MAT 242: Introduction to Proofs

Spring 2025

Basic Information

Instructor: Jed Yang, CC221,
Office hours: Tuesday 10:55–11:55 and Friday 11:55–12:55; or by appointment (instructions)
Lectures: Mod B (MWF 09:00–10:10) in RC229
Course website: https://www.mathcs.bethel.edu/yang/mat242.25s/

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. After class, start doing the 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

1 1. 02/03 M
introduction
hw02: Getting started
2. 02/05 W
1.1–7 sets
hw03:
1.1 # 32, 36.
1.2 # 4, 12, 18.
1.3 # 2, 4, 10.
1.4 # 4, 8, 14, 16.
1.5 # 2, 6.
1.7 # 4, 6, 8.
2.1 # 2, 4. (hint: read p.35)

remember to fill out survey1.

3. 02/07 F
2.1–2 propositional logic
hw04:
1.3 # 12, 14.
1.4 # 10, 18.
1.6 # 2.
2.1 # 6, 14.
2.2 # 2, 6, 8, 12.
2.3 # 6.
2.6 # 4.

2 4. 02/10 M
2.3–6 implications
hw05:
2.3 # 1*, 2, 3*, 4, 8, 10. (hint: think critically, write down your answer, and only then check against p.44)

* For 1 and 3, also write down and label the 3 derived implications like we did in class.

2.4 # 2, 4.
2.5 # 2, 6, 8. (hint: use standard row ordering like on p.49)
2.6 # 8, 12, 10. (hint: do 12 first before 10)
2.7 # 4.

5. 02/12 W
2.7–10 predicate logic
hw06:
2.7 # 8, 9, 10.
2.9 # 4, 10.
2.10 # 2, 4.
worksheet # 9, 10.
4 # 2.

6. 02/14 F
(chapters 1 and 2 wrap-up)
hw07:
2.10 # 6, 10.
worksheet # 7, 8, 9, 10.
4 # 4, 6. (hint: read and use Def 4.4 on p.116)

3 7. 02/17 M
4.1–3 direct proof
hw08:
4 # 8, 10, 14, 20, 26.
5 # 2.

(reminder: carefully read chapter 4, at least up to 4.3; you are responsible for all the textbook content, as always; to do the two preview problems, you would want to at least read the first page or two of 4.4 and 5.1.)

8. 02/19 W
4.4–5 proof by cases
5.1 contrapositive proof
hw09:
4 # 16.
5 # 4, 10, 12, 14, 18.

(reminder: finish reading chapter 4 and read chapter 5 at least up to 5.1; to do the preview problem, read at least the first page of 5.2.)

9. 02/21 F
5.2 congruence of integers
5.3 mathematical writing
hw10:
4 # 28.
5 # 16, 22, 24, 30, 32.

(reminder: finish reading chapter 5, paying extra attention to 5.3.)

4 10. 02/24 M
6 proof by contradiction
hw11: none; study for exam.
11. 02/26 W
exam1 (topics and tips)
hw12:
6 # 4, 10, 12, 16.
7 # 2.

(hint: cite and use Prop 10.1.)

12. 02/28 F
7 proving non-conditional statements
hw13:
6 # 14, 18.
7 # 6, 12, 30.
worksheet # 4.
10 # 2.

5 13. 03/03 M
10.1 mathematical induction
hw14:
7 # 8, 14, 32.
10 # 4, 10, 16, 18.

14. 03/05 W
10.5 sequences and series
- OEIS

hw15:
7 # 34.
10 # 8, 14, 26, 34.
worksheet # 3, 6.

Please remember to bring both hw14 and hw15 to class on Friday.

15. 03/07 F
10.2 strong induction
hw16:
10 # 28, 42.
worksheet # 4, 5.
8 # 2.

6 (Spring break) (Spring break) (Spring break)
7 16. 03/17 M
8 proofs involving sets
hw17:
8 # 6, 8, 22, 28.
9 # 2, 8.

17. 03/19 W
9 disproof
hw18:
8 # 10, 12*, 18.
9 # 12, 14, 16, 18, 20.

* Hint for 8 # 12: use laws and 2-column approach.

18. 03/21 F
10.3,4 more induction
hw19:
7 # 16, 22.
9 # 26, 28, 30, 34.
10 # 6*, 12*.

* Try to use proof by smallest counterexample. If you cannot, then use induction for partial credit.

8 19. 03/24 M
(wrap-up)
hw20: none; study for exam.
20. 03/26 W
exam2 (topics and tips)

Week	Monday	Wednesday	Friday
1	1. 02/03 M introduction hw02: Getting started	2. 02/05 W 1.1–7 sets hw03: 1.1 # 32, 36. 1.2 # 4, 12, 18. 1.3 # 2, 4, 10. 1.4 # 4, 8, 14, 16. 1.5 # 2, 6. 1.7 # 4, 6, 8. 2.1 # 2, 4. (hint: read p.35) remember to fill out survey1.	3. 02/07 F 2.1–2 propositional logic hw04: 1.3 # 12, 14. 1.4 # 10, 18. 1.6 # 2. 2.1 # 6, 14. 2.2 # 2, 6, 8, 12. 2.3 # 6. 2.6 # 4.
2	4. 02/10 M 2.3–6 implications hw05: 2.3 # 1, 2, 3, 4, 8, 10. (hint: think critically, write down your answer, and only then check against p.44) * For 1 and 3, also write down and label the 3 derived implications like we did in class. 2.4 # 2, 4. 2.5 # 2, 6, 8. (hint: use standard row ordering like on p.49) 2.6 # 8, 12, 10. (hint: do 12 first before 10) 2.7 # 4.	5. 02/12 W 2.7–10 predicate logic hw06: 2.7 # 8, 9, 10. 2.9 # 4, 10. 2.10 # 2, 4. worksheet # 9, 10. 4 # 2.	6. 02/14 F (chapters 1 and 2 wrap-up) hw07: 2.10 # 6, 10. worksheet # 7, 8, 9, 10. 4 # 4, 6. (hint: read and use Def 4.4 on p.116)
3	7. 02/17 M 4.1–3 direct proof hw08: 4 # 8, 10, 14, 20, 26. 5 # 2. (reminder: carefully read chapter 4, at least up to 4.3; you are responsible for all the textbook content, as always; to do the two preview problems, you would want to at least read the first page or two of 4.4 and 5.1.)	8. 02/19 W 4.4–5 proof by cases 5.1 contrapositive proof hw09: 4 # 16. 5 # 4, 10, 12, 14, 18. (reminder: finish reading chapter 4 and read chapter 5 at least up to 5.1; to do the preview problem, read at least the first page of 5.2.)	9. 02/21 F 5.2 congruence of integers 5.3 mathematical writing hw10: 4 # 28. 5 # 16, 22, 24, 30, 32. (reminder: finish reading chapter 5, paying extra attention to 5.3.)
4	10. 02/24 M 6 proof by contradiction hw11: none; study for exam.	11. 02/26 W exam1 (topics and tips) hw12: 6 # 4, 10, 12, 16. 7 # 2. (hint: cite and use Prop 10.1.)	12. 02/28 F 7 proving non-conditional statements hw13: 6 # 14, 18. 7 # 6, 12, 30. worksheet # 4. 10 # 2.
5	13. 03/03 M 10.1 mathematical induction hw14: 7 # 8, 14, 32. 10 # 4, 10, 16, 18.	14. 03/05 W 10.5 sequences and series - OEIS hw15: 7 # 34. 10 # 8, 14, 26, 34. worksheet # 3, 6. Please remember to bring both hw14 and hw15 to class on Friday.	15. 03/07 F 10.2 strong induction hw16: 10 # 28, 42. worksheet # 4, 5. 8 # 2.
6	(Spring break)	(Spring break)	(Spring break)
7	16. 03/17 M 8 proofs involving sets hw17: 8 # 6, 8, 22, 28. 9 # 2, 8.	17. 03/19 W 9 disproof hw18: 8 # 10, 12, 18. 9 # 12, 14, 16, 18, 20. Hint for 8 # 12: use laws and 2-column approach.	18. 03/21 F 10.3,4 more induction hw19: 7 # 16, 22. 9 # 26, 28, 30, 34. 10 # 6, 12. * Try to use proof by smallest counterexample. If you cannot, then use induction for partial credit.
8	19. 03/24 M (wrap-up) hw20: none; study for exam.	20. 03/26 W exam2 (topics and tips)

Course Information

Official course description: An introduction to mathematical reasoning skills with a focus on proof techniques such as direct and indirect proof, proof by contradiction, and mathematical induction. Also includes examination of sets, logic, and elementary number theory. Emphasis on mathematical communication.
Prerequisites: MAT124M (Calculus 1) with C- or higher.
Textbook: Richard Hammack, Book of Proof, 3rd edition, 2018, ISBN: 9780989472128.
The textbook can be downloaded for free or purchased at-cost from Amazon.

Objectives

Introduction to Proofs is a course in Bethel's curriculum designed to help math and CS students transition from computation-based courses (such as Calculus) to proof-based courses (many upper-level math/CS courses). Topics include sets, propositional logic, predicate logic, and various proof techniques. Along the way, students will practice using precise mathematical definitions, do mathematics by problem-solving, and communicate mathematics through writing and presenting ideas, proofs, and solutions.

By the end of the course, a successful student should be able to say:

I can read, understand, and construct proofs.
I can write and speak about mathematics using precise mathematical language.
I understand the role of definitions in mathematics and can use them effectively.
I can perform basic set operations.
I can convert logical statements from informal language to propositional and predicate logic expressions.
I can use rules of inference to construct proofs in propositional and predicate logic.
I can employ proof techniques including direct proof, proof by contraposition, proof by contradiction, and proof by induction.

(These objectives are adapted from and in alignment with the MAA curriculum guide for math majors and the ACM curriculum guide for CS majors.)

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%

The total score will be converted to a letter grade whose lower bounds are: 93% A, 90% A-, 87% B+, 83% B, 80% B-, 77% C+, 72% C, 69% C-, 66% D+, 60% D, 0% F.

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.
- Colossians 3:23–24 NIV

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.

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. Read the book! You should prepare for class by looking over the sections we will cover. Your aim is not to understand every detail, but to get a sense of where we are headed. Even a few minutes of pre-reading can help with class time. We will not have time to cover every single detail in class. As such, after class, read the sections carefully again to fill in the gaps. Keep up with the reading: reading large sections right before an exam is less effective.

Homework. Homework will be assigned most days. 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 two (2) 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 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 3, 2, 1, or 0 times to get an A, B, C, or D, respectively. I reserve the right to decrease the number of required presentations.

Exams. There are several in-class exams (see calendar for a tentative schedule).

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 about two to three 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 14 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;
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

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).

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, online repositories, or AI chatbots.
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.

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.
The book. Our text is very readable, particularly at explaining the intuition behind proofs and proof techniques; it's a great place to turn if you are having any trouble.
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. (See Bethel University's full policy on Academic Honesty in the catalog.)

Accessibility statement. Bethel University is committed to accessibility for students with disabilities and the Office of Accessibility Resources & Services (OARS) is a resource to ensure students experience access. Reasonable accommodations are approved after an interactive process with the student and OARS. The instructor will provide accommodations, but the student is required to initiate 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.
OARS recommends the student and faculty discuss how accommodations may apply in the specific course. Accommodations cannot modify essential requirements or fundamentally alter the nature of the course. Consultation with OARS may be necessary to clarify reasonable accommodations based on the course. If there are any questions or concerns, connect with OARS at accessibility-services@bethel.edu or 651.638.6833.

Multilingual learners 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.