MATH 251, Section 1 Discrete Mathematics Spring 2020
Syllabus
Welcome! The course we’re about to take together will guide us through some of the fundamental mathematics underpinning computer science and related fields. We will investigate the rudiments of formal logic and methods of mathematical proof and, in turn, we will apply these skills to learn more about number theory, graph theory, and basic combinatorics.
Before we get too deep into these topics, though, let’s take care of a few “bureaucratic” matters:
• Who’s the teacher? Patrick Bahls...please feel free to call me “Patrick.”
• Where and when will class meet? We will meet in Rhoades Hall, Room 106, from 8:00 to 8:50 on Mondays, Wednesdays, and Fridays.
• What text will we be using? We will work through several chapters of Discrete Mathematics: An Introduction to Mathematical Reasoning (Brief Edition), by Susanna Epps, Boston, MA: Brooks/Cole, 2011. Please procure a copy of this text (new or used, bought or rented) as cheaply as you can, and let me know if you have any problems in so doing.
• What should I bring to class? You should always have something to write with (an old-fashioned pen or pencil will do just fine) and take notes on. You will be required to use the LATEX typesetting environment (more about this below) to submit weekly problem set
write-ups and take-home exams, and it wouldn’t be a bad idea for you to get in the habit of bringing a laptop with LATEX at the ready.
• When and where are your office hours? I will generally be free on Monday, Wednes-day, and Friday from 9:00 a.m. to noon and most of the day (9:00 a.m. to 3:00 p.m.) on Tuesdays and Thursdays. Please feel free to make an appointment if you’d like to guarantee a meeting with me. Incidentally, all of my office hours will be held in my Honors Program office, 140 Karpen Hall.
• How can I get a hold of you? My campus phone number is 828-232-5190 and my e-mail addresses are [email protected] and [email protected]. E-mail’s typically the best way to get a hold of me.
• What about other electronic media? My website is at faculty.unca.edu/˜pbahls, and there’s a link on that website to our class page. There, you can access just about any course-related information you need (a schedule of homework assignments, LATEX resources, a copy
A few words about accessibility, equity, and inclusion. I would like to make every attempt to address any learning disabilities or other accessibility issues you might have that could affect your participation in our course. To that end, if you believe your participation may be so affected, I encourage you to get in touch with the Office of Academic Accessibility by writing to [email protected]. Furthermore, I would like to make our classroom as equitable and inclusive as possible. To that end, at the very least, I ask that you ensure that I address you in the manner in which you wish to be addressed. This means not only that I invite you to inform me of the name you wish to go by, but that you also, for instance, inform me of the gender(s) (if any) with which you wish to be identified so that I might make appropriate use of pronouns. I invite you to inform me of any other accommodations I might be able to make in order to improve our class’s accessibility and your experience in our class.
A few words about sexual misconduct and sexual harassment. All members of the university community are expected to engage in conduct that contributes to the culture of integrity and honor upon which the University of North Carolina, Asheville is grounded. Acts of sexual misconduct, sexual harassment, dating violence, domestic violence and stalking jeopardize the health and welfare of our campus community and the larger community as a whole and will not be tolerated. The university has established procedures for preventing and investigating allegations of sexual misconduct, sexual harassment, dating violence, domestic violence and stalking that are compliant with Title IX federal regulations. To learn more about these procedures or to report an incident of sexual misconduct, go to titleix.unca.edu. Students may also report incidents to an instructor, faculty or staff member, who are required by law to notify the Title IX Office.
Back to math...what will we do in this course? As noted above, the purpose of this class is to help you develop the mathematical knowledge, skills, and habits of mind to succeed in future coursework in computer science and closely related fields. Specifically, we will begin with a careful consideration of propositional logic, using this to come up with careful definitions of mathematical statements and arguments. We will spend a fair amount of time learning how to craft formal arguments, also known asproofs, and in so doing we will learn a lot about properties of the natural and rational numbers (number theory), basic methods for counting (combinatorics) and computing probabilities, and modeling networks using graphs (graph theory).
Sound like a lot? Well, it is...but we’re in this together, and you’ll have not just my help in the work we’ll do, but the help of everyone else in our class. I cannot emphasize enough thatmathematics is a social endeavor: we learn mathematics best by doing it and by doing it in the company of others. Thus, I have taken care to design our course so that it is highly collaborative, encouraging your engagement with one another (and with me) both in and outside of class.
What do you expect me to get from this course? I hope that after this course is over you should be able to
• solve elementary problems in number theory, combinatorics, or graph theory; • write clear, complete, correct proofs of basic mathematical claims;
• demonstrate proficient use of the LATEX typesetting environment, and
How will our class be organized? Though the topics we’ll deal with will change from day to day, a typical class will feature each of the following:
• Quiz (5 minutes)
• Presentation of solutions to homework problems (5 minutes)
• Discussion of, and activities related to, the topic of the day (30-35 minutes) • Wrap-up and preview of coming attractions (5 minutes)
I’ll have more to say about each of these as I answer the following question:
What kind of work will I be responsible for in this course? Your grade will be based upon the following activities:
• completing daily quizzes on the topic for the given day,
• presenting (at least twice) a solution to a homework problem in class, • completing regular write-ups of solutions to homework problems, • completing three take-home exams, and
• delivering a collaborative presentation on a mathematical topic of your choice (at the end of the semester).
Reading quizzes. Before coming to class on a given day, you will be expected to (1) read the section (or, in rare cases, sections) of the text assigned for that class meeting and (2) craft rough solutions to a small number of problems corresponding to that reading. To encourage active engagement with the reading, we will begin each class period with a quiz on the reading for that day. The quizzes will be brief, no more than 2-3 minutes, and they should be easy, as long as you’ve done the reading for that day.
I would like to propose that quizzes, altogether, be worth 15% of your overall grade.
In-class problem presentations. Further encouragement to do the readings in advance (seri-ously...do them, y’all) will be given by the requirement that at least twice during the semester you each present a solution to one of several problems assigned for each reading. These presentations do not have to be polished, but they should demonstrate a firm understanding of the problem and its solution. Using LATEX (yes, yes, more on that below) for these presentations is encouraged but
not required. We will generally have time for at least one or two presentations per day.
I would like to propose that your presentation of at least two problems’ solutions be worth 12% of your overall grade.
Homework write-ups. As noted above, I will assign several problems from each section you’re expected to read. For one (or, in rare cases, two) of these problems you will be expected to write up a clear, complete, correct solution using LATEX. These write-ups will be due to me (in a single
.pdf file per week) via email by 11:59 p.m. on a designated Friday, roughly 2-4 problems per week...totally doable! See the schedule of readings, problems, and write-up due dates near the end of this document for more info.
I would like to propose that your homework write-ups be worth 30% of your overall grade.
I hope that it goes without saying that these exams are to be completed individually and without consulting any mathematical resources other than your own notes. (You may, however, come to ask me for guidance or clarification, as needed...but I promise to not be very helpful!) For the first two exams, after I have graded them I will offer you the chance to revise your solutions and the opportunity to earn up to half credit back on any problems that you miss. For instance, if you were to initially receive 3 out of 5 points on a problem whose solution you revised to near perfection, you could receive 1 point (1 = 2 points missed÷2) back for that revised solution.
Though I do not want to belabor the point, I will mention thatNOTHING makes me grumpier as a teacher than academic dishonesty. If I see evidence of cheating on take-home exams (and I’ve been at this for 20 years; I can see it if it’s there), rest assured that I will take action on the matter. ‘Nuff said? ‘Nuff said.
I would like to propose that your take-home exams be worth 30% of your overall grade (10% apiece).
Collaborative end-of-semester presentations. Sometime around the 10th week of class I will ask you to form groups of 3-4 folks to work together on a miniature “research project” into a topic related to the content of our course. You might choose, for example, to solve a particularly challenging problem from number theory or combinatorics...or to prove a more complicated theorem than those we’ve dealt with in class...or maybe you’d like to supplement our understanding of any one of the topics we’ve talked about in class with a deeper dive into the literature. I will give you a great deal of latitude, as long as you clear your group’s topic with me before working on it.
Once you’ve settled on a topic, your group will work to put together a 10-12 minute presentation on the topic, which you will deliver at the end of the semester. These presentations will take place on our final day of class (Monday, April 27) and during our scheduled final exam time slot,from 8:00 a.m. to 10:30 a.m. on Friday, May 1. I’ll share a feedback form for these presentations nearer to the midpoint of the semester, so that you can plan your presentation accordingly.
I would like to propose that your collaborative presentations be worth 13% of your overall grade.
The astute reader will note that I’ve listed the percentage values for the various components of our course as “proposals.” This is because I feel strongly that you all should have a hand in designing this course (it is, after all, our course and not merely mine), and so later this semester I will lead us in a discussion on the matter of assessment and grades...we might decide, as a class, on different point values; those listed above are meant as a starting point for negotiations.
Okay...can you please say a few more words about LATEX? Certainly! LATEX is a
variation of an earlier (and harder-to-use) software distribution called TEX, the purpose of both of which is to give the user complete control of the typesetting environment, enabling them to manipulate not only ordinary things as spacing, indentation, pagination, etc. but also special characters, symbols, and so forth that pertain to technical fields like mathematics. In LATEX for
instance, you can typeset simply beautiful mathematical expressions like
f(x) = sin(x)⇒F(x) = Z
f(x)dx= Z
sin(x) dx=−cos(x) +C
or
d dx
You will find that LATEX will be more than up to the challenge of this class. (Note: nearly every
document I’ll use in this class, including this syllabus, was created using LATEX.)
Using LATEX well takes practice, and it’s most difficult at first. We will spend a full class period
(Wednesday, January 15) near the beginning of the semester on a crash course on the software, and I urge you to dive into it in order to surmount the learning curve as quickly as you can. As I’ve mentioned above, I willREQUIREyou to complete all of your solutions to problem set and exam problems in LATEX and I encourage you to use it in in-class presentations as well. Once you
get used to it, you’ll be producing indescribably beautiful mathematical works with relatively little effort!
Tentative schedule of class activities, readings, and problems. Please note that the reading listed for a given date is the reading to be completed for theNEXTclass meeting, and the problems listed are those eligible for presentation at the next meeting. The problems indicated with a star, ∗, are those for which you must submit LATEXed write-ups (due dates for these write-ups
are listed for each problem set...these solutions are due to me, in the form of LATEXed .pdf files,
sent via email, no later than 11:59 p.m. on the given date).
Please note also that this schedule is subject to change, should we need to adjust course to account for inclement weather, lingering conversations, unexpected teacher absences,etc.
Week 1; no problem sets due Friday.
• Monday, January 13. First day of class! We will get to know one another and talk a little bit about the class. Reading: please read over the syllabus to familiarize yourself with the work we’ll be doing this term. Also, please read over the LATEX resources available on
the class website and be ready to work with LATEX (either through Overleaf or through
some offline distribution) in class on Wednesday.
• Wednesday, January 15. Today will be spent on an in-class crash course on LATEX!
Reading for Friday: Section 1.1 (4,9∗).
• Friday, January 17. We begin talking about math in earnest, considering variables and quantifiers. Reading for Wednesday, January 22: Section 1.2 (2, 7∗, 9).
Week 2; problem sets due Friday: 1.1 and 1.2.
• Monday, January 20. Martin Luther King, Jr.’s birthday; no class! No reading for Wednesday.
• Wednesday, January 22. Let’s talk about sets, subsets, and ordered pairs. Reading for Friday: Section 1.3 (1, 2, 6∗, 10, 15).
• Friday, January 24. We begin talking about relations and functions. No reading for Monday.
Week 3; problem set due Friday: 1.3.
• Monday, January 27. We continue to talk about relations and functions. Reading for Friday: Section 2.1 (8, 15∗, 24, 30, 35).
• Wednesday, January 29. I will be out of town, and we will not have class.
Week 4; problem sets due Friday: 2.1, 2.2, and 2.3.
• Monday, February 3. We talk about conditional statements, the basis of mathematical claims. Reading for Wednesday: Section 2.3 (7, 12∗, 22, 27, 39).
• Wednesday, February 5. We talk about mathematical arguments today. Reading for Friday: Section 3.1 (2, 3, 6∗, 18, 19, 32).
• Friday, February 7. Our first discussion of predicate logic! Reading for Monday: Section 3.2 (1, 12, 15∗, 21, 31, 38).
Week 5; problem sets due Friday: 3.1, 3.2, and 3.3.
• Monday, February 10. Predicate logic, continued! Reading for Wednesday: Section 3.3 (3, 8, 12, 20, 29, 44∗).
• Wednesday, February 12. We talk today about statements with multiple quantifiers. Reading for Monday, February 17: Section 3.4 (4, 7, 11, 18, 22∗, 26).
• Friday, February 14. I will be out of town, so we will have no formal class meeting today. No new readings!
Week 6; problem set due Friday: 3.4
• Monday, February 17. How do we deal with arguments involving quantifiers? Let’s talk! Reading for Wednesday: Section 4.1 (6, 13, 18, 28, 32∗, 48, 51, 60∗).
• Wednesday, February 19. We begin to talk about the method of direct proof today. Reading for next Monday: Section 4.3 (5, 13, 18, 31, 37, 42∗).
• Friday, February 21. We didn’t have class today, as it was a “snow” day! Moreover, we just decided to roll Section 4.2 into 4.3...you will not be responsible for the problems originally assigned from Section 4.2.
Week 7; problem sets due Friday: 4.1 and 4.3.
• Monday, February 24. Now that we’ve introduced rational numbers, we can talk about divisibility. Reading for Wednesday: Section 4.4 (6, 15, 21, 33∗, 38).
• Wednesday, February 26.After wrapping up our treatment of Section 4.3, we move onto the Quotient and Remainder Theorem. Reading for Friday: Section 4.5 (4, 12∗, 19, 24, 30). • Friday, February 28. Today’s discussion focuses on proof by contradiction. No new
readings for Monday!
Week 8; problem sets due Friday: 4.4, 4.5, and 4.6.
• Monday, March 2. Today we talk about proofs by contraposition. Reading for Wednes-day: Section 4.6 (8, 12, 32∗).
• Wednesday, March 4. We look at a couple of classic theorems today: the irrationality of √2 and the infinitude of primes. Reading for Monday, March 16: Section 5.1 (2, 7, 11∗, 23, 40, 45, 56, 63, 72∗).
• Friday, March 6. I will be out of town for a gymnastics meet (you can’t make this stuff up), so let’s cancel class. Have a lovely spring break!
Week 10; problem sets due Friday: 5.1 and 5.2.
• Monday, March 16. Welcome back! Today, we’ll talk about on sequences and sums. Reading for Wednesday: Section 5.2 (3, 7, 11∗, 26, 31).
• Wednesday, March 18. We begin talking about mathematical induction! No new read-ings for now.
• Friday, March 20. I will be out of town at a conference (the last time I anticipate being gone this term), so we will not meet. No new readings!
Week 11; problem set due Friday: 5.3.
• Monday, March 23. We continue our conversation on mathematical induction. Reading for Wednesday: Section 5.3 (2, 9, 25∗, 39).
• Wednesday, March 25. Yet more on induction...as you can tell from the amount of time we’re devoting to it, induction is pretty important! Reading for Friday: Section 5.4 (2, 6∗, 17).
• Friday, March 27. Today’s topic is a slight variation on the last few days’ theme: strong induction! Reading for Monday: Section 5.5 (6, 12∗, 15, 26).
Week 12; problem sets due Friday: 5.4, 5.5, and 5.6.
• Monday, March 30. Today we talk about sequences that are defined recursively. Reading for Wednesday: Section 5.6 (4∗, 8, 25, 33).
• Wednesday, April 1. We begin talking about solving recurrence relations. No new readings for Friday!
• Friday, April 3. More on solving recurrence relations. Reading for Monday: Section 9.1 (1, 7, 8∗, 19).
Week 13; problem sets due Friday: 9.1 and 9.2.
• Monday, April 6. Changing gears, let’s look at the basics of combinatorics. Reading for Wednesday: Section 9.2 (2, 4, 10, 15, 17∗, 30).
• Wednesday, April 8. It’s on to permutations today! Reading for Friday: Section 9.3 (3, 6, 17, 24, 31∗).
Week 14; problem sets due Friday: 9.3, 9.4, and 9.5.
• Monday, April 13. Another useful tool: The Pigeonhole Principle! Reading for Wednes-day: Section 9.5 (2, 7, 16∗, 21, 25).
• Wednesday, April 15. Our discussion of combinatorics wraps up with a conversation on combinations. Reading for Friday: Section 10.1 (2, 7, 19, 23∗, 37, 41).
• Friday, April 17. Today we hit the basics of graph theory. Reading for Monday: Section 10.2 (4, 8, 15, 22, 36∗, 49).
Week 15; problem sets due Friday: 10.1, 10.2, and 10.3.
• Monday, April 20. Paths, trails, and circuits! Reading for Wednesday: Section 10.3 (8, 9, 12, 17, 25∗).
• Wednesday, April 22. Today we introduce trees (and not the green and leafy kind). Reading for Friday: Section 10.4 (1, 6, 11, 19; note that none of these need to be written up).
• Friday, April 24. Our final topic of the term is rooted trees, about which we talk today. No new readings...for the rest of the term!
Week 16
• Monday, April 27. On this last day of class a few groups will give their end-of-semester presentations!
• Wednesday, April 29. Reading Day; no class!
• Friday, May 1 (8:00 a.m. to 10:30 a.m.). This is our designated final exam time, when we’ll meet in our usual classroom for the remaining groups’ end-of-semester presentations.