Instructor: Tim Chumley
Office: Clapp 423
Phone: 413-538-2525
e-mail: tchumley
Office Hours: Mondays and Tuesdays 4:00-5:00, Wednesdays and Thursdays 1:00-2:00; additional availability by appointment
Textbook: Reading, Writing, and Proving, by Ulrich Daepp and Pamela Gorkin, ISBN: 9780387008349;
available as a free e-text
Announcements
Announcements will be posted on the Course Announcements Moodle forum throughout the semester, but essentially all other materials will be posted on this page.
Syllabus
Check the syllabus for all the important class policies (grades, attendance, etc.).
Homework
There will be weekly homework assignments throughout the semester to be turned in. Please read these guidelines for writing in our class.
- General information. A selection of problems will be assigned to be written up individually and turned in each week.
- These problems will be due Fridays at 5 pm.
- You may work with others but the writing should be done on your own.
- Gradescope. Homework will be turned in through Gradescope.
- You should be enrolled automatically. Please let me know if you have any issues logging in.
- Gradescope has made a short tutorial on submitting homework.
- Collaboration. I want you to work together on the homework! The process of explaining your ideas to one another really helps in learning math. However, you must write up the assignments on your own and avoid copying others’ work directly. Also, please only write what you understand so that I know where to help, and please avoid using online forums like Math StackExchange, solutions manuals, or similar resources. A huge part of learning in this class comes from figuring out how to get unstuck by spending time thinking on your own or talking to me and classmates; these other resources can end up being counter-productive in the long term.
- Rewrites. Homework is for practice, and you are not expected to write perfect answers from the start!
- You will be allowed to submit revisions of any problems for full credit each week.
- Your revisions will be due on Fridays at 5 pm. This means each week you’ll have two things to turn in on Fridays: an initial submission and a redo submission.
- Please resubmit (only the problems you’re revising) on Gradescope by using the resubmit function. I’ll be able to see your submission history in order to see what was initially correct or incorrect.
| Assignment | Due |
|---|---|
| Homework 0 | Sep 6 |
| Homework 1 | Sep 13 |
| Homework 2 | Sep 20 |
| Homework 3 | Sep 27 |
| Homework 4 | Oct 4 |
| Homework 5 | Oct 18 |
| Homework 6 | Oct 25 |
| Homework 7 | Nov 1 |
| Homework 8 | Nov 15 |
| Homework 9 | Nov 22 |
| Homework 10 | Dec 6 |
Quizzes
There will be quizzes most weeks that will be given on Fridays. The purpose of these is to check in to see that you’re comfortable with fundamental material and homework problems. Problems will always be related to the previous homework and class topics.
| Quiz | Date | Material |
|---|---|---|
| Quiz 1 | Sep 13 | Homework 0 |
| Quiz 2 | Sep 20 | Homework 1 |
| Quiz 3 | Sep 27 | Homework 2 |
| Quiz 4 | Oct 4 | Homework 3 |
| Quiz 5 | Oct 25 | Homework 5 |
| Quiz 6 | Nov 1 | Homework 6 |
| Quiz 7 | Nov 22 | Homework 8 |
| Quiz 8 | Dec 6 | Homework 9 |
Exams
There will be three exams. The dates for the exams are subject to change slightly.
| Exam | Due Date | Format | Material |
|---|---|---|---|
| Exam 1 | Oct 11 | in-class | Homework 0-4 |
| Exam 2 | Nov 8 | in-class | Homework 5-7 |
| Exam 3 | Dec 13-17 | self-scheduled | Homework 8-10 |
Course plan
Our plan is to cover most of chapters 1-20 in the textbook, with possibly some sections omitted due to time constraints or other sections added. Please check back regularly throughout the semester for precise details on what is covered. Day by day class information and materials will be posted and updated below. A note about files: LaTeX and LyX source code is available for nearly all typeset documents. Simply change the file extension from .pdf to .tex or .lyx in the URL.
Chapters 1-2
Tuesday
- Topic: Convocation Day, no class.
Thursday
- Topic: Introduction. We’ll introduce ourselves, talk a little about class structures, logistics, and goals, and begin talking about mathematical statements.
- Class materials: Lecture notes, worksheet
- After class: Please read the first two pages of Chapter 1 and finish today’s worksheet.
Friday
- Topic: Chapter 2: Logic and truth tables. We discuss forming new mathematical statements using negation, and, or, and implication. We discuss a little bit about the algebra of logical statements as well.
- Class materials: Lecture notes, worksheet
- After class: Please read the first three pages of Chapter 3 and finish today’s worksheet. Start working on Homework 1. Post a message on the class online forum introducing yourself. Some things you might want to include are: your name, pronouns, what classes you are taking, and what are some things you like to do outside of academics. Please create your own introduction post and feel free to read and reply to your classmates’ posts.
Chapters 3-5
Tuesday
- Topic: Chapter 3: Contrapositive, converse, and first proofs. We discuss contrapositive and converse statements and begin to write our first proofs. Along the way, we discuss some guidelines or rules of thumb for writing good proofs.
- Class materials: Lecture notes, worksheet
- After class: Finish today’s worksheet. If you have time, try Problems 3.4, 3.5, 3.15.
Thursday
- Topic: Chapter 4: Set notation and quantifiers. We discuss “for all” and “there exists” statements and their negations.
- Class materials: Lecture notes, worksheet
- After class: Study for Friday’s quiz. I want you to be able to identify the antecedent and conclusion in an implication and work out truth tables for simple statements involving and, or, negation, and implication.
Friday
- Topic: Chapter 5: Proof techniques. We get more practice writing proofs and do examples of direct proofs and proofs by contradiction.
- Class materials: Lecture notes, worksheet
- After class: We will go over Problem 2 on today’s worksheet in our next class. Make sure to try that before Tuesday. Also, get started on Homework 2.
Chapters 5-6
Tuesday
- Topic: Chapter 5: Proof techniques, II. We discuss proofs by cases and pay particular attention to the absolute value function and inequalities.
- Class materials: Lecture notes, worksheet
- After class: Finish today’s worksheet.
Thursday
- Topic: Mountain day, no class.
- After class: Study for this week’s quiz, which will ask you to state DeMorgan’s Laws and the Distributive Laws for conjunction and disjunction. See page 4 of the day 2 lecture notes for these laws. I will also ask you to negate a statement in natural language.
Friday
- Topic: Chapter 6: Sets. We discuss some more basics of set theory. In particular, we focus on set containment, how to prove sets are equal, and how to prove a set is a proper subset of another.
- Class materials: Lecture notes, worksheet
- After class: Finish today’s worksheet and get started on Homework 3.
Chapters 7-8, 26
Tuesday
- Topic: Chapter 7: Operations on sets. We discuss general formulas involving set containment and operations like intersection, union, and complements.
- Class materials: Lecture notes, worksheet
- After class: Write up your proof to Problem 2 on today’s worksheet and read the first two pages of Chapter 8. Note that solutions to today’s worksheet problems are in the class textbook.
Thursday
- Topic: Chapter 8: More operations on sets. We discuss arbitrary and indexed collections of sets and how to think about their intersections, unions, and complements.
- Class materials: Lecture notes, worksheet
- After class: Study for tomorrow’s quiz. I’ll ask you to state the definition of \(a\) divides \(b\), negate a statement involving quantifiers, and write a proof (either direct, possibly contrapositive, or by contradiction).
Friday
- Topic: Open sets. We discuss the notion of open sets, which is not covered in our textbook but is nevertheless important an analysis. The goal of open sets is to generalize the notion of an open interval and to generalize the notion of nearness in mathematical systems without a way to measure distance.
- Class materials: Lecture notes, worksheet
- After class: Finish today’s worksheet and start on Homework 4.
Chapters 26, 9
Tuesday
- Topic: Closed sets. We continue our discussion on basic topology of \(\mathbb R\) by discussing closed sets which are sets whose complement is an open set.
- Class materials: Lecture notes, worksheet
- After class: Finished today’s worksheet and read the first two pages of Chapter 9.
Thursday
- Topic: Open and closed sets. We conclude our discussion on open and closed sets by giving proofs and counterexamples to statements about unions and intersections of open and closed sets.
- Class materials: Lecture notes
- After class: Study for tomorrow’s quiz. It will include one problem where I ask you to compute the union, intersection, and set differences of two given sets. It will also include one problem in the style of Problem 4 from Homework 3.
Friday
- Topic: Chapter 9: Power sets and Cartesian products. We discuss two more ways to construct new sets from given sets. The power set of a set is the collection of all subsets of that set. The Cartesian product of two sets is the set of all ordered pairs whose components come from the respective given sets.
- Class materials: Lecture notes, worksheet
- After class: Begin studying for next week’s exam. Start by making sure you know the definitions we’ve learned so far and review old worksheet and homework problems. If you have time, begin Homework 5 which will be due in two weeks.
Chapter 10
Tuesday
- Topic: Chapter 10: Relations. We discuss relations and, more specifically, equivalence relations. As we will see, equivalence relations give us a way to classify related mathematical objects and partition a given class of objects into disjoint equivalence classes.
- Class materials: Lecture notes, worksheet
- After class: Study for this Friday’s exam. See Thursday’s review sheet for some comments on what to expect.
Thursday
- Topic: Review. We spend the day reviewing for Exam 1.
- Class materials: Review sheet, solutions
- After class: Continue brushing up on material but make sure to get some rest before tomorrow’s exam.
Friday
- Topic: Exam 1.
- After class: Enjoy your break!
Chapters 11-12
Tuesday
- Topic: Fall break, no class.
Thursday
- Topic: Chapter 11: Partitions. We discuss the concept of a partition of a nonempty set. This will be tied closely to the notion of equivalence relation that was introduced last time. Our main result will be that equivalence relation gives rise to a partition, as well as the converse that every partition gives rise to an equivalence relation.
- Class materials: Lecture notes, worksheet
- After class: Read the definitions at the end of Chapter 12. Finish Homework 5.
Friday
- Topic: Chapter 12: Order in the reals. We discuss upper bounds, lower bounds, maxima, minima, suprema, and infima of sets. Looking ahead, our goal is to discuss what distinguishes the real numbers from the rationals, and this will be formulated in terms of the notion of supremum.
- Class materials: Lecture notes, worksheet
- After class: Read from the bottom of page 124 (the paragraph preceding the Completeness Axiom) through to the end of Chapter 12. Start on Homework 6.
Chapters 12-14
Tuesday
- Topic: Chapter 12: Order in the reals. We discuss the Completeness Axiom of \(\mathbb R\), the Archmidean principle, and the well-ordering principle of \(\mathbb N\).
- Class materials: Lecture notes, worksheet
- After class: Read the first page of Chapter 13 and try Exercise 13.1.
Thursday
- Topic: Chapter 13: Consequences of the completeness of \(\mathbb R\). We discuss why it is the case that between any two real numbers there is always a rational number. We also define what it means for a set to be complete.
- Class materials: Lecture notes, worksheet
- After class: Study for tomorrow’s quiz. I’ll ask you to find a power set, sketch a Cartesian product, give the definition of equivalence relation, and explain why a certain example fails to be an equivalence relation.
Friday
- Topic: Chapter 14: Functions, Domain, and Range. We discuss the definition of a function using the notion of relations, and then study examples of relations to determine whether they are functions. Along the way, we define the terms domain, codomain, and range and discuss proving a given function has a certain range.
- Class materials: Lecture notes
- After class: Start on Homework 7.
Chapter 15-16
Tuesday
- Topic: Chapter 15: Functions, One-to-one, and Onto. We discuss what it means for a function to be one-to-one, what it means to be onto, and what it means to be bijective.
- Class materials: Lecture notes, worksheet
- After class: Read the first two pages of Chapter 16.
Thursday
- Topic: Chapter 16: Inverses. We continue our discussion of bijective functions and discuss function composition and inverses.
- Class materials: Lecture notes
- After class: For tomorrow’s quiz, I want you to know the definition of a partition and of supremum and infimum. You will also be given a set and asked to prove that its supremum is a given value.
Friday
- Topic: Chapter 16: Inverses. We continue our discussion of inverses and work on a worksheet mean to help our understanding of Theorem 16.4.
- Class materials: Lecture notes, worksheet
- After class: Read Theorem 16.8 and think about what distinguishes part 3 of this theorem from part 4 of today’s theorem. Begin studying for next week’s exam. It will cover material from Homework 5 to 7.
Chapter 16
Tuesday
- Topic: Chapter 16: Inverses. We wrap up our discussion on inverses and prove Theorem 16.8.
- Class materials: Lecture notes
- After class: Continue studying for Exam 2.
Thursday
- Topic: Review. We spend the day reviewing for Exam 2.
- Class materials: worksheet, solutions
- After class: Study for Exam 2. Get a good night’s sleep!
Friday
- Topic: Exam 2.
- After class: Work on Homework 8.
Chapters 18-19
Tuesday
- Topic: Chapter 18: Mathematical induction. We discuss a proof technique used to prove that a sequence of statements are all true.
- Class materials: Lecture notes, worksheet
- After class: Finish today’s worksheet.
Thursday
- Topic: Chapter 18: Mathematical induction. We get some more practice doing proofs by induction and discuss strong induction.
- Class materials: Lecture notes, worksheet
- After class: Finish Homework 8. Read the first three pages of Chapter 19.
Friday
- Topic: Chapter 19: Sequences. We discuss properties of sequences, which are defined to be real-valued functions of the natural numbers or, informally, infinite ordered lists of real numbers.
- Class materials: Lecture notes, worksheet
- After class: Work on Homework 9.
Chapter 20
Tuesday
- Topic: Chapter 20: Convergence of sequences. We continue our discussion of sequences, focusing on what it means for a sequence to converge or diverge.
- Class materials: Lecture notes, worksheet, Desmos demo
- After class: Read the proofs of Theorems 20.7, 20.8 and 20.9.
Thursday
- Topic: Chapter 20: Convergence of sequences. We discuss general properties of convergent sequences and algebraic limit theorems for sequences.
- Class materials: Lecture notes, worksheet
- After class: Read Problem 20.17. For tomorrow’s quiz, I want you to know the definitions of one-to-one and onto and the proof of Theorem 16.7.
Friday
- Topic: Chapter 20: Convergence of sequences. We discuss the monotone convergence theorem, a useful theorem for proving convergence of a sequence even when its limit is not explicitly known.
- Class materials: Lecture notes, worksheet
- After class: Work on Homework 10.
November break
Tuesday
- Topic: November break, no class. I will hold office hours during class time and the usual 4-5 pm office hour is cancelled.
- After class: Enjoy the break!
Thursday
- Topic: November break, no class.
Friday
- Topic: November break, no class.
Chapter 27
Tuesday
- Topic: Chapter 27: Congruence modulo \(n\). We introduce the integers modulo \(n\) and prove the Division Algorithm.
- Class materials: Lecture notes, worksheet
- After class: Work on Homework 10. Read pages 303-304.
Thursday
- Topic: Chapter 27: Euclidean algorithm. We discuss greatest common divisors and an algorithm for computing them. We also begin our discussion of modular arithmetic.
- Class materials: Lecture notes, worksheet
- After class: Study for the quiz. I want you to know the statements of the Principle of Mathematical Induction and the Second Principle of Mathematical Induction (Strong Induction). You will also be asked to write a basic induction proof.
Friday
- Topic: Chapter 27: Modular arithmetic. We discuss the algebra of the integers modulo \(n\).
- Class materials: Lecture notes
- After class: Begin studying for Exam 3 and take a moment to relax before finals!
Getting help
Here are a few ways to get help:
- Office hours: Mondays and Tuesdays 4:00-5:00, Wednesdays and Thursdays 1:00-2:00; additional availability by appointment
- Evening help: Our class will have a TA who will be holding evening help. Details will be posted on Moodle.
- Study groups: Other students in the class are a wonderful resource. I want our class to feel like a community of people working together. Please get in touch if you’d like me to help you find study partners, and please reach out if you’d like others to join your group. You may work together on homework, explain and check answers, but make sure you write what you know on homework in order to get good feedback.
- Message board: I’ve set up an online forum for asking and answering questions about class material, including homework problems. My plan is to check the forum regularly and answer questions as they come up, but my hope is everyone in the class will pitch in and answer others’ questions as another form of participation in the class.
Resources
- I’ve collected some resources to help you with some basics of LaTeX.
- Here is a LaTeX template file for writing nicely formatted documents, along with its pdf output.
- A LaTeX quick reference is available for commonly used symbols.
- Overleaf is a cloud service that lets you edit and compile LaTeX through a web browser so that no local installation is needed. The free version will be sufficient for our class.
- To install LaTeX locally on your personal computer, I recommend installing MacTex if you use a Mac, MikTeX for Windows, or TeXLive for Linux.
- There are many proof writing and introduction to pure math books out there. Reading our required textbook will be enough to get a strong understanding of the material in class, but if you’re curious, here are some other books.
- Book of Proof by Richard Hammack. Fundamentally similar in focus as our textbook, but uses more discrete math topics.
- How to Prove It by Daniel Velleman. I have less experience with this book, but past students have told me they found it very nice to read.
- Adobe has a Merge PDF tool that is free to use over the web.