last Thursday, keep working on it. If you have trouble with any of the questions, try to write out what confuses you in words.

INDIVIDUAL Collected Homework (Due Friday, April 1 at 3:00pm):

• Remember that this assignment should be done on your own. (Of course, you are welcome to ask me questions!)
• Print out and complete this worksheet. Note that there are no proofs in this assignment!

WEEK 9

Have a great spring break!!!

Due to various committee meetings and the colloquium, I will need to change my office hours this Friday. They will be NOON-1:30. Please let me know if you have questions and cannot make those hours.

Journal Homework for class Tuesday, March 22:

• Read/reread pages 57-61 of Chapter 3 very carefully. Then try to tackle Problem 3.2.2 on page 62. If you can, also try any one of the next four (or all of them!). We will do 3.2.2 as a class and then we will be presenting the other four in groups in another class. You will have some time to work in groups on these this week. The more you have worked with them/become familiar with them before class, the better we will be at tackling the real questions during class.
• Start reading Chapter 4, especially Section 4.1 (pages 65-69). There will be groupwork on these concepts as well.
• Complete Exercise 8(a) at the end of Chapter 2 (page 55) in your journal. First give an example that illustrates the statement, then prove that the statement is true in general.

COLLABORATIVE Collected Homework (Due Tuesday, March 22 at 5:00pm):

• Remember that although you may discuss this assignment with others, your write up should be your own.
• Draw a Venn Diagram representing the symmetric difference of A and B.
• Prove Theorem 2.4.11 part 4 (page 50) TWO ways: first using an element argument, then using algebraic identities.

Journal Homework for class Thursday, March 24:

• Start working on your group's induction proof. At the very least you should figure out what P_k is and do a base case, so that you are prepared to discuss it in your group on Thursday.
• Reread Section 4.1 (pages 65-69). If you did not finish the worksheet in class on Tuesday, try to do so now. We will begin class by working on these problems a little bit more and then discussing them. Start reading Section 4.2 (pages 69-70).
• Write proofs of parts 2, 3 and 4 of Problem 4.1.10 in your journal. I will be looking for a volunteer to present a proof of at least one of these parts. (Note that each of these requires essentially four smaller proofs.)

INDIVIDUAL Collected Homework (Due Friday, March 25 at 3:00pm):

• Remember that this assignment should be done on your own. (Of course, you are welcome to ask me questions!) Here is a proof to keep the "old" ideas fresh!
• Complete Problem 6 at the end of Chapter 2, pages 54-55. Note that there are two separate statements to prove or disprove in part (a). Also note that there is a typo. It should say that "In Theorem 2.4.11 you proved that set difference does NOT distribute over union and intersection."

WEEK 8

Journal Homework for class Tuesday, March 8:

• Groups 1-3: Jot down some notes about working with your groupmates that you can use when filling out your group evaluation form on Tuesday. All Groups: Review your work on the proof of Theorem 2.4.9 part 2 to make sure you are prepared to present (if you are in Group 4) or evaluate the presentation of that proof.
• Reread Section 2.4. In addition to the exercises in the reading, make sure you have completed Problem 2.4.8, and also write TWO proofs of Theorem 2.4.11 part 1 (see note in book!). Theorem 2.4.11 deals with set differences. Remember the power of definitions!!! Then return to Section 2.5 and power sets. Work more on Problem 2.5.7. Reread Section 2.6 and review your work on Problem 2.6.1. All of this (except for the proof of 2.4.11 part 1) has been assigned previously. Hopefully you already have some work for each and can bring new insight with the work we have done since then.
• I will be looking for a volunteer to present an element argument proof of 2.4.11 part 1, and a volunteer to present a proof of 2.4.11 part 1 using algebraic identities.

COLLABORATIVE Collected Homework (Due Tuesday, March 8 at 5:00pm):

• Remember that although you may discuss this assignment with others, your write up should be your own.
• Prove Theorem 2.4.6 part ONE (page 49) using an element argument. Use this handout to do your work. Note that I have revised this so that you only need to show one example. You should not write in the "final draft" section of the handout until you are finished asking me questions.

Journal Homework for class Thursday, March 10:

• Complete two proofs for the second and third parts of Theorem 2.4.11 (one that is an element argument and one that uses identities we have proved). Then return to section 2.5 and power sets. Work more on Problem 2.5.7. Reread Section 2.6 and review your work on Problem 2.6.1.
• Start reading Chapter 3!
• I will be looking for a volunteer to present BOTH 2.5.5 part 2 and 2.5.6 (these go together!).

INDIVIDUAL Collected Homework (Due Friday, March 4 at 3:00pm):

• Remember that this assignment should be done on your own. (Of course, you are welcome to ask me questions!) Here is a proof to keep the "old" ideas fresh!
• Prove: deMorgan's FIRST law (Theorem 2.4.9 part 1, pages 49-50) using an element argument. Use this handout to do your work.

WEEK 7

Field Assignment (Due Tuesday, March 1 at 1:30):

• Read the third chapter of Letters to a Young Mathematician, The Breadth of Mathematics (pages 18-32).
• Write a short essay on this reading (2-3 pages). Your essay should include your reactions to this chapter. Is there anything that surprises you? What is mathematics? Is it scientific? Is it an art? What was Hilbert upset about? Are you too? Is mathematics discovered or is it invented? Defend your opinion. You don't have to agree with Dr. Stewart on these questions. And for another angle, what do you think it should mean to address the "quantitative goal" (one of your requirements for graduation)?
• Be prepared to discuss this chapter in class on Tuesday, March 1st.

Journal Homework for class Tuesday, March 1:

• Read/reread Sections 2.5-2.6 (pages 50-53).
• In addition to the exercises in the reading, also review your proof of 2.5.4, write a proof of Theorem 2.5.5 and work Problems 2.5.7 and 2.6.1.
• Bowen will finish presenting Theorem 2.5.4 and I will be looking for a volunteer to present BOTH 2.5.5 part 2 and 2.5.6 (these go together!).
• For a portion of class we will be working in groups on proofs of theorems to be presented in the next couple of classes. Each group will focus on one of the following: Theorem 2.4.5 part 1, Theorem 2.4.5 part 2, Theorem 2.4.6 part 2 and 2.4.9 part 2. Take a look at these Theorems and work on ideas to prove them.

Since we just had an exam on Thursday evening, there will be no collected homework due on Tuesday. Concentrate on preparing your field assignment and your journal work.

Journal Homework for class Thursday, March 3:

• Reread Section 2.4 and prepare for your group presentations. These proofs are similar to proofs we did earlier in this section, but now our indexing set is not necessarily finite. Thus we will need ideas from earlier proofs as well as definition 2.3.13.
• In addition to the exercises in the reading, also complete Problem 2.4.8. Note that this problem will be especially helpful to the group presenting 2.4.9.
• Take a look at what the other groups will be presenting as well so that you are prepared to evaluate their work. The problems that will be presented are: Theorem 2.4.5 part 1, Theorem 2.4.5 part 2, Theorem 2.4.6 part 2 and 2.4.9 part 2.

INDIVIDUAL Collected Homework (Due Friday, March 4 at 3:00pm):

• Remember that this assignment should be done on your own. (Of course, you are welcome to ask me questions!) Here is a proof to keep the "old" ideas fresh!
• Prove that 3 divides x if and only if x can be written as a sum of three consecutive integers.

WEEK 6

Remember our first exam is Thursday (February 24th) 7PM-9PM in Napier 101. Remember to bring your journal to be turned in!

Journal Homework for class Tuesday, February 22:

• Reread pages 47-48 of Section 2.4 in the text. Remember that I will be looking for volunteers for a proof of 2.4.4.
• We are going to skip around a little. Read Section 2.5 (pages 50-52). In addition to the exercises, work on a proof of Theorem 2.5.4 in your journal.
• Problem 4 end of Chapter 2 (page 54). For parts (b) and (c), assume r is an element of the POSITIVE rationals instead of the rationals in general (i.e. there is a typo/omission here).
• Catch up on journal work in preparation for your exam. On Tuesday I will post a list of the problems that I expect to see in your journal. Review the journal guidelines in the syllabus to check that you have followed them.

COLLABORATIVE Collected Homework (Due Tuesday, February 22 at 5:00pm):

• Remember that although you may discuss this assignment with others, your write up should be your own.
• Problem 5 end of Chapter 2 (page 54). Parts (a) and (b) will be done on separate paper. For part (b), use this handout. This is like what you needed to do for last Tuesday's assignment, and we will use this handout for future proofs as well (thus spaces for you to fill in!). The idea is to encourage the process behind writing a proof and not to expect that the first things you write will be the final proof. At the top of the page you can fill in "Proof of Chapter 2 Problem 5". And so on. Note for part (a) that you have two different statements and for each you want to give the two step process for all three proof methods. So that is six different parts! Note for part (b) you do not have to prove the two parts the same way. It might help your intuition to draw a Venn diagram in your scratch work (which you could include as part of (ii) in your handout) for this problem.

Journal Homework for class Thursday, February 24:

• Finish the power set groupwork sheet from class. Complete as much of it as you are able and bring questions and solutions to class. In other words, you should have either a solution or a question for each exercise.
• Reread Section 2.5 (pages 51-52). Write a proof or revise your proof of Theorem 2.5.4. I will be looking for a volunteer to present it.
• Prepare for your exam! Catch up on journal work. (Check out this list for a condensed version of what should be in your journal.) Work on the problem on the back of the review sheet. Bring questions to class.

Since we have an exam on Thursday night, there will be NO collected assignment due on Friday.

WEEK 5

Journal Homework for class Tuesday, February 15:

• Reread/read Sections 2.2 and 2.3 in the text (pages 42-47).
• I will be looking for a volunteer to present a proof of Exercise 2.3.8.
• Problem 1 at the end of Chapter 2 (page 53). Think carefully of your answer and use full sentences with good vocabulary words to respond.

COLLABORATIVE Collected Homework (Due Tuesday, February 15 at 5:00pm):

• Remember that although you may discuss this assignment with others, your write up should be your own.
• Complete handout.

Journal Homework for class Thursday, February 17:

• Finish Reading Section 2.3 in the text (pages 44-47). Also read page 47-48 of Section 2.4.
• Start working on a proof of Theorem 2.4.5 on page 49.
• I will be looking for a volunteer to present a proof of Exercise 2.3.8, as well as a volunteer for Exercise 2.4.1 (Note that this one should involve using different colors and step by step explanations, not just a final product), and a volunteer to prove Exercise 2.4.4.

INDIVIDUAL Collected Homework (Due Friday, February 18 at 3:00pm):

• Remember that this assignment should be done on your own. (Of course, you are welcome to ask me questions!)
• Complete Problems 2 and 3 at the end of Chapter 2, pages 53-54. Use interval notation where appropriate. Note for Problem 2 you have three different pairs of sets and you are asked to figure out two different things for each pair. Be sure to label your Venn diagrams for Problem 3 clearly.

WEEK 4

Due to the department colloquium on Wednesday, my Wednesday office hours will be shortened to 2:00-3:30. Please let me know if you have questions.

Journal Homework for class Tuesday, February 8:

• Reread Sections 1.9-1.15 in the text (this covers pages 28-36). Be prepared to evaluate other groups' presentations as well as to be able to coherently discuss your own.
• Prepare your group presentations for class. Let me know if you have any questions. I highly recommend that at least one member of your group meet with me on Monday about your proof and presentation.

COLLABORATIVE Collected Homework (Due Tuesday, February 8 at 5:00pm):

• Exercise 1.8.11 on page 27. Use the hint!
• Bonus: What is the statement in exercise 1.8.11?
• Negate the following statements. It may be helpful to rewrite the statement in an equivalent form before negating it. Make your statements positive where possible. Think about assumed quantifiers and whether or not you should explicitly state them.
  1. Everyone may be seated.
  2. Some people eat at the Water Street Cafe on Sundays.
  3. If f has a local maximum or minimum at c, then c is a critical number of f.
  4. Some children do not like to play in the snow.
  5. There exists a color k, such that for all textbooks b, if b is k, then Philipp is happy and Tyler is sad.
  6. If there is a largest natural number, then it is 1.
  7. If x is divisible by 6, then x is divisible by 3.
• Remember that although you may discuss this assignment with others, your write up should be your own.

Journal Homework for class Thursday, February 10:

• Review Sections 1.10, 1.12, 1.13 and your notes from the group presentations on Tuesday. Make sure you are clear about how to organize proofs of uniqueness theorems, direct proofs, and proofs by contraposition.
• Group 4 review your presentation which you will give on Thursday. Groups 1-3 review their section and be prepared to ask questions.
• After Group 4's presentation, we will have the tools of proof techniques and we will get to talk about sets! Read pages 39-44, sections 2.1, 2.2 and the beginning of section 2.3 in your text. In addition to working through the exercises and examples in the reading, complete a proof of Theorem 2.2.2. I will be looking for volunteers to present proofs of Theorem 2.2.2 and Exercise 2.2.4. Remember to use your definitions!
• Complete problem 9 at the end of chapter 1 (page 38).

INDIVIDUAL Collected Homework (Due Friday, February 11 at 3:00pm):

• Remember that this assignment should be done on your own. (Of course, you are welcome to ask me questions!)
• Use direct proof to prove: Let a, b, c, and d be integers. If a divides b and c divides d, then ac divides bd.
• Review Proof Writing and Presentation Tips to remind yourself of the details you should be considering as you write your proof.
• Complete problem 8 at the end of chapter 1 (page 38).

WEEK 3

Journal Homework for class Tuesday, February 1:

• In your journal, write a "best" proof for the problem on the worksheet in class Thursday.
• Read/reread Sections 1.5-1.8 in the text (this covers pages 15-28). Remember that an assignment to read the text includes working through all exercises and examples in the reading as well as making a list of new vocabulary so that you are ready to ask questions about, discuss and present the material.

COLLABORATIVE Collected Homework (Due Tuesday, February 1 at 5:00pm):

• Problem 6 at the end of Chapter 1 (pages 37-38).
• Rewrite the following sentences as if-then statements. (Note that you don't have to understand what they mean to do this!)
  1. The fish bite only when the moon is full.
  2. A topological space is disconnected whenever there is a non-empty proper subset of the space that is both open and closed.
  3. William Shakespeare had a high-speed internet connection is sufficient for a pig to fly.
  4. Two functions are one-to-one only if their composition is one-to-one.
  5. A time of 3 minutes, 48 seconds or less is necessary to qualify for the Olympic team.
  6. All mathematics students visit Lansing Hall.
• Write a statement that cannot be written as an if-then statement.
• Remember that although you may discuss this assignment with others, your write up should be your own.

Field Assignment (Due Thursday, February 3 at 1:30):

• Read the first chapter of Letters to a Young Mathematician (pages 1-10). This book is on reserve in the library. It is written as letters from an experienced mathematician to someone who, like you, has just begun a path toward studying higher level mathematics. I don't agree with everything Ian Stewart says, but I agree with most of it and think he has some really interesting points.
• Write a short essay on this reading (2-3 pages). Your essay should include your reactions to this chapter. Is there anything that surprises you? Do you think that mathematics is beautiful? Why or why not? If so, is there a particular mathematical concept that strikes you as beautiful? The author mentions several mathematical ideas at the end of the chapter, which he does not go into detail about. Which of these sounds the most intriguing to you? Feel free to use the library or internet to explore these ideas.
• Be prepared to discuss this chapter in class on Thursday.

Journal Homework for class Thursday, February 3:

• Read Sections 1.8-1.15 in the text (this covers pages 27-36). Remember that an assignment to read the text includes working through all exercises and examples in the reading as well as making a list of new vocabulary so that you are ready to ask questions about, discuss and present the material.
• We will have group presentations on the latter sections of this chapter next Tuesday. You will have an opportunity to work with your group on Thursday, so the more work you have done on this before class, the more you will have to contribute.

INDIVIDUAL Collected Homework (Due Friday, February 4 at 3:00pm):

• Remember that this assignment should be done on your own. (Of course, you are welcome to ask me questions!)
• Construct a truth table for: [(A implies not B) and (B or C)]. (Note this is one statement. Please let me know if you have trouble interpreting what I mean here.)
• Consider the statement: If p is a prime that divides ab, then p divides a or p divides b. Rephrase this statement in TWO equivalent ways using ideas we have discussed.
• Exercise 1.8.5 parts 2, 4, 6 and Exercise 1.8.10 parts 2 and 3 (pages 26-7). For each of these statements, assign variables to each substatement as in Example 1.8.6 and show the original statement and then the negated statement in symbolic form. Then write out the negated statement in words. Note that you should break down the statement as much as possible so that you have variables that represent each simple (i.e. not compound) predicate.

WEEK 2

Remember to keep your appointments!!!

Note that due to individual appointments, my open office hours on Wednesday are shortened this week. The revised hours for this week are: Monday 1:00-3:00, Wednesday 2:00-3:30pm, and Friday 1:30-3:00. If you cannot make these times and need to see me, please make an appointment.

Journal Homework for class Tuesday, January 25:

• Read the syllabus. Note there is a link to it at the top of our class website. The correct dates are in this version of the syllabus.
• Check that you have no conflict with the dates and times of the midterm exams. Speak with me ASAP if you do have conflicts.
• Read the Proof Writing and Presentation Tips website. This may be revised from time to time. Use this as a reference when you are preparing your homework and presentations for class. Some of this will not mean anything to you yet, but it will give you an idea of what I will be looking for as I grade your work. Note that there is a link to the Proof Writing and Presentation Tips website at the top of this website as well.
• Read A Note to the Student, Chapter 0: Introduction-An Essay, and Sections 1.1-1.4 in the text. (This covers pages xiii-xiv, and 1-15.) Be sure to finish the "Thought experiment" we started in class, i.e. think a bit more about whether you think each statement (not just the six we focused on) is true or false, experiment with proving your conviction, and divide your convictions into the four catagories listed (a-d). Remember that an assignment to read the text includes working through all exercises and examples in the reading as well as making a list of new vocabulary (like free variable, predicate, etc.) so that you are ready to ask questions about, discuss and present the material.
• Work problem 1 at the end of Chapter 1 (page 37). Note there are two parts for each of (a), (b) and (c).
• Additional exercise: Describe what you would need to do to show that a statement "For all z, A(z)" is false. Give an example (it need not be mathematical).
• Remember to follow the journal guidelines in the syllabus. Work from the exercises and a vocabulary list should be in your journal along with the two additional problems/questions noted above, and any notes you have from the reading.

COLLABORATIVE Collected Homework (Due Tuesday, January 25 at 5:00pm):

• This is a short one! :-)
• Consider the statement: If x is a positive real number, then ln(x) is positive. Is this true or false? Prove your assertion. Although we have only introduced the idea of proof, try writing a formal proof given what we discussed in class. (I will not grade these as formally as I will grade later proofs.)
• Remember that although you may discuss this assignment with others, your write up should be your own.

Journal Homework for class Thursday, January 27:

• Read Sections 1.5-1.7 in the text (this covers pages 15-24). Remember that an assignment to read the text includes working through all exercises and examples in the reading as well as making a list of new vocabulary so that you are ready to ask questions about, discuss and present the material.

INDIVIDUAL Collected Homework (Due Friday, January 28 at 3:00pm):

• Remember that this assignment should be done on your own. (Of course, you are welcome to ask me questions!)
• Note that Section 1.5 will be very helpful for this assignment.
• Problem 2 at the end of chapter 1 (page 37). Explain your answer carefully. Provide an example. Hint: Use the note and don't necessarily go with your first instinct.
• Problem 3 at the end of chapter 1 (page 37). Explain your answers; do not just write "True" or "False".
• Consider the following. Suppose Jill said, "If I do not get my car fixed, I will miss my job interview." Later you come to know that Jill's car was repaired but that she missed her job interview. Was Jill's statement true or false? Explain.

WEEK 1

Welcome to First Steps into Advanced Mathematics!!!

Collected Homework (Due Friday, January 21st at 3:00pm):

• Write an autobiographical essay or poem as assigned on the syllabus.

Last modified: Tuesday 3 May 15:40:30 EDT 2011