MAT 241: Discrete Mathematics
Spring 2023
Basic Information
- Instructor: Jed Yang, CC221, 651-638-6405,
- Office hours: Tuesday and Friday 13:30–14:30; or by appointment (instructions)
- Lectures: Mod C (MWF 11:10–12:00) in RC229
- Course website: https://www.mathcs.bethel.edu/yang/mat241.23s/
Calendar
Daily/weekly schedule to be updated throughout the term; topics and exam dates are tentative and subject to change.Before class, please read the textbook section(s) to be covered. Do the preview homework exercises. After class, start doing the review homework assigned that day as soon as possible. Unless otherwise stated, homework will be due at the beginning of next class.
| Week | Monday | Wednesday | Friday |
|---|---|---|---|
| Unit 0: sets, logic, proofs | |||
| 1 | 1. 02/03 F introduction hw02: Getting started | ||
| 2 | 2. 02/06 M 1.1–1.3 introduction hw03: A more typical homework | 3. 02/08 W 2.1 sets hw04: 2.1.2 # 1, 15bd, 16f, 18d. 2.1.4 # 1, 6ace, 11bc, 12d, 17a. Fill out DMGN: Chapter 3, Logic. 2.2.4 # 1bdegkm. 2.3.4 # 6a, 11, 13ac. 2.3.9 # 1, 2, 3, 4a, 7b. | 4. 02/10 F 2.2–2.3 propositional logic hw05: 2.2.4 # 1aln. 2.3.4 # 12. 2.3.9 # 4e, 8bc, 9bd, 10, 11, 13, 19, 21adf. Read Why should I read a math book? on Moodle. 2.4.3 # 1, 2, 5a, 10ab. Read Appendix G xor extended version. |
| 3 | 5. 02/13 M 2.4 equivalence, inference hw06: 2.4.3 # 8, 9, 10d, 11, 12b, 14. Read Why are definitions important? on Moodle. 2.5.3 # 2, 3, 5, 9, 11, 14ab. | 6. 02/15 W 2.5 boolean algebras hw07: 2.5.3 # 4, 6 (idempotence only), 13, 15, 18abe, 23, 25. Watch What good is boolean algebra? on Moodle. 2.6.2 # 1ac, 2b, 3bd, 9bc, 13af, 18b. | 7. 02/17 F 2.6 predicate logic hw08: 2.6.2 # 1df, 2d, 4b, 9d, 14c, 19de, 21. Read Why should I memorize? on Moodle. DMGN: Chapter 4, Elementary Number Theory. 3.1.3 # 1, 4ac. 3.2.6 1ad, 8c. |
| 4 | 8. 02/20 M 3.1 proofs 3.2 elementary number theory hw09: 3.1.3 # 4b. 3.2.6 # 1b, 8a, 10, 11, 13, 14. DMGN: Chapter 5, Proof. Re-read Appendix G xor extended version. 3.3.10 # 1a, 4, 10, 15, 20, 22. | 9. 02/22 W 3.3 proof strategies hw10: none; study for exam. | 10. 02/24 F exam1 (topics and tips) hw11: 3.3.10 # 9, 12, 16, 19, 27, 30, 37, 39*, 42, 44. * 39: extra credit part will not be graded. DMGN: Chapter 6, Mathematical Induction. 3.5.6 # 1a*. * 1a: use the same formatting style as in textbook examples (proof of Theorem 3.54, Example 3.35, etc.). Clearly label the step where you use the inductive hypothesis. |
| 5 | 11. 02/27 M 3.5.{1,2} induction hw12: 3.5.6 # 1c, 3, 5*, 9*, 11, 27a. Continue to signpost your induction proofs as in the textbook, with clearly defined $P(n)$, clearly labelled base step, inductive step, and a conclusion sentence; also clearly label where you use the inductive hypothesis. * 5: think and then read the solution; do not hand in. * 9: show details (scratch work) of your discovery process. Install and become familiar with Mathematica. | 12. 03/01 W 3.5.{3,4} more induction hw13: 3.5.6 # 8, 15, 25, 27cd.
3.4.7 # 1b, 7, 15ad. | 13. 03/03 F 3.4.1 Euclidean algorithm 3.4.4 Linear Congruence hw14: 3.4.7 # 1ad, 8, 10, 15bc, 19, 21. |
| 6 | 14. 03/06 M 3.4.4 Chinese Remainder Theorem 3.4.5 Fermat's Theorems hw15: 3.2.6 # 15 (induction on $k$), 18, 20. 3.4.7 # 11*, 17. * 11: must use the Euclidean Algorithm at least twice to find inverses (instead of trial and error). There are no preview problems. You may pre-read or not, up to you. We will prove Thm 3.50 in class. | 15. 03/08 W 3.4.6 RSA cryptosystem hw16: Handout.
3.6.3 # 6, 9, 11, 12, 13, 15, 16, 21. | 16. 03/10 F 3.6 creating proofs hw17: 3.6.3 # 3, 4, 7, 14, 29, 34, 41.
DMGN: Counting 1, Chapter 8 (but skip Counting 2). 5.1.6 # 2, 3, 5, 8, 10, 12. |
| Unit 1: counting | |||
| 7 | (Spring break) | (Spring break) | (Spring break) |
| 8 | 17. 03/20 M 5.1 counting hw18: 5.1.6 # 19, 21, 22, 26, 27, 28, 30, 31, 33, 40. 5.1.8 # 2, 4, 7, 10, 18c. | 18. 03/22 W (catch-up) hw19: none; study for exam. | 19. 03/24 F exam2 (topics and tips) hw20: 5.1.6 # 44a. 5.1.8 # 15, 25, 29, 30. Read the vcp overview; download the binaries, unzip, and run 5.2.3 # 1, 2, 4, 15. |
| 9 | 20. 03/27 M 5.2 double counting hw21: Read (and follow) the advice in items 11–13 of extended version of Appendix G. 5.2.3 # 3, 5, 7, 8, 16. DMGN: Counting 2, Chapter 9. 5.3.3 # 1, 2, 3, 8, 11, 17, 26cd. | (advising day) | 21. 03/31 F 5.3.1 pigeonhole principle 5.3.2 inclusion-exclusion hw22: 5.2.3 # 10, 11*, 17, 18*. * 11: use a combinatorial proof. * 18: do twice; (a) use an algebraic proof (induction) and (b) use a combinatorial proof. |
| 10 | 22. 04/03 M (catch-up) hw23: 5.3.3 # 9, 10, 12, 15, 16*, 18, 28. * Here and always, provide enough details and justification. 7.2.5 # 1, 3bd, 4b. | 23. 04/05 W 7.2.{0,1} recurrence relations hw24: 7.2.5 # 3a (simplify fully), 5a, 7. Read. 7.2.5 # 11ab, 12ac, 16*. * 16: produce (as always, with detailed justification) the recurrence relation, but do NOT solve the recurrence relation. | (Good Friday) |
| Unit 2: theory of computation | |||
| 11 | (Easter Monday) | 24. 04/12 W 7.2.{2,3,4} LHRRwCCs hw25: 7.2.5 # 12d, 15, 16, 18bd, 19b, 21. DMGN: Chapter 11, Finite-State Machines. 9.2.3 # 1abc, 2, 3, 4, 12ab. | 25. 04/14 F class cancelled hw26: none; please turn in hw25 by Monday. |
| 12 | 26. 04/17 M theory of computation 9.2 deterministic finite automata hw27: none; study for exam. | 27. 04/19 W exam3 (topics and tips) hw28: 9.2.3 # 5ab, 6c, 9, 15, 17cd. 9.3.2 # 1abc, 5acd, 7ab. | 28. 04/21 F 9.3 regular grammars hw29: 9.3.2 # 6e, 8a, 10, 11. Worksheet # 2cde. Do not memorize Table 9.8. 9.4.3 # 1, 2, 3, 4. Bring laptop next class if possible. Bring hw23 if you want to get the grade recorded. |
| 13 | 29. 04/24 M 9.4 regular expressions hw30: 9.4.3 # none. Worksheet # 4abcde. 11.3.4 # 1, 2. | 30. 04/26 W 9.6.1 context-free grammars 11.3.1 parse trees hw31: 11.3.4 # 4, 5, 6. DMGN: Chapter 12, Graphs. 10.1.3 # 2acd, 3a, 5a, 6f, 7, 8, 9*. * 9: there may be other even-degree vertices. | 31. 04/28 F 10.1 graphs hw32: 10.1.3 # 4c, 11*, 13, 15, 17, 19, 20. * 11: explicitly transform this to a graph theory problem and then solve. 10.2.3 # 4 (no explanation necessary), 5a, 6a (explain your answer). 10.4.3 # 10 (we defined "isomorphism" in class, so you need NOT read 10.4). |
| Unit 3: graph theory | |||
| 14 | 32. 05/01 M 10.2 walks and connectivity hw33: none; study for exam. | 33. 05/03 W (catch-up) hw34: none; study for exam. | 34. 05/05 F exam4 (topics and tips) hw35: 10.2.3 # 1, 7c, 11*, 13, 15, 16, 17. * 11a: you may use a calculator or Mathematica. 10.3.3 # 1, 3, 5, 7. |
| 15 | 35. 05/08 M 10.3 Euler and Hamilton hw36: 10.3.3 # 8, 12, 13, 14, 15, 16. 10.5.4 # 3*, 9, 18. * 3: Recommendation: use Sandbox; first add all vertices in alphabetical order as the textbook so the labels are right; then add in all edges; then start dragging. When done, you may print it out or draw it by hand. To aid grading, your graph must have correct vertex labels. As always, feel free to help each other understand the problem, but execute your solution yourself: it is unlikely that two students will arrive at exactly the same configuration. Do not plagiarize. | 36. 05/10 W 10.5 major graph theorems - Proper drawings of planar graphs hw37: 10.5.4 # 1, 2, 7, 8, 10ab, 13ab. | 37. 05/12 F 10.5.3 chromatic number hw38: 10.5.4 # 10e, 11, 12, 16, 17. DMGN: Chapter 13, Trees. 11.1.1 # 6, 12, 15 (do not assume the graph is a tree), 16 (read proof, do not hand in). |
| 16 | 38. 05/15 M 11.1 trees hw39: 11.1.1 # 1, 2, 4, 5, 18, 19. Fill out | 39. 05/17 W (wrap-up) hw40: none; study for exam. | 40. 05/19 F (catch-up)
|
| Final Exam: 05/26 Friday 09:00–10:10 (topics and tips) | |||
Course Information
- Official course description: Covers a collection of topics useful to mathematics and computer science majors. The unifying factor is that topics deal mainly with finite collections of mathematical objects (graphs, trees, finite state machines, etc.). Also includes examination of sets, logic, Boolean algebras, proof techniques, algorithm analysis, counting, and recursion.
- Prerequisites: MAT124M (Calculus 1) with C- or higher.
- Textbook:
- (required) Eric Gossett, Discrete Mathematics with Proof, 2nd edition, 2009, ISBN: 9780470457931.
- (optional) Eric Gossett, Discrete Math: The Graphic Novel, 2016, ISBN: 9781524948566.
Overview
Goals. Discrete Mathematics is a course in Bethel's curriculum designed with three related driving goals:
- to help develop process skills to learn, do, and communicate mathematics.
- content: to introduce the wonderful mathematical world of discrete (as opposed to continuous) objects.
- to provide foundational knowledge, skills, and maturity required to succeed in subsequent math and computer science courses. In fact, this course is a (transitive) prerequisite for roughly half of the required Math or CS courses in our majors.
Topics. We will consider topics such as proofs, counting, models of computation, graphs, and trees.
Objectives. We will use learning the content (secondary goal) as an opportunity to develop mathematical skills (primary goal). Specifically, I will guide you in learning how to:
- Learn mathematics: through reading and interrogating a mathematical textbook.
- Do mathematics: through problem-solving, not rote practice.
- Communicate mathematics: through writing and presenting ideas, proofs, and solutions.
Grading
Your grade will be determined by a weighted arithmetic mean of various components with weights listed in the table on the right.| component | weight |
|---|---|
| Participation | ±3% |
| Homework and quizzes | 31% |
| Exams | 66% |
Each letter grade has an associated number of required presentations (see below).
Note that there is no preset curve of how many of each letter grade will be given. If you all do A-level work, you will each get an A. As such, you are encouraged to help each other in the pursuit of perfection.
In many courses I intentionally make one exam harder than others, which gives me information (in a mathematical sense) in separating an A performance from an A- performance. Typically, I will let you know and adjust that exam's scores upward. What this means is that you should NOT care about how hard an exam is. If you do A-level work, you will get an A, regardless of the raw numerical score prior to adjustment.
Besides possibly adjusting scores upward for difficult exams, I also reserve the right to lower the grade cutoffs. Both of these help you. I will not hurt you by adjusting your exam scores downward or increasing the grade cutoffs.
Requirements
Whatever you do, work at it with all your heart, as working for the Lord, not for human masters, since you know that you will receive an inheritance from the Lord as a reward. It is the Lord Christ you are serving.I will be trying to make these verses true for me as I work with you throughout this course, and I hope that you will, too.- Colossians 3:23–24 NIV
Attendance and participation. I expect you to attend class. You may not notice me taking attendance during class meetings, but I will notice if you are not in class. Occasional absences will not impact your grade because what I look for is not mere attendance, but engagement and participation.
Indeed, coming to class is not just about showing up; it is also about being fully engaged in the learning experience. If you have a question, others in the class may also be wondering the same thing. So, please speak up and ask questions anytime you need to. Not only will you be helping yourself, but also you will be helping your peers. Attending office hours is another great opportunity to ask questions.
Be mindful of others. Refrain from using mobile phones or laptops for activities unrelated to the learning process. If you prefer to use laptops to take notes, please kindly sit in the back, as the screen may distract others. There is research that suggests taking notes by hand is better for long-term retention (P. A. Mueller and D. M. Oppenheimer, The pen is mightier than the keyboard, Psychological Science 25 (2014), 1159–1168).
Silence and put away mobile phones and do not use laptops for anything other than class-related activities.
It is my sincere hope that every one of you get all the points for attendance and participation.
Reading. The first objective is not to learn mathematics (content), but to learn how to learn mathematics (skill). Specifically, you will learn how to read a mathematical textbook.
Reading a math textbook is different from reading a novel. You must interrogate it. Ask questions. Work through exercises. Think about every sentence.It will take time to get used to reading properly, but it will pay dividends if you do not give up. To hold you accountable to the reading, there will often be preview exercises due as part of daily homework based on the reading, before the topic is discussed in class.
Homework. Homework will be assigned most days. One of the three course objectives is to learn how to do mathematics. The goal of the homework is to give you an opportunity to continuously engage directly with the material. Some of the homework questions are meant to be challenging and to stretch you; simply put, I believe that the homework is where you will do the vast majority of your learning in this class. Grapple with the questions; talk to classmates about solution strategies if you are feeling stuck; do the homework.
Please staple your homework before coming to class and write your name, PO number, and homework number in the top right corner.
Homework is due at the beginning of the next class after it was assigned, unless otherwise stated. In general, late work is not accepted. If there are special circumstances, talk to the instructor. To alleviate your anxiety from accidentally forgetting to bring your homework to class, illness, emergencies, or other situations beyond your control, the lowest three (3) assignments will be dropped.
Because communicating results to others is an important skill, showing your work is as important as getting an answer. In many instances, credit will only be given if your work accompanies your answer. You are encouraged to collaborate, but what you turn in must be your own work. See "Learning integrity" and the collaboration policy below.
Presentations. Learning how to communicate mathematics is one of the three course objectives. You will communicate by writing (homework) and verbally (presentations). We will have opportunities for presentations on most class days. Presentations are required but will not be graded. Instead, students are asked to put in a genuine effort. The required minimum number of presentations is 4, 3, 2, 1, or 0 times to get an A, B, C, D, or F, respectively. I reserve the right to decrease the number of required presentations.
Exams. There are several in-class midterm exams (see calendar for a tentative schedule). Subsequent exams will mainly focus on the material covered since the previous exam, but can include previous material too. There will be a final exam during the official final exam period covering the entire course.
There are no make-up exams except in circumstances recognized by the instructor as beyond the control of the student. To receive this consideration, the instructor must be notified of the problem before the exam unless this is impossible, in which case as soon as possible.
Time outside of class. I expect a typical student to spend at least two to four hours outside of class for each hour in class. Some students need to spend a bit more than that (which is okay). If you are spending more than 15 hours per week on this course outside of class time, please come talk to me so we can find ways to help you learn the material without spending so much time.
Illness. You should make every effort to attend class when you are healthy. If you become ill, for your well-being and the well-being of the rest of the class, you should not come to class. (Nor should you show up to my office with your germs!) Yes, this sounds like common sense, but it is tempting to try and power through as normal so as not to fall behind. If you become ill, or know that you will need to miss class for some reason, please contact me as soon as you are able, and we will work together to plan how you will keep up and/or make up any missed work.
Learning integrity.
Search me, O God, and know my heart;Collaborative work is an integral part of many successful ventures. As such, I expect that you should collaborate with your classmates a lot during your time in this course. However, it is important to understand that there is a big difference between thinking about and solving a problem as part of a group (which is good, both educationally and morally) and copying an answer or letting someone else copy your answer (which is bad, educationally and morally, and has punitive consequences).
Try me, and know my anxieties;
And see if there is any wicked way in me,
And lead me in the way everlasting.- Psalm 139:23–24 NKJV
In short, I trust you to maintain the utmost level of academic integrity in this course. Please do not break this trust; if you do, there will be repercussions. The formal policy below lays this out explicitly, and supplements Bethel's academic honesty policy.
Collaboration policy.
- You may collaborate on the homework assignments to the extent of formulating ideas as a group, but you may not collaborate in the actual writing of solutions (unless explicitly allowed in the instructions).
- In particular, you may not work from notes taken during collaborative sessions.
- You may not consult any materials from any previous offerings of this course or from any other similar course offered elsewhere unless explicitly permitted.
- You may not look up solutions in any form, including from solution manuals or online repositories.
- You are required to completely understand any solution that you submit and, in case of any doubt, you must be prepared to orally explain your solution to me. If you have submitted a solution that you cannot verbally explain to me, then you have violated this policy.
Adjustments. Due to the uncertain and ever-changing conditions the world is in, adjustments may need to be made. Thus:
- The instructor reserves the right to make reasonable adjustments to the syllabus if necessary. These changes, if any, will be communicated to students in writing. The instructor will make a good-faith effort to help students adversely affected by such changes.
- Each student is asked to make a good-faith effort to try to adapt.
Getting Help
If you need help there are multitude of resources you can use:- Yourself. If you're stuck on a problem or struggling with a concept from class, take a break and think about something else (e.g., your Hebrew assignment, the economics of Star Trek) for a few hours and then try a fresh start.
- Your classmates. You are each other's best resource: talking through the course material with someone else who is also trying to master it is a great way for you both to learn. (And don't discount the learning that you will do while trying to explain to a classmate an idea covered during class that you think you understand; I can't count the number of times that I've discovered that I didn't really understand something until I tried to teach it to someone.) The homework assignments are meant to challenge you, and figuring some of them out together is a great approach.
- Math Lab. The Math Department offers support for students enrolled
in math classes by providing a Math Lab.
Hours, as well as logistics for attending will be provided on Moodle.
If you are having any difficulty with your homework in this class,
please seek help from the tutors in Math Lab.
The Math Lab is not only a great place to get help from tutors,
but also is the perfect place to meet other students from class, do homework, and check your work.
Plan Math Lab hours into your weekly schedule and develop this habit early on in the course.
- The instructor. Come to my office hours or email to make an appointment. Please read and follow instructions.
Bethel Policies
The following are policies that apply to every course at Bethel.Academic honesty policy. Violation of honesty standards can result in denial of credit (U or F) in a course, as well as dismissal from the university. Penalties are given at the discretion of the faculty member, and offenders will be referred to the associate provost of the College of Arts & Sciences.
Accommodation policy. Bethel University is committed to accessibility for students with disabilities and the Office of Accessibility Resources and Services (OARS) is a resource to ensure students experience access. The instructor will provide accommodations after the student initiates the process.
- Students with a documented disability may contact OARS to learn more about how to register for accommodations. Reasonable accommodations are approved after an interactive process with the student and OARS.
- Students registered with OARS are responsible for logging in to their AIM Accessibility Accommodation portal (via MyBethel) each term to request their Faculty Notification Letter of Accommodations. Accommodations cannot be applied prior to the faculty’s receipt of the letter.
Multilingual student support. If you are a multilingual student and believe you would benefit from support for this course, please see your instructor. Possible supports include access to lecture notes, additional time for completing assignments and/or tests, vocabulary lists, use of translation dictionaries, additional time for writing assignments.
- When you notify your instructor, s/he may refer you to the AESC office (HC324) so that you can meet with an academic counselor. The academic counselor will help determine the supports that could contribute to your success in the course and will notify your instructor to suggest these supports be made available to you.
- In addition to specific supports for this course, one-on-one writing support is available for multilingual students. Stop by HC324 or schedule an appointment for Multilingual Support. More information on multilingual support is available.
Concerns and appeals. If you have any concerns regarding the course, your grades, or the instructor, see the instructor first. If needed, see Bethel's academic appeals policy.