# MATH 135 - Spring 2017 First Steps into Advanced Mathematics

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

Office Hours: M: 1:30-3:00pm, T: 1:00-2:30pm, W: 10:30-11:30am, Th: 4:00-5:15pm, and by appointment
Class Schedule: held TTh 10:20-11:45am in Gulick 223
Course Syllabus
Proof Writing and Presentation Tips

### READING/EXAM WEEK: May 3 - May 9

One Last Journal Homework to Review Chapter 15:

• Read Section 15.5 (pages 450-453).
• In your journal, complete Check Yourself problems at the end of Section 15.5 on page 453 of your textbook.
• Ponder the Ping Pong Ball Conundrum. Discuss it with friends and family. Isn't infinity wild!?!?!?

Review Session: Friday, May 5th from 2:00pm until 3:00pm in Napier 201. Attendance is optional and you are welcome to come for any portion of the review session. Bring lots of questions.

Office Hours:

• Wednesday, May 3rd: 1:30pm-3:00pm
• Thursday, May 4th: 2:00pm-3:30pm
• Friday, May 5th: 3:30pm-4:30pm
• Monday, May 8th: 11:30am-1:30pm
• By appointment

In-Class Final Exam: The in-class final exam is on Sunday, May 7th from 1:30pm until 4:30pm in Gulick 223. Remember to bring your journals! (Check out this list for a condensed version of what should be in your journal.)

Take-Home Final Exam Due: The take-home final exam is due Sunday, May 7 at 1:30pm, though I am allowing you to have a free extension until 3:00pm on Monday, May 8. You must turn in the take-home final in person. Everyone will need to fill out a partner evaluation after you have completed the take-home final. You may do this in my office any time after you turn in your take-home final, or after you complete the in-class final. This evaluation is a requirement for the course.

Thanks for an great class! Have a great winter break! Keep in touch!

### WEEK 15: May 1 - May 2

There will be no quiz this week!

Partners for the Take-Home Portion of the Final Exam (Due MONday, May 1 at 3:00pm):

• Since we have 13 people in the class, twelve of you will be working in pairs and one of you will be working individually on the take home exam.
• On a piece of paper, list your top three choices for whom you would like to work with on the final exam. If one of your choices is to work by yourself, please write a short paragraph explaining why you wish to work individually, and why that would be particularly beneficial for you.
• Turn in your paper to me by Monday at 3pm.

Note that BOTH GROUPS get to do an extra rewrite for this Tuesday!

Journal Homework for class Tuesday, May 2:

• Review our work from class on Thursday and make any revisions to your journal you think necessary. In particular add the answer to your question for that Thursday or plan to come to an office hour if you haven't figured the answer out yet!
• Reread/Read Sections 15.2-15.4 in the text. This covers pages 429-450 in our text. Take notes on your thoughts about the answers to the questions in Section 15.4 we have not yet addressed. You can put these in your class notes, they need not be in your journal. Be ready to articulate your thoughts to the class. This is really wild and really cool stuff!
• In your journal, complete problems 3 and 9 in Section 15.8 on pages 455-456 of your textbook.
• In your journal, complete problem 11 from Section 1.3 (Problems) on page 14 of your Relations handout.
• In your journal, complete problems 16 and 22 on page 114 of your textbook. This should help with the task on the next line.
• Keep reviewing for the final exam!
• Remember to include a question you have about the reading (or something else relevant to class) in your journal.

REWRITES (Due TUESday, May 2 at 2:30pm):

• Rewrites should always be done on your own. Of course, you are welcome to ask me questions, but do not discuss them with others or use outside resources (anything other than YOUR notes and YOUR book and YOUR brain and me is considered "outside").
• Rewrites are optional, but highly encouraged!
• This TUESday BOTH GROUPS!!! may submit a rewrite. That IS you!
• You may rewrite one PROOF from one previous assignment that I have graded. (Note that this does include problems for which you had to do counterexamples, as you had to justify them with a proof.) Read my comments carefully and ask me if you have questions. Staple your new work to the original work that you turned in. Do NOT staple this to the new assignment above. You can earn back all lost points on these questions! Note that since this is a rewrite/bonus, I will expect higher standards than I did when I first collected it. Be sure to read it outloud to see if it sounds right and check the Proof Writing and Presentation Tips website for ideas about how to make your proof GREAT!
• Do not write on your previous submission to revise or correct it! It should appear as it did when I returned it to you.

### WEEK 14: April 24 - April 28

Quiz 11 will take place on Tuesday at the beginning of class. It will cover Journal assignments due Thursday, April 20th and Tuesday, April 25th. Please let me know if you have any questions with your journal assignments and come prepared!

Journal Homework for class Tuesday, April 25:

• Review our work from class on Thursday and make any revisions to your journal you think necessary. In particular add the answer to your question for that Thursday or plan to come to an office hour if you haven't figured the answer out yet!
• Review the work that we did on problem 8 from Section 1.3 (Problems) on page 14 of your Relations handout; complete any parts of Problem 8 you did not finish in class.
• Reread the Bounded Sets subsection of the Relations handout (this covers pages 11-12).
• UNIQUENESS: Complete this for your journal. Assume that $4x^3-53$ has a real root. Prove that it has only one. (This is a uniqueness proof!)
• Presentation Opportunities: I will be looking for a volunteer to present You Try It 1.37 on page 12. If you want to present this question, you should come to office hours to discuss it, or email me a copy of your work at least 24 hours in advance so that we have time to correspond. Be sure that you come to office hours with your presentation at least outlined and as much of the details you need as possible! Note that these may be the last opportunity during the regular class sessions to present. I will offer additional opportunities during a review session over Reading Period for those who have not presented, but you should be ready to present this Tuesday...there may be some competition for the spots this time!
• Read Sections 15.1 and 15.2 in the text to prepare for our play reading! This covers pages 429-445 in our text. Practice your parts and speaking clearly. Forget which part you have? Check the list here. This is a very silly play, but it contains some really deep and beautiful mathematics.
• REVIEW: In your journal, complete problems 4 and 11 on pages 113-114 of your textbook.
• Remember to include a question you have about the reading (or something else relevant to class) in your journal.

Journal Homework for class Thursday, April 27:

• REMEMBER to bring a laptop or other devise on which you could complete evaluations!
• Review our work in Tuesday's class. Make any revisions to your journal you think necessary. In particular add the answer to your question for Tuesday or plan to come to an office hour if you haven't figured the answer out yet!
• In your journal, complete You Try It 1.37 on page 12 of your Relations handout. Hint: Use the Lemma Mathias proved!
• LAST CHANCE! Presentation Opportunity: I will be looking for a volunteer to present You Try It 1.37 on page 12. If you want to present this question, you should come to office hours to discuss it, or email me a copy of your work in advance so that we have time to correspond. Be sure that you come to office hours with your presentation at least outlined and as much of the details you need as possible! Note that this is the last opportunity during the regular class sessions to present!
• Note you should prepare the presentation problems even if you have already presented! You can earn bonus points for doing more than one presentation!
• Reread Section 15.2 in the text. This covers pages 429-445 in our text. Then read Section 15.3 (page 445) and start reading Section 15.4 (pages 447-450). Take notes on your thoughts about the answers to the questions in Section 15.4. You can put these in your class notes, they need not be in your journal. Be ready to articulate your thoughts to the class. This is really wild and really cool stuff!
• In your journal, complete problem 4 from Section 15.8 on page 455 of your textbook.
• In your journal, complete problems 12, 15 and 24 on pages 113-114 of your textbook. This should help with the task on the next line.
• Start reviewing for the final exam!
• Remember to include a question you have about the reading (or something else relevant to class) in your journal.

Collected Homework (Due Friday, April 28 at 2:00pm):

• Remember that this assignment should be done on your own. (Of course, you are welcome to ask me questions!) DO NOT USE ONLINE RESOURCES!!!
• Get out your Strategies for Problem Sets and Proofs laminated handout. Use it as a guide as you solve the following questions.
• Remember to pay careful attention to your base cases in your induction proofs!
• Question 1: Complete Problem 9 from Section 1.3 (Problems) on page 14 of your Relations handout.
• Question 2: Complete Problem 10 from Section 1.3 (Problems) on page 14 of your Relations handout. For part (a), you will want to do the proof in three parts, one for each inequality.
• Submit a rewrite! Note below - both groups can!
• Before turning in your work, remember to review the points on the Proof Writing and Presentation Tips website to help make your proof great! Especially review tips 4, 5, 6, 11, and 12!

REWRITES (Due Friday, April 28 at 2:00pm):

• Rewrites should always be done on your own. Of course, you are welcome to ask me questions, but do not discuss them with others or use outside resources (anything other than YOUR notes and YOUR book and YOUR brain and me is considered "outside").
• Rewrites are optional, but highly encouraged!
• This Friday BOTH GROUPS!!! may submit a rewrite. That IS you!
• You may rewrite one PROOF from one previous assignment that I have graded. (Note that this does include problems for which you had to do counterexamples, as you had to justify them with a proof.) Read my comments carefully and ask me if you have questions. Staple your new work to the original work that you turned in. Do NOT staple this to the new assignment above. You can earn back all lost points on these questions! Note that since this is a rewrite/bonus, I will expect higher standards than I did when I first collected it. Be sure to read it outloud to see if it sounds right and check the Proof Writing and Presentation Tips website for ideas about how to make your proof GREAT!
• Do not write on your previous submission to revise or correct it! It should appear as it did when I returned it to you.

### WEEK 13: April 17 - April 21

Quiz 10 will take place on Tuesday at the beginning of class. It will cover Journal assignments due Thursday, April 13th and Tuesday, April 18th. Please let me know if you have any questions with your journal assignments and come prepared!

Journal Homework for class Tuesday, April 18:

• Review our work from class on Thursday and make any revisions to your journal you think necessary. In particular add the answer to your question for that Thursday or plan to come to an office hour if you haven't figured the answer out yet!
• Carefully reread the Maximal Elements, Greatest Elements, and Successors and Predecessors subsections of the Relations handout (this covers pages 9-10). Then read the Bounded Sets subsection of the Relations handout (this covers pages 11-12).
• Put "You Try It" problems 1.25 (the second lattice only since we did the first one in class), 1.26, 1.27, 1.29, 1.30, 1.31 and 1.32 in your journal.
• Presentation Opportunity: I will be looking for a volunteer to present You Try It 1.27 on page 10. If you want to present this question, it is preferable that you come to office hours to discuss it, or email me a copy of your work in advance.
• Presentation Opportunities not all for Tuesday: I will be looking for a volunteer to guide us through the Successors and Predecessors section of the handout on page 10. This volunteer would discuss the definitions, work through the You Try It questions (1.31 and 1.32) and mention Theorem 1.17. I will also be looking for a volunteer to present You Try It 1.36 on page 12. If you want to present either of these questions, it is preferable that you come to office hours to discuss it, or email me a copy of your work in advance.
• Remember you should prepare the presentation problems even if you have already presented! People who have not yet presented have priority, but you can earn bonus points for doing more than one presentation!
• Remember to include a question you have about the reading (or something else relevant to class) in your journal.

Journal Homework for class Thursday, April 20:

• Review our work in Tuesday's class. Make any revisions to your journal you think necessary. In particular add the answer to your question for Tuesday or plan to come to an office hour if you haven't figured the answer out yet!
• Reread the Bounded Sets subsection of the Relations handout (this covers pages 11-12).
• Put "You Try It" problems 1.33, 1.35 and 1.36 in your journal.
• Presentation Opportunities: I will be looking for a volunteer to present You Try It 1.36 and possibly a second volunteer to present 1.37 on page 12. If you want to present either of these questions, you should come to office hours to discuss it, or email me a copy of your work at least 24 hours in advance so that we have time to correspond. Be sure that you come to office hours with your presentation at least outlined and as much of the details you need as possible! Note that these may be the last two opportunities during the regular class sessions to present. I will offer additional opportunities during a review session over Reading Period for those who have not presented, but you should be ready to present this Thursday...there may be some competition for the spots this time!
• Note you should prepare the presentation problems even if you have already presented! You can earn bonus points for doing more than one presentation!
• Complete problems 3 and 6 from Section 1.3 (Problems) on page 13 of your Relations handout.
• Start reading Chapter 15 in your textbook (this begins on page 429). Section 15.2 is a play! We will be reading it aloud in class, so think about what character you would like to be! In fact, rank your top three choices! We will assign parts on Thursday! :-)
• Remember to include a question you have about the reading (or something else relevant to class) in your journal.

Collected Homework (Due Friday, April 21 at 2:00pm):

• Remember that this assignment should be done on your own. (Of course, you are welcome to ask me questions!) DO NOT USE ONLINE RESOURCES!!!
• Get out your Strategies for Problem Sets and Proofs laminated handout. Use it as a guide as you solve the following questions.
• Remember to pay careful attention to your base cases in your induction proofs!
• Question 1: Complete Problem 4 from Section 1.3 (Problems) on page 13 of your Relations handout. One way of thinking about this question is to translate each partial ordering from their lattice diagram into ordered pairs, and then figure out what ordered pairs you might add to create a total ordering. Note that there are A LOT of ordered pairs! There IS more than one way to do this! If you discuss this question with someone else, see if you can come up with at least two ways to answer this question (then you can each submit one!). However you approach this question, be sure to create a lattice diagram illustrating your new ordering. Correct solutions need only include your final lattice diagrams.
• Question 2: Prove that every non-zero real number has a unique multiplicative inverse. (This is both an existence AND a uniqueness proof!) (Note that a real number $y$ is said to be the multiplicative inverse of a real number $x$, if $xy=1$. Also note that we will discuss uniqueness proofs in class on Tuesday the 18th.)
• Question 3: Complete problem 7 from Section 1.3 (Problems) on page 13 of your Relations handout. You need not justify your answers, but be sure to label everything clearly. Each question (and some questions have several parts!) should be applied to all four of the given subsets.
• Before turning in your work, remember to review the points on the Proof Writing and Presentation Tips website to help make your proof great! Especially review tips 4, 5, 6, 11, and 12!

REWRITES (Due Friday, April 21 at 2:00pm):

• Rewrites should always be done on your own. Of course, you are welcome to ask me questions, but do not discuss them with others or use outside resources (anything other than YOUR notes and YOUR book and YOUR brain and me is considered "outside").
• Rewrites are optional, but highly encouraged!
• This Friday Group 2 may submit a rewrite. Click here to find out if that is you.
• You may rewrite one PROOF from one previous assignment that I have graded. (Note that this does include problems for which you had to do counterexamples, as you had to justify them with a proof.) Read my comments carefully and ask me if you have questions. Staple your new work to the original work that you turned in. Do NOT staple this to the new assignment above. You can earn back all lost points on these questions! Note that since this is a rewrite/bonus, I will expect higher standards than I did when I first collected it. Be sure to read it outloud to see if it sounds right and check the Proof Writing and Presentation Tips website for ideas about how to make your proof GREAT!
• Do not write on your previous submission to revise or correct it! It should appear as it did when I returned it to you.

### WEEK 12: April 10 - April 14

Remember our first exam is Tuesday, April 11, 7:15PM-9:15PM in Napier 101. Remember to bring your journal to be turned in at the beginning of the exam!

Since we have an exam this week, we will have no quiz on Tuesday!

Journal Homework for class Tuesday, April 11:

• Review our work from class on Thursday and make any revisions to your journal you think necessary. In particular add the answer to your question for that Thursday or plan to come to an office hour if you haven't figured the answer out yet!
• Be sure to review the Introduction to Relations worksheet from March 23rd we discussed in class, and finish the last two questions if you have not already.
• Read Sections 4.7-4.10 (pages 105-109) in the text. Be sure to sing Section 4.10.
• Reread AGAIN! Section 1.1 (Introduction) of your Relations handout (this covers pages 1-4), and the Partial Orderings subsection of Section 1.2 (Orderings) of your Relations handout (this covers pages 4-5). Read the Total Orderings subsection of Section 1.2 in your Relations handout (this covers pages 5-7). Work through all the examples, not in your journal, but be sure that the details make sense to you.
• Put "You Try It" problems 1.9, 1.11-1.15 in your journal.
• Put Problem 2 from Section 1.3 (Problems) on page 13 of your Relations handout in your journal.
• Presentation Opportunity: I will be looking for a volunteer to explain Definition 1.7 and its connection to Example 1.9, and then present You Try It 1.13 and You Try It 1.14 on pages 5-6. These are not that long, but do require an understanding of these definitions! If you want to present this question, it is preferable that you come to office hours to discuss it, or email me a copy of your work for the presentation. The end of the semester is fast approaching, and I hope to see that several people are ready to present!!!
• Prepare for your exam! Read the review sheet review sheet. Note that the review sheet has practice problems for induction on the second side. Catch up on journal work. Bring questions to class.
• Check out this list for a condensed version of what should be in your journal. If you find something missing (i.e. it is on the website, but did not make it on the list), inform me immediately; your classmates will thank you.
• Remember to include a question you have about the reading (or something else relevant to class) in your journal.

Journal Homework for class Thursday, April 13:

• Review our work in Tuesday's class. Make any revisions to your journal you think necessary. In particular add the answer to your question for Tuesday or plan to come to an office hour if you haven't figured the answer out yet!
• Reread the Total Orderings and read the Lattice (Hasse) Diagrams subsections of Section 1.2 in your Relations handout (this covers pages 5-8). Then read the Maximal Elements, Greatest Elements subsection of the Relations handout (this covers page 9 and part of 10). Write down any questions you have on this material and ask them in class!!!
• Carefully reread the uniqueness proof of Theorem 1.14 at the top of page 10. What are the main steps that are used in this proof (and in uniqueness proofs in general)? That is, for each style of proof in the past, we gave an outline (for example, for proof by contraposition we (1) assume the conclusion is false and (2) show that the hypothesis must then be false as well), what is the outline for this one? Write it in your journal. (The headline you highlight in your journal for this could be "Uniqueness Proof Outline".)
• Put "You Try It" problems 1.16-1.18, 1.21 and 1.23 in your journal.
• Remember to include a question you have about the reading (or something else relevant to class) in your journal.

Collected Homework (Due Friday, April 14 at 2:00pm):

• Remember that this assignment should be done on your own. (Of course, you are welcome to ask me questions!) DO NOT USE ONLINE RESOURCES!!!
• Get out your Strategies for Problem Sets and Proofs laminated handout. Use it as a guide as you solve the following questions.
• Remember to pay careful attention to your base cases in your induction proofs!
• Question 1: (The question I meant to write on the exam!) Let $G$ be a graph with eight vertices and ten edges such that all of $G$'s vertices have degree two or three. How many vertices of degree two does $G$ have?
• Question 2: (a) Write down the power set of the set $S=\{h, \&, w, s\}$. Then (b) draw the lattice diagram for the poset on the power set of $S$ under the relation subset. (c) Is this relation totally ordered? Why or why not?
• Before turning in your work, remember to review the points on the Proof Writing and Presentation Tips website to help make your proof great! Especially review tips 4, 5, 6, 11, and 12!

REWRITES (Due Friday, April 14 at 2:00pm):

• Rewrites should always be done on your own. Of course, you are welcome to ask me questions, but do not discuss them with others or use outside resources (anything other than YOUR notes and YOUR book and YOUR brain and me is considered "outside").
• Rewrites are optional, but highly encouraged!
• This Friday Group 1 may submit a rewrite. Click here to find out if that is you.
• You may rewrite one PROOF from one previous assignment that I have graded. (Note that this does include problems for which you had to do counterexamples, as you had to justify them with a proof.) Read my comments carefully and ask me if you have questions. Staple your new work to the original work that you turned in. Do NOT staple this to the new assignment above. You can earn back all lost points on these questions! Note that since this is a rewrite/bonus, I will expect higher standards than I did when I first collected it. Be sure to read it outloud to see if it sounds right and check the Proof Writing and Presentation Tips website for ideas about how to make your proof GREAT!
• Do not write on your previous submission to revise or correct it! It should appear as it did when I returned it to you.

### WEEK 11: April 3 - April 7

Quiz 9 will take place on Tuesday at the beginning of class. It will cover Journal assignments due Thursday, March 30th and Tuesday, April 4th. Please let me know if you have any questions with your journal assignments and come prepared!

Journal Homework for class Tuesday, April 4:

• Review our work from class on Thursday and make any revisions to your journal you think necessary. In particular add the answer to your question for that Thursday or plan to come to an office hour if you haven't figured the answer out yet!
• If you took pictures of the proofs on the board, copy the proofs from your phone to your class notebook. As you do so (or if you already have them in your notebook go back to them and...), make the revisions that we discussed but did not write down that would make the proofs even more precise and beautiful.
• Prepare for the presentation of Section 4.3 problem 2. The presenting group should make sure their presentation is close to flawless (you have had lots of extra time to prepare!!!), and the rest of the class should become very comfortable with the question, trying to make your own proof ahead of time in your class notes, so that you are prepared to ask questions, confirm good form, etc.
• Finish the Introduction to Relations worksheet from March 23rd. We will be discussing side two; be ready to share your answers and/or questions.
• Reread Section 1.1 (Introduction) of your Relations handout (this covers pages 1-4), and the Partial Orderings subsection of Section 1.2 (Orderings) of your Relations handout (this covers pages 4-5). Then start reading the Total Orderings subsection of Section 1.2. Remember that definitions should go in your journal!!!
• Note: when working with a new relation, it is very valuable to list out a few ordered pairs (elements that are related to other elements) before you start doing anything else!
• Add "You Try It" problems 1.6 parts 1 and 2, and 1.7 and 1.8 (complete 1.8 with respect to the parts of 1.5 and 1.6 you were required to do) in your journal.
• REVIEW: Put Problems 20 (Review!) and 25 (induction!) from Section 4.12 (page 114) in your journal.
• Remember to include a question you have about the reading (or something else relevant to class) in your journal.

Journal Homework for class Thursday, April 6:

• Review our work in Tuesday's class. Make any revisions to your journal you think necessary. In particular add the answer to your question for Tuesday or plan to come to an office hour if you haven't figured the answer out yet!
• ***Finish the Introduction to Relations worksheet from March 23rd. We will be discussing side two; be ready to share your answers and/or questions right at the beginning of class!!!***
• Read Section 4.5 (pages 103-104) in the text. Note that you should NOT yet read Section 4.7 or materials elsewhere that would give it away! Section 4.5 is a version of the proof we did at the end of class on Thursday. Talk about it with a classmate and see if you can figure out what is going on. Be ready to discuss this at the beginning of Tuesday's class.
• Complete the Check Yourself problems 1 and 3 at the end of Section 4.5 (page 104). (You can do Check Yourself problem 2 as well, but it is not necessary.)
• Complete problem 1 in Section 4.6 (page 104).
• Review all our work on induction and write down any remaining questions you have. Be ready to share your questions in class.
• Presentation Opportunity: I will be looking for a volunteer to present You Try It 1.5 part 2 on page 3. Note that this involves four (albeit short) proofs. Since we have not worked with these ideas much in class, you are required to discuss it with me before class if you wish to present! We will discuss part 1 as a class and then I will look for volunteers for part 2.
• Reread AGAIN! Section 1.1 (Introduction) of your Relations handout (this covers pages 1-4), and the Partial Orderings subsection of Section 1.2 (Orderings) of your Relations handout (this covers pages 4-5). Read the Total Orderings subsection of Section 1.2 in your Relations handout (this covers pages 5-7). Work through all the examples, not in your journal, but be sure that the details make sense to you.
• Put "You Try It" problems 1.9, 1.11-1.15 in your journal.
• Remember to include a question you have about the reading (or something else relevant to class) in your journal.

Collected Homework (Due Friday, April 7 at 2:00pm):

• Remember that this assignment should be done on your own. (Of course, you are welcome to ask me questions!) DO NOT USE ONLINE RESOURCES!!!
• Get out your Strategies for Problem Sets and Proofs laminated handout. Use it as a guide as you solve the following questions.
• Remember to pay careful attention to your base cases in your induction proofs!
• Use induction to prove that for all natural numbers $n$, 8 divides $5^{2n}-1$. (Do you forget the definition of divides? Check out the assignment for February 10th!)
• (a) Complete You Try It 1.5 part 6 (Note that this involves four (albeit short) proofs! Make outlines before Thursday and then finalize after Thursday's class). (b) Is this relation an equivalence relation? Why or why not? (c) Is this relation a partial ordering? Why or why not?
• Before turning in your work, remember to review the points on the Proof Writing and Presentation Tips website to help make your proof great! Especially review tips 4, 5, 6, 11, and 12!

REWRITES (Due Friday, April 7 at 2:00pm):

• Rewrites should always be done on your own. Of course, you are welcome to ask me questions, but do not discuss them with others or use outside resources (anything other than YOUR notes and YOUR book and YOUR brain and me is considered "outside").
• Rewrites are optional, but highly encouraged!
• This Friday Group 2 may submit a rewrite. Click here to find out if that is you.
• You may rewrite one PROOF from one previous assignment that I have graded. (Note that this does include problems for which you had to do counterexamples, as you had to justify them with a proof.) Read my comments carefully and ask me if you have questions. Staple your new work to the original work that you turned in. Do NOT staple this to the new assignment above. You can earn back all lost points on these questions! Note that since this is a rewrite/bonus, I will expect higher standards than I did when I first collected it. Be sure to read it outloud to see if it sounds right and check the Proof Writing and Presentation Tips website for ideas about how to make your proof GREAT!
• Do not write on your previous submission to revise or correct it! It should appear as it did when I returned it to you.

### WEEK 10: March 27 - March 31

Quiz 8 will take place on Tuesday at the beginning of class. It will cover Journal assignments due Thursday, March 23rd and Tuesday, March 28th. Please let me know if you have any questions with your journal assignments and come prepared!

Journal Homework for class Tuesday, March 28:

• Review our work from class on Thursday and make any revisions to your journal you think necessary. In particular add the answer to your question for that Thursday or plan to come to an office hour if you haven't figured the answer out yet!
• Finalize your group presentation problem. Meet with your group to practice! Review the Proof Writing and Presentation Tips website for reminders about how to make a good presentation even better. Remember that these presentations will be graded. Your group should be ready to present at the beginning of Tuesday's class. I encourage each group to check in with me before presenting to be sure you are confident with your proof; at least one group member should come to office hours on Monday or make an appointment to see me.
• Work through the questions from Section 4.3 that the other groups will be presenting. Remember that you are responsible for critiquing the other groups' presentations, so you should be sure to come up with solutions or questions for each one. This need not be in your journal; you may keep it in your class notes.
• REVIEW: Complete problem 18 in Section 3.13 (page 87).
• Read Section 1.1 (Introduction) of your Relations handout (this covers pages 1-4) that I passed out on Thursday.
• Work through the examples in the Relations handout reading (in your head or on the handout is fine), and put "You Try It" problems 1.1-1.4 in your journal.
• Remember to include a question you have about the reading (or something else relevant to class) in your journal.

Journal Homework for class Thursday, March 30:

• Review our work in Tuesday's class. Make any revisions to your journal you think necessary. In particular add the answer to your question for Tuesday or plan to come to an office hour if you haven't figured the answer out yet!
• Groups presenting questions 1, 2, 3 and 4 should take this opportunity to be sure your proofs are ready to present. Since you have extra time, you have a great opportunity to make your presentation even better! Don't forget to read the directions and review the Proof Writing and Presentation Tips website for reminders about how to make a good presentation even better.
• The group who presented Tuesday should take this opportunity to be sure you have worked through the other questions from Section 4.3 so that you can ask questions and make suggestions during the presentations. Everyone should remember that your grade depends partially on your participation in responding to the presentations of other groups as well as your own presentation.
• Complete problems 5, 6 and 7 in Section 4.11 (page 109). Hint: Note in problem 7 that Prof. belcastro uses $k$ for the index. This means that you will not want to use $k$ for your assumption value! Try using $j$.
• Reread Section 1.1 (Introduction) of your Relations handout (this covers pages 1-4) that I passed out last Thursday.
• Add "You Try It" problems 1.5 parts 1, 2, 4 and 5 from the Relations handout to your journal.
• Remember to include a question you have about the reading (or something else relevant to class) in your journal.

Collected Homework (Due Friday, March 31 at 2:00pm):

• Remember that this assignment should be done on your own. (Of course, you are welcome to ask me questions!) DO NOT USE ONLINE RESOURCES!!!
• Get out your Strategies for Problem Sets and Proofs laminated handout. Use it as a guide as you solve the following questions.
• Remember to pay careful attention to your base cases in your induction proofs!
• Using induction, prove that every tree on at least two vertices has at least two leaves. (We proved this directly, but now we will prove it with induction. In your proof do you see why we need to say that the tree has at least two leaves instead of has one? You should!)
• Using induction, prove that the derivative of $f(x)=x^n$ is equal to $f'(x)=nx^{n-1}$ for every natural number $n$. (Hint: Use other derivative rules!)
• Before turning in your work, remember to review the points on the Proof Writing and Presentation Tips website to help make your proof great! Especially review tips 4, 5, 6, 11, and 12!

REWRITES (Due Friday, March 31 at 2:00pm):

• Rewrites should always be done on your own. Of course, you are welcome to ask me questions, but do not discuss them with others or use outside resources (anything other than YOUR notes and YOUR book and YOUR brain and me is considered "outside").
• Rewrites are optional, but highly encouraged!
• This Friday Group 1 may submit a rewrite. Click here to find out if that is you.
• You may rewrite one PROOF from one previous assignment that I have graded. (Note that this does include problems for which you had to do counterexamples, as you had to justify them with a proof.) Read my comments carefully and ask me if you have questions. Staple your new work to the original work that you turned in. Do NOT staple this to the new assignment above. You can earn back all lost points on these questions! Note that since this is a rewrite/bonus, I will expect higher standards than I did when I first collected it. Be sure to read it outloud to see if it sounds right and check the Proof Writing and Presentation Tips website for ideas about how to make your proof GREAT!
• Do not write on your previous submission to revise or correct it! It should appear as it did when I returned it to you.

### WEEK 9: March 20 - March 24

Have a great spring break!!!

Quiz 7 will take place on Tuesday at the beginning of class. It will cover Journal assignments due Thursday, March 9nd and Tuesday, March 21st.

Journal Homework for class Tuesday, March 21:

• Review our work from class on Thursday, March 9th and make any revisions to your journal you think necessary. In particular add the answer to your question for that Thursday or plan to come to an office hour if you haven't figured the answer out yet!
• Review the induction outline we discussed in class before break. Does the idea behind induction make sense to you?
• Complete Problem 16 in Section 3.13 in your journal.
• Reread Sections 4.1-4.2 in the text. (This covers pages 91-99.)
• Start working on the questions in Section 4.3 in the text (pages 100-101). Groups will be presenting these Thursday the 23rd, so start thinking about how to approach these questions. Your notes on this can go in your class notes and need not be in your journal.
• Read Section 4.4 in the text. Here we see two more examples of proof by induction! (This covers pages 101-102.)
• Complete the Check Yourself problems at the end of Section 4.4 (page 103).
• Remember to include a question you have about the reading (or something else relevant to class) in your journal.

EXAM BONUS (Due Tuesday, March 21 at 2:30pm):

• This assignment should be done on your own; do not discuss it with anyone except Prof. King!
• Complete a nice solution for the Bonus problem on the exam.
• You may earn up to five bonus points on your exam for this assignment. Note that earning all five points will be extremely challenging!
• You may just turn in your solution/proof. You need not resubmit your exam.

Journal Homework for class Thursday, March 23:

• Review our work in Tuesday's class. Make any revisions to your journal you think necessary. In particular add the answer to your question for Tuesday or plan to come to an office hour if you haven't figured the answer out yet!
• Review the induction example we did in class on Tuesday. Look how all the features of an induction proof we outlined are used in the proof.
• If you did not finish the worksheet I handed out in class on Tuesday, do so now and let me know if you have questions.
• Complete Problems 14 and 17 in Section 3.13 in your journal.
• Reread Sections 4.2 and 4.4 in the text. (This covers pages 91-99 and 101-102.)
• Complete Problems 2 and 3 in Section 4.11 in your journal.
• Start working on the questions in Section 4.3 in the text (pages 100-101). If you read side two of the worksheet you will see that each group will be presenting one of these, and everyone should be ready to discuss and critique all of the exercises. We will do these in class next Tuesday since we did not have time to start them in class Tuesday.
• Presentation Opportunity! I will looking for someone to present Problem 2 from Section 4.11 (page 109). Remember you are welcome (and highly encouraged!) to discuss it with me before class!
• Remember to include a question you have about the reading (or something else relevant to class) in your journal.

Collected Homework (Due Friday, March 24 at 2:00pm):

• Remember that this assignment should be done on your own. (Of course, you are welcome to ask me questions!) DO NOT USE ONLINE RESOURCES!!!
• Get out your Strategies for Problem Sets and Proofs laminated handout. Use it as a guide as you solve the following questions.
• Complete Problem 13 in Section 3.13 (page 87).
• Complete Problem 18 in Section 4.11 (page 110). (Hint: Use the theorem we proved in class on Tuesday!)
• Complete Problem 21 in Section 4.11 (page 110).
• Before turning in your work, remember to review the points on the Proof Writing and Presentation Tips website to help make your proof great! Especially review tips 4, 5, 6, 11, and 12!

REWRITES (Due Friday, March 24 at 2:00pm):

• Rewrites should always be done on your own. Of course, you are welcome to ask me questions, but do not discuss them with others or use outside resources (anything other than YOUR notes and YOUR book and YOUR brain and me is considered "outside").
• Rewrites are optional, but highly encouraged!
• This Friday Group 2 may submit a rewrite. Click here to find out if that is you.
• You may rewrite one PROOF from one previous assignment that I have graded. (Note that this does include problems for which you had to do counterexamples, as you had to justify them with a proof.) Read my comments carefully and ask me if you have questions. Staple your new work to the original work that you turned in. Do NOT staple this to the new assignment above. You can earn back all lost points on these questions! Note that since this is a rewrite/bonus, I will expect higher standards than I did when I first collected it. Be sure to read it outloud to see if it sounds right and check the Proof Writing and Presentation Tips website for ideas about how to make your proof GREAT!
• Do not write on your previous submission to revise or correct it! It should appear as it did when I returned it to you.

### WEEK 8: March 6 - March 10

Quiz 6 will take place on Tuesday at the beginning of class. It will cover Journal assignments due Thursday, March 2nd and Tuesday, March 7th.

Journal Homework for class Tuesday, March 7:

• Review our work in Thursday's class and make any revisions to your journal you think necessary. In particular add the answer to your question for Thursday or plan to come to an office hour if you haven't figured the answer out yet!
• Groups who did not present on Thursday should be prepared to put problems 3 and 4 from Section 3.8 (pages 78-79) on the board as soon as everyone has finished their quiz on Tuesday. These should remain in your class notes.
• Complete problems 1, 6, 7 and 8 in Section 3.8 (pages 78-79).
• Read Sections 3.9-3.11 in the text. (This covers pages 80-84.) In Section 3.9 Prof. belcastro reviews the secret to the Dot Game we discussed in class. See if you can explain to a friend why there will never be a tie in the Dot Game if you perform the game on six or more vertices.
• Complete the Check Yourself problems at the end of Section 3.9 (page 81).
• Start reading Chapter 4. In particular, read pages 91-94. This is an introduction to a new type of proof! Cool stuff!
• Complete problems 1, 9 and 12 in Section 3.13 (pages 85-86).
• Presentation Opportunity! I will looking for someone to discuss Ramsey numbers! The presenter would explain the definition of Ramsey number from Section 3.9 and further explain the definition by discussing our work with the Dot game, presenting Check Yourself question 1 from Section 3.9, and explain how to show $R(k,m)=N$ (note this is similar to Check Yourself question 2 from 3.9, but not exactly the same). Remember you are welcome (and highly encouraged!) to discuss it with me before class!
• Remember to include a question you have about the reading (or something else relevant to class) in your journal.

On Thursday, March 9th at 4:45pm in Napier 201, there will be a colloquium given by Yu Cai, Taylor Mancini and Zhaoyi Ren, about their trip to the Nebraska Conference for Undergraduate Women in Mathematics. Refreshments will be served at 4:30pm. I hope you can make it!

Due to the colloquium on Thursday, March 9, my office hours will be moved to 3:30-4:30 on Thursday of this week!

Journal Homework for class Thursday, March 9:

• Review our work in Tuesday's class. Make any revisions to your journal you think necessary. In particular add the answer to your question for Tuesday or plan to come to an office hour if you haven't figured the answer out yet!
• We saw a presentation of a solution to Problem 4 from Section 3.8 that used degrees of vertices. In your journal, write another solution to the problem that uses the formula from Problem 3 of Section 3.8 about edges. That is, show that the only cycle that is isomorphic to its complement is $C_5$.
• In your journal, using the ideas we discussed at the end of Tuesday's class, write a NICE proof of the theorem that states: Every tree with at least two vertices has at least one leaf.
• Take Two: Presentation Opportunity! I will looking for someone to discuss Ramsey numbers! The presenter would explain the definition of Ramsey number from Section 3.9 and further explain the definition by discussing our work with the Dot game, presenting Check Yourself question 1 from Section 3.9, and explain how to show $R(k,m)=N$ (note this is similar to Check Yourself question 2 from 3.9, but not exactly the same). Remember you are welcome (and highly encouraged!) to discuss it with me before class!
• Complete problems 4, 5 and 6 in Section 3.13 (page 86).
• Complete the Check Yourself problems at the end of Section 4.2 (page 100).
• Remember to include a question you have about the reading (or something else relevant to class) in your journal.

Collected Homework (Due Friday, March 10 at 2:00pm):

• Remember that this assignment should be done on your own. (Of course, you are welcome to ask me questions!)
• Get out your Strategies for Problem Sets and Proofs laminated handout. Use it as a guide as you solve the following questions.
• Degree Sequences Exercise: Determine if the following are degree sequences of some simple graph $G$. If a sequence is a degree sequence, give an example of a graph $G$ that has that degree sequence. If it is not a degree sequence, explain why it cannot be one (note that "I couldn't make a graph with that sequence" is not an explanation).
• (a) (1, 1, 2, 2, 3, 4, 7)
• (b) (1, 1, 2, 2, 3, 4, 4, 5)
• (c) (1, 2, 2, 2, 3, 3, 3, 4, 5)
• Another Degree Sequence Exercise: Find two non-isomorphic graphs with the degree sequence (1, 1, 1, 1, 2, 2, 3, 3). Draw them and then prove why they are non-isomorphic.
• Complete problem 25 in Section 3.13 (page 88). Be sure to explain your work carefully. This is a proof.
• Complete Problem 11 in Section 3.13 (page 86). Your answers to the last two parts of this question should be proofs!
• Before turning in your work, remember to review the points on the Proof Writing and Presentation Tips website to help make your proof great! Especially review tips 4, 5, 6, 11, and 12!

REWRITES (Due Friday, March 10 at 2:00pm):

• Rewrites should always be done on your own. Of course, you are welcome to ask me questions, but do not discuss them with others or use outside resources (anything other than YOUR notes and YOUR book and YOUR brain and me is considered "outside").
• Rewrites are optional, but highly encouraged!
• This Friday Group 1 may submit a rewrite. Click here to find out if that is you.
• You may rewrite one PROOF from one previous assignment that I have graded. (Note that this does include problems for which you had to do counterexamples, as you had to justify them with a proof.) Read my comments carefully and ask me if you have questions. Staple your new work to the original work that you turned in. Do NOT staple this to the new assignment above. You can earn back all lost points on these questions! Note that since this is a rewrite/bonus, I will expect higher standards than I did when I first collected it. Be sure to read it outloud to see if it sounds right and check the Proof Writing and Presentation Tips website for ideas about how to make your proof GREAT!
• Do not write on your previous submission to revise or correct it! It should appear as it did when I returned it to you.

### WEEK 7: February 27 - March 3

Quiz 5 will take place on Tuesday at the beginning of class. It will cover Journal assignments due Thursday, February 23rd and Tuesday, February 28th.

Journal Homework for class Tuesday, February 28:

• Review our work in Thursday's class and make any revisions to your journal you think necessary. In particular add the answer to your question for Thursday or plan to come to an office hour if you haven't figured the answer out yet!
• Finish working on the worksheet: Properties of Functions, from class. Make a list of any questions you have about functions and the properties one-to-one and onto.
• DeMorgan's First Law: For homework on Friday you proved the Generalized Version of DeMorgan's Second Law. Now prove the Generalized Version of DeMorgan's First Law.
• Read Section 3.3.2, pages 63-65, in the text. Put the answers to questions 1-4 on pages 64-65 in your journal.
• Read Section 3.3.3 pages 65-66, and play the Dot Game with your friends and/or relatives. Try to answer the questions on page 65 about the Dot Game. After you have played and thought about the Dot Game for a few days, read the hints on page 66. Does this influence your conclusions?
• Read Sections 3.4 in the text. (This covers page 66 and the very beginning of page 67.) Do NOT read Section 3.5. Section 3.4 confirms what we were discussing about the number of possible functions.
• Complete problem 7 in Section 3.13 (page 86) in your journal.
• Remember to include a question you have about the reading (or something else relevant to class) in your journal.

Journal Homework for class Thursday, March 2:

• Review our work in Tuesday's class. Make any revisions to your journal you think necessary. In particular add the answer to your question for Tuesday or plan to come to an office hour if you haven't figured the answer out yet!
• Continue to play the Dot Game with your friends, and see if you can come up with any new conjectures, ideas to support your claims, or counterexamples to show your claims were false. All three are important mathematical tasks! Add these attempts to your work in your journal reflecting on the questions on page 65 about the Dot Game. We discuss your results and will play the game in Tuesday's class.
• Reread Section 3.5 in the text (the material you read in class). This covers pages 67-70. Be sure to add the vocabulary words to your list - there are lots!
• Finish the Check Yourself problems at the end of Sections 3.5 that you were working on in class (pages 70-71). Put problems 6 - 10 in your journal.
• Read Sections 3.6-3.7 in the text. (This covers pages 71-77.)
• Complete the Check Yourself problems at the end of Sections 3.6 and 3.7 (pages 74 and 78 respectively).
• Complete problems 2 and 5 in Section 3.8 (pages 78-79).
• Presentation Opportunity! I will looking for someone to discuss isomorphisms! The presenter would introduce the definition of isomorphism and illustrate the definition by presenting questions 2 and 5 from Section 3.8. Remember you are welcome (and highly encouraged!) to discuss it with me before class!
• Remember to include a question you have about the reading (or something else relevant to class) in your journal.

Collected Homework (Due Friday, March 3 at 2:00pm):

• Remember that this assignment should be done on your own. (Of course, you are welcome to ask me questions!)
• Get out your Strategies for Problem Sets and Proofs laminated handout. Use it as a guide as you solve the following questions.
• Let $f:\mathbb R\to \mathbb R$ be the function $f(x)=17-\frac{3}{2}x^3$.
• (a) Determine whether or not $f$ is one-to-one. Prove your assertion without graphing the function.
• (b) Determine whether or not $f$ is onto. Prove your assertion without graphing the function.
• How many simple 5-regular graphs are there with seven vertices? Draw them and prove that you have found them all. Be sure to make it clear that you understand the definition of 5-regular.
• Let $v$ be a vertex of a graph $G$. Prove that if $G-v$ is 3-regular, then $G$ has odd order.
• Before turning in your work, remember to review the points on the Proof Writing and Presentation Tips website to help make your proof great! Especially review tips 4, 5, 6, 11, and 12!

REWRITES (Due Friday, March 3 at 2:00pm):

• Rewrites should always be done on your own. Of course, you are welcome to ask me questions, but do not discuss them with others or use outside resources (anything other than YOUR notes and YOUR book and YOUR brain and me is considered "outside").
• Rewrites are optional, but highly encouraged!
• This Friday Group 2 may submit a rewrite. Click here to find out if that is you.
• You may rewrite one PROOF from one previous assignment that I have graded. (Note that this does include problems for which you had to do counterexamples, as you had to justify them with a proof.) Read my comments carefully and ask me if you have questions. Staple your new work to the original work that you turned in. Do NOT staple this to the new assignment above. You can earn back all lost points on these questions! Note that since this is a rewrite/bonus, I will expect higher standards than I did when I first collected it. Be sure to read it outloud to see if it sounds right and check the Proof Writing and Presentation Tips website for ideas about how to make your proof GREAT!
• Do not write on your previous submission to revise or correct it! It should appear as it did when I returned it to you.

### WEEK 6: February 20 - February 24

Remember our first exam is Tuesday, February 21, 7:15PM-9:15PM in Napier 101. Remember to bring your journal to be turned in at the beginning of the exam!

Since we have an exam this week, we will have no quiz on Tuesday!

Journal Homework for class Tuesday, February 21:

• Review our work in Thursday's class and make any revisions to your journal you think necessary. In particular add the answer to your question for Thursday or plan to come to an office hour if you haven't figured the answer out yet!
• Meet with your group to work on your presentation on power sets and element arguments for Tuesday's class (this is from the third page of this worksheet). Your presentation should not be in your journal. You should think about the nicest way to present it as well as making sure your proofs (etc.) make sense. Look back at the Proof Writing and Presentation Tips website for ideas. All tips are important, but especially review presentation tips 4, 6 and 10! Please come to my office hours or email me if you have any questions. Ideally at least one of your group will check in with me before Tuesday. We will have a brief amount of time at the beginning of class to review these with your group and write them on the board before we start presenting, but you should be prepared before class.
• Work through the proofs that the other groups will be presenting as well so that you have the ability to ask the other groups questions and critique their work.
• In your journal, prove the following Theorem: Let $A$, $B$ and $X$ be sets. If $X\subseteq A$ or $X\subseteq B$, then $X\subseteq A\cup B$. (Note we proved this idea in class for the intersection a couple of weeks ago.)
• Presentation Opportunity! I will ask someone to present Part III (3) b and c of this worksheet. Remember you are welcome (and highly encouraged!) to discuss it with me before class!
• Presentation Opportunity! I will be looking for a volunteer to present problem 15 from Section 2.9 (page 54) during Tuesday's class.
• Presentation Opportunity! It would also be great if someone volunteered to present problem 21 from Section 2.9 (page 54) during Tuesday's class.
• Complete problems 15 (note that the parenthetical remark in the problem is important!), 21 and 23 (this should be a review from class) in Section 2.9 (page 54).
• Prepare for your exam! Read the review sheet here. Catch up on journal work. Bring questions to class.
• Remember to include a question you have about the reading (or something else relevant to class) in your journal.

Check out this list for a condensed version of what should be in your journal. If you find something missing (i.e. it is on the website, but did not make it on the list), inform me immediately; your classmates will thank you.

On Thursday, February 23rd at 4:45pm in Napier 201, there will be a colloquium given by Mark Curiel, Hobart class of 2017, mathematics and music major, and Jacque Kane, William Smith class of 2018, mathematics and computer science major, about their work researching graph theory with me last summer. Refreshments will be served at 4:30pm. I hope you can make it!

Due to the colloquium on Thursday, February 23, my office hours will be moved to 3:30-4:30 on Thursday of this week!

Journal Homework for class Thursday, February 23:

• Review our work in Tuesday's class. Make any revisions to your journal you think necessary. In particular add the answer to your question for Tuesday or plan to come to an office hour if you haven't figured the answer out yet! If you photographed the proofs that were presented, be sure to copy those proofs into your class notes by hand!
• Reread Sections 3.1-3.2 in the text. (This covers pages 57-62.)
• Work through this worksheet and the questions in 3.3.1 on page 63. Be ready to compare your results with your group partners. You may keep the worksheet in your class notes (i.e. not in your journal), but put Section 3.3.1 numbers 3, 4 and 5 in your journal. If you would like a print out of the worksheet, feel free to come by office hours Wednesday.
• Remember to include a question you have about the reading (or something else relevant to class) in your journal.

Collected Homework (Due Friday, February 24 at 2:00pm):

• Remember that this assignment should be done on your own. (Of course, you are welcome to ask me questions!)
• Get out your Strategies for Problem Sets and Proofs laminated handout. Use it as a guide as you solve the following questions.
• Prove the generalized version of DeMorgan's Second Law! That is, show given an arbitrary indexing set $\Lambda$ for a collection of sets $\{B_{\alpha}\}_{\alpha\in\Lambda}$, that $\overline{\cap_{\alpha\in\Lambda}B_{\alpha}}$ $=\cup_{\alpha\in\Lambda}$ $\overline{B_{\alpha}}$. Look back at our worksheet, Practicing with Indexing Sets, from February 16 to review vocabulary and definitions needed for this proof.
• Before turning in your work, remember to review the points on the Proof Writing and Presentation Tips website to help make your proof great! Especially review tips 4, 5, 6, 11, and 12!

REWRITES (Due Friday, February 24 at 2:00pm):

• Rewrites should always be done on your own. Of course, you are welcome to ask me questions, but do not discuss them with others or use outside resources (anything other than YOUR notes and YOUR book and YOUR brain and me is considered "outside").
• Rewrites are optional, but highly encouraged!
• This Friday Group 1 may submit a rewrite. Click here to find out if that is you.
• You may rewrite one PROOF from one previous assignment that I have graded. (Note that this does include problems for which you had to do counterexamples, as you had to justify them with a proof.) Read my comments carefully and ask me if you have questions. Staple your new work to the original work that you turned in. Do NOT staple this to the new assignment above. You can earn back all lost points on these questions! Note that since this is a rewrite/bonus, I will expect higher standards than I did when I first collected it. Be sure to read it outloud to see if it sounds right and check the Proof Writing and Presentation Tips website for ideas about how to make your proof GREAT!
• Do not write on your previous submission to revise or correct it! It should appear as it did when I returned it to you.

### WEEK 5: February 13 - February 17

I need to visit another faculty member's class on Monday during my normal office hours. Thus my office hours Monday will be 3:00-4:00pm. If you would like to see me and cannot make that time, please make an appointment.

Quiz 4 will take place on Tuesday at the beginning of class. It will cover Journal assignments due Thursday, February 9th and Tuesday, February 14th.

Journal Homework for class Tuesday, February 14:

• Review our work in Thursday's class and make any revisions to your journal you think necessary. In particular add the answer to your question for Thursday or plan to come to an office hour if you haven't figured the answer out yet!
• On Thursday, Christian presented the first half of the proof for part II number three on this worksheet. In your journal, write the second half of the proof!
• Presentation Opportunity! I will ask someone to present part (g) of the Types of Proof worksheet from Thursday's class. This is the proof by contraposition. Remember you are welcome (and highly encouraged!) to discuss it with me before class!
• Reread Section 2.3 of Richard Hammack's chapter on Logic (this is on pages 41-44, or pages 9-12 of the pdf). Pay special attention to the different forms of "if-then" on page 43 (page 11 of the pdf). In your journal, write all ten ways of expressing the statement: If $f$ is a differentiable function, then $f$ is a continuous function. (That is, this one and nine more ways!)
• Complete problems 4(a)-(c) (Note that you have actually already done most of (a)! Also the word "Prove" in part (c) should really be something like "Illustrate".), 8 and 9 in Section 2.4 (page 45) in your journal.
• Read Section 2.7 in the text. (This covers pages 49-50.) Note that part of this section asks you to reread some other material! Do that too! When she says "reread Section 3", she means of the Preface back on pages xxxiii and xxxiv.
• Turn to page three of your Equal Sets and Power Sets handout. Start working on solving these theorems. You do NOT need to put these in your journal. You may leave your work on your worksheet or in your class notes.
• Remember to include a question you have about the reading (or something else relevant to class) in your journal.

Journal Homework for class Thursday, February 16:

• Review our work in Tuesday's class. Make any revisions to your journal you think necessary. In particular add the answer to your question for Tuesday or plan to come to an office hour if you haven't figured the answer out yet!
• Click here to see the worksheet I handed out in class last Thursday. Turn to page 3 and work on the proofs on that page. You need not put these in your journal. Work on an outline for the proofs in your class notes.
• Presentation Opportunity! Present question 1 of Part III on the worksheet mentioned above. Before presenting your proof, give the class an outline of the proof including main steps and goals. If you wish to present this question, you are required to communicate with me first either via email or in office hours.
• You should already have been experimenting with (a)-(c) of problem 4 in Section 2.4. Do you have any questions with this? You actually did half of part (d) in class and for homework already! Do the other half! Using notation from page 30, now complete part (e).
• Review your work on Section 2.4 problem 7. Think about the connections between this notation and the summation notation you saw in calculus II. Does this help you understand the question? Start by listing what $A_1$, $A_2$ and $A_7$ are. Then consider the union of these sets. Afterwards, work on the more general notation. (The notation you need for problem 7 is the same idea as that which you need for part (e) of problem 4.)
• Check out what Richard Hammack has to say about indexed sets on pages 24-27 of his chapter on Sets, which is pages 22-25 of the pdf. Do exercises 1, 4 and 5 of Section 1.8 on page 28, which are on page 26 of the pdf.
• Read Sections 3.1-3.2 in the text. (This covers pages 57-62.)
• Complete the Check Yourself problems at the end of Section 3.2 (page 62).
• Remember to include a question you have about the reading (or something else relevant to class) in your journal.
• Also remember that our first exam is next Tuesday NIGHT! Be sure to check that you are caught up on your journal work.

Collected Homework (Due Friday, February 17 at 2:00pm):

• Remember that this assignment should be done on your own. (Of course, you are welcome to ask me questions!)
• Get out your Strategies for Problem Sets and Proofs laminated handout. Use it as a guide as you solve the following questions.
• Complete this handout.
• (a) Use proof by contradiction to prove: For all integers $a$ and $b$, if $ab$ is an even number, then either $a$ or $b$ is an even number. (b) Explain why proof by contradiction was a good method to use.
• Before turning in your work, remember to review the points on the Proof Writing and Presentation Tips website to help make your proof great! Especially review tips 4, 5, 6, 11, and 12!

REWRITES (Due Friday, February 17 at 2:00pm):

• Rewrites should always be done on your own. Of course, you are welcome to ask me questions, but do not discuss them with others or use outside resources (anything other than YOUR notes and YOUR book and YOUR brain and me is considered "outside").
• Rewrites are optional, but highly encouraged!
• This Friday Group 2 may submit a rewrite. Click here to find out if that is you.
• You may rewrite one PROOF from one previous assignment that I have graded. (Note that this does include problems for which you had to do counterexamples, as you had to justify them with a proof.) Read my comments carefully and ask me if you have questions. Staple your new work to the original work that you turned in. Do NOT staple this to the new assignment above. You can earn back all lost points on these questions! Note that since this is a rewrite/bonus, I will expect higher standards than I did when I first collected it. Be sure to read it outloud to see if it sounds right and check the Proof Writing and Presentation Tips website for ideas about how to make your proof GREAT!
• Do not write on your previous submission to revise or correct it! It should appear as it did when I returned it to you.

### WEEK 4: February 6 - February 10

Quiz 3 will take place on Tuesday at the beginning of class. It will cover Journal assignments due Thursday, February 2nd and Tuesday, February 7th.

Journal Homework for class Tuesday, February 7:

• Review our work in Thursday's class and make any revisions to your journal you think necessary. In particular add the answer to your question for Thursday or plan to come to an office hour if you haven't figured the answer out yet!
• Reread Section 2.3 in the text. (This covers pages 35-44.) Also (re)read (in full or at least some of) Richard Hammack's chapter on Logic. Be sure to especially look at Section 2.8 of his text (this is on pages 54-55 of his text which is pages 22-23 of the pdf).
• Complete problem 5 in Section 2.9 (page 53) of our text.
• Start working on the problems in Section 2.4 (pages 44-45). Put numbers 1, 2 and 3 in your journal and start thinking about 4.
• Review power sets from page 28 in the text and our groupwork sheets in class. We will do another worksheet on these as well, so be sure you have reviewed the definitions, reworked some examples, and write down any questions you have about power sets.
• Remember to include a question you have about the reading (or something else relevant to class) in your journal.

Due to a doctor's appointment, my office hours on Tuesday, February 7th will be moved. They will be Noon-1:15. If you would like to see me and cannot make that time, please make an appointment.

Journal Homework for class Thursday, February 9:

• Review our work in Tuesday's class. Make any revisions to your journal you think necessary. In particular add the answer to your question for Tuesday or plan to come to an office hour if you haven't figured the answer out yet!
• At the end of class we were looking at the following two statements: (1) For every poison there is a chemical that is the antidote. (2) There is a chemical that is the antidote for every poison. We translated the first statement to symbols. Do the same for the second statement, negate both statements in symbol form, then translate the negated statements back into words. Be ready to share your work at the beginning of class. In fact, it would be great if you all put these on the board as soon as you got to class so we could discuss them right away. This work need not go in your journal, but can stay on the worksheet or in your class notes.
• Click here to see the worksheet I was hoping we would work on in class on Tuesday. Turn to page 2 and work on the questions about proving one of DeMorgan's Laws.
• Presentation Opportunity! Consider the symbolic statement $(P\land Q)\Rightarrow (R\vee Q)$. Negate this statement. Then illustrate a truth table for the negation. Please feel free to come and discuss this with me before Thursday!
• Another Presentation Opportunity! Referring to the worksheet above. Present questions 1 and the first half of 3 of Part II. I highly encourage you to come and discuss these with me before Thursday!
• Continue working on the problems in Section 2.4 (pages 44-45). Add numbers 5, 6 and 7 to the previous assignment in your journal. Keep thinking about problem 4.
• Read Section 2.5 in the text. (This covers pages 46-47.) This section introduces two new types of proof: proof by contraposition and proof by contradiction! Woo-hoo! Complete the Check Yourself problems at the end of Section 2.5 (page 48).
• In your journal, write down the two main steps for a proof by contraposition, then do the same for contradiction. (Hint: the two steps should start with "Assume" and "Show"!)
• Remember to include a question you have about the reading (or something else relevant to class) in your journal.

Collected Homework (Due Friday, February 10 at 2:00pm):

• Remember that this assignment should be done on your own. (Of course, you are welcome to ask me questions!)
• Get out your Strategies for Problem Sets and Proofs laminated handout. Use it as a guide as you solve the following questions.
• Here is a new definition. "Definition: Let $x$ and $y$ be integers. Then we say that $x$ divides $y$ if there exists an integer $k$ such that $xk=y$. (Note that this is defined in terms of multiplication, NOT division. Thus there should not be division in your proof!)" Using this new definition, prove the following: For all integers $a$, $b$ and $c$ with $a\neq 0$, if $a$ divides $b$ and $a$ divides $c$, then $a$ divides $3b-4c$. (Hint: Be sure to use all your hypotheses!)
• Consider the Theorem: For any sets $A$ and $B$, $(A\backslash B)\cup(B\backslash A)=(A\cup B)\backslash (A\cap B)$.
1. Carefully draw Venn diagrams illustrating these sets in a step by step fashion (like in Figure 2.9 on page 33 of our textbook).
2. Using an element argument, prove that the theorem is true.
• Negate the following statements. It may be helpful to rewrite the statement in an equivalent form (using if-then, there exists, etc.) before negating it. Make your statements positive where possible. You may, but do not need to, show me a translation of the original sentence into symbols.
1. All HWS students prefer Oba Express over Ichiro.
2. If there is a largest natural number, then it is 1.
3. Some children like to play basketball.
4. For all colors $k$, there exists a textbook $b$ such that, if $b$ is $k$, then Philipp is happy and Tyler is sad.
5. If $G$ is a bipartite graph, then $G$ has no odd cycles.
6. If $x$ is divisible by 10, then $x$ is divisible by 2 and $x$ is divisible by 5.
• Before turning in your work, remember to review the points on the Proof Writing and Presentation Tips website to help make your proof great! Especially review tips 4, 5, 6, 11, and 12!

REWRITES (Due Friday, February 10 at 2:00pm):

• Rewrites should always be done on your own. Of course, you are welcome to ask me questions, but do not discuss them with others or use outside resources (anything other than YOUR notes and YOUR book and YOUR brain and me is considered "outside").
• Rewrites are optional, but highly encouraged!
• This Friday Group 1 may submit a rewrite. Click here to find out if that is you.
• You may rewrite one PROOF from one previous assignment that I have graded. (Note that this does include problems for which you had to do counterexamples, as you had to justify them with a proof.) Read my comments carefully and ask me if you have questions. Staple your new work to the original work that you turned in. Do NOT staple this to the new assignment above. You can earn back all lost points on these questions! Note that since this is a rewrite/bonus, I will expect higher standards than I did when I first collected it. Be sure to read it outloud to see if it sounds right and check the Proof Writing and Presentation Tips website for ideas about how to make your proof GREAT!
• Do not write on your previous submission to revise or correct it! It should appear as it did when I returned it to you.

### WEEK 3: January 30 - February 3

Quiz 2 will take place on Tuesday at the beginning of class. It will cover Journal assignments due Thursday, January 26th and Tuesday, January 31st.

If you haven't had your individual appointment with me, be sure to make it ASAP. Also, if you forgot to bring me your picture at your appointment, be sure to bring it to class or office hours to increase your grade on the First Collected Homework.

Journal Homework for class Tuesday, January 31:

• Review our work in Thursday's class and make any revisions to your journal you think necessary. In particular add the answer to your question for Thursday or plan to come to an office hour if you haven't figured the answer out yet!
• In groupwork this week we will be looking at sets of real numbers. Hence we will be using interval notation. To review interval notation, check out the book that Prof. belcastro recommends for "further reading", Richard Hammack's book, Book of Proof. In particular, check out page 6 of his chapter on Sets, which is page 4 of the pdf. To review how we find intersections and unions of sets expressed in interval notation, check out page 17, which is page 15 of the pdf.
• Reread Section 2.2 and read Section 2.3 in the text. (This covers pages 26-44.)
• Complete Check Yourself problems 1, 4, 5, and 8 at the end of Section 2.3 (page 44).

Last chance to bring me your picture if you forgot to bring it to your appointment! Be sure to bring it to class or office hours to increase your grade on the First Collected Homework!

Journal Homework for class Thursday, February 2:

• Review our work in Tuesday's class. Make any revisions to your journal you think necessary. In particular add the answer to your question for Tuesday or plan to come to an office hour if you haven't figured the answer out yet!
• Finish the salmon worksheet we were working on at the end of Tuesday's class. Make sure exercises 7-13 make sense and/or write down questions you have about them. We will spend a short time on this at the beginning of class, and you should come to class with work to share with your group. This work need not go in your journal, but can stay on the worksheet or in your class notes.
• Reread Sections 2.2 and 2.3 in the text. (This covers pages 26-44.) There is a lot of information here. Any new questions? Does it make more sense now that we worked on 2.2 in class?
• Recall that Prof. belcastro recommends checking out Richard Hammack's book, Book of Proof, which is free online, as a supplement. Read (in full or at least some of) his chapter on Logic. Then do the following exercises from his text: Section 2.6 numbers 3 and 13 (this is on page 51 of his text which is page 19 of the pdf), and Section 2.10 numbers 1, 2, 4 and 8 (this is on pages 60-61 of his text which are pages 28-29 of the pdf). Put these in your journal.
• Remember to include a question you have about the reading (or something else relevant to class) in your journal.

Collected Homework (Due Friday, February 3 at 2:00pm):

• Remember that this assignment should be done on your own. (Of course, you are welcome to ask me questions!)
• Get out your Strategies for Problem Sets and Proofs laminated handout. Use it as a guide as you solve the following questions.
• Complete problems 10, 11, 13 and 14 in Section 2.9 (page 54). For Problem 10, use a sentence to explain what the truth table shows you. Note that both Problem 11 and 11 involve proofs. For Problem 13, be sure to show the step by step process, that is Venn diagrams illustrating each set needed like in Figure 2.9 on page 33. Also assume that $A$, $B$ and $C$ are subsets of some universal set (also like in Figure 2.9).
• Prove the following theorem:

${\bf Theorem}$: Let $A$, $B$ and $C$ be sets. Then $A\cup(B\cap C) = (A\cup B)\cap(A\cup C)$.

In other words, show that union distributes over intersection using an element argument. (Hints: Reread Section 2.2.3 on page 29 of our text. Note that there are two parts to a proof that proves two sets are equal! Note that in Example 2.3.5, Prof. belcastro shows that intersection distributes over union. We are NOT proving this theorem that way. Here we are using an element argument. Also, I think Prof. belcastro makes an assumption not given in the text in her proof, so I claim the proof is incomplete anyway!)
• Before turning in your work, remember to review the points on the Proof Writing and Presentation Tips website to help make your proof great! Especially review tips 4, 5, 6, 11, and 12!

### WEEK 2: January 23-27

Quiz 1 will take place on Tuesday at the beginning of class. It will cover Journal assignments due Thursday, January 19th and Tuesday, January 24th.

Journal Homework for class Tuesday, January 24:

• Review our work in Thursday's class and make any revisions to your journal you think necessary. In particular add the answer to your question for Thursday or plan to come to an office hour if you haven't figured the answer out yet!
• Read Sections 1.5-1.6 in the text. (This covers pages 14-20.)
• Complete the Check Yourself problems at the end of Section 1.5 (page 19).
• Complete problem 17 in Section 1.7 (page 21).
• On page 18, Prof. belcastro introduces existence proofs/theorems that I referred to a couple of times in class on Thursday. In your journal, describe what they are and what makes them different from other proofs/theorems.
• Here is an existence theorem: "There exists an integer whose cube equals its square." Try writing a nice proof of this theorem.
• Prepare for our first quiz! This mostly means that you should make sure your journal assignments are complete. Anything that was required in the journal for Thursday the 19th and today is fair game for the quiz! The next note is helpful for this as well.
• Remember to follow the journal guidelines in the syllabus (including highlighting, etc.). Labeling each problem clearly is important for the quiz. Sections can be labeled by page or problem, but make sure they are clear to you in case I refer to a specific section and problem number on the quiz. Work from the exercises mentioned above, a vocabulary list and a question you have about the reading should be in your journal. Be sure to leave some space after your question where you can put the answer once you figure out what that is! You might be able to answer the question just from our work in class the next time, or you might ask the question specifically during class, or you might figure out the answer by coming to office hours to ask me.

Journal Homework for class Thursday, January 26:

• Review our work in Tuesday's class. Make any revisions to your journal you think necessary. In particular add the answer to your question for Tuesday or plan to come to an office hour if you haven't figured the answer out yet!
• Read Sections 2.1-2.2 in the text. (This covers pages 25-34.)
• Complete Check Yourself problems 1-6 at the end of Section 2.2 (page 34).
• Remember to put at least one question about the reading (or other material) in your journal and be ready to ask it at the start of class!
• On page 34, Prof. belcastro suggests three websites to try out Venn diagrams. The second one no longer seems to be in existence, but check out this one and this one!
• Complete problem 13 and 15 in Section 1.7 (page 21).
• Remember to follow the journal guidelines in the syllabus (including highlighting, etc.). Work from the exercises mentioned above, a vocabulary list and a question you have about the reading should be in your journal. Be sure to leave some space after your question where you can put the answer once you figure out what that is!

Collected Homework (Due Friday, January 27 at 2:00pm):

• Remember that this assignment should be done on your own. (Of course, you are welcome to ask me questions!)
• Get out your Strategies for Problem Sets and Proofs laminated handout. Use it as a guide as you solve the following questions.
• Prove or disprove: The sum of two primes is even.
• Prove or disprove: If $n$ is an odd integer, then $3n+7$ is an odd integer.
• Complete problem 20 in Section 1.7 (page 22). Be sure to explain your answer.
• Cell Phone Question: As of June, 2013 there were 327,577,529 cell phones in the US. How many must have the same last 7 digits in their phone numbers, i.e., *** - ****? Explain.
• Complete problem 22 in Section 1.7 (page 22). Be sure to give details and explain why.
• Tip: It would be good to reread Prof. belcastro's tips for writing mathematics on pages xxxvi-xxxvii before you wrote your final draft of the proof in the first question.

### WEEK 1: January 17-20

Welcome to First Steps into Advanced Mathematics!!!

Collected Homework (Due Wednesday, January 18 by 4:00pm):

Journal Homework for class Thursday, January 19:

• Read the syllabus again. In fact, read it two more times. Although we spent some time on this in class, we did not discuss every detail. You should be sure you have read all of it and understand what is expected. Please ask if you have questions. Note the paper copy I gave you is blue so that you can easily find it. Refer to it often.
• Check that you have no conflict with the dates and times of the midterm exams as listed in the syllabus. Speak with me ASAP if you do have conflicts. Put the dates on your calendar.
• Read the laminated strategies (for both problem sets and reading) handout you received today in class. Use this as a constant reference as you work on your homework.
• 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. You should refer to it regularly throughout the semester.
• Read the Preface for Students and Other Learners, and Sections 1.3-1.4 in the text. (This covers pages xxix-xxxvi and 6-14.) This reading assignment is longer than usual because I think reading the Preface is important! Notice how Prof. belcastro states that it is important to start assignments early at the bottom of page xxxvi! She notes that one benefit to this is that it makes you more efficient...i.e. you actually spend less conscious time working on the assignment! (It isn't just me that says this!) She also has great tips for reading mathematics on pages xxxiii-xxxiv, ideas for attacking problems on page xxxv, and tips for writing mathematics on pages xxxvi-xxxvii! You might want to re-read these throughout the semester!
• Start working through the problems in Section 1.2 (pages 4-5 in the text). In your journal, show your work for problems 2, 4, 6 and 7. You should either copy the question in your journal or make a clear outline of it so that when you look back at your journal to review for an exam you know what the original question was asking. Write a short explanation and/or make a diagram to accompany each answer.
• In your journal, complete the Check Yourself problems at the end of Section 1.3 (page 9) and also at the end of Section 1.4 (page 14). In addition, write the "nice version" of Example 1.4.7 as requested on page 13. As mentioned above and for the whole semester, you should either copy the question in your journal or make a clear outline of it so that when you look back at your journal to review for an exam you know what the original question was asking; and write a short explanation and/or make a diagram to accompany each answer. (Note that the answers or parts of answers to the Check Yourself problems are at the back of the book...hence the name. Be sure that you work on the problems before you check your answers, and realize that the back does not always give complete solutions.
• Remember to follow the journal guidelines in the syllabus (including highlighting, etc.). This will help you find the problems you need to refer to during quizzes! Work from the exercises mentioned above, a vocabulary list and a question (labeled with "Question 1" and that label highlighted) you have about the reading should be in your journal. Be sure to leave some space after your question where you can put the answer once you figure out what that is!

Collected Homework (Due Friday, January 20 at 2:00pm):

• Remember that this assignment should be done on your own. (Of course, you are welcome to ask me questions!)
• Get out your Strategies for Problem Sets and Proofs laminated handout. Use it as a guide as you solve the following questions.
• Is the sum of an even number and an odd number even or odd? Write a theorem that gives the answer to this question, then prove it. In one sentence, after your proof, describe your strategy for discovering the statement of the theorem. (Note that this question has three parts: stating a theorem, proving the theorem, and stating your strategy.)
• Complete the following problems from Section 1.7 in the textbook (pages 20-21): 6 and 9.
• Notes: The last two problems are counting problems. Be sure to explain your work and not just show me calculations. The first problem includes a proof. Use the ideas from your reading and we will also discuss this in class.
• Tip: It would be good to reread Prof. belcastro's tips for writing mathematics on pages xxxvi-xxxvii before you wrote your final draft of the proof in the first question.

Hobart and William Smith Colleges: Department of Mathematics and Computer Science
Erika L.C. King