Math 135:
First Steps into Advanced Math

   Department of Mathematics and Computer Science
   Hobart and William Smith Colleges

   Fall 2019.

   Instructor:  David J. Eck  (
   Monday, Wednesday, Friday, 11:00–12:00
        Room: Gearan Center 228 (but might be moved)

About The Course and the Textbook

Math 135 is meant as an introduction to mathematical rigor. The idea is that high school and even early college math courses present mathematics as a set of rules for calculation. But real math is not about following rules. It is about discovering ideas and proving that they are correct. Proof, in particular, is a central concept. To succeed in mathematics at an advanced level, you need to understand proofs, and you need to be able to create your own proofs. The course is called "First Steps into Advanced Mathematics" (which some people have said they find condescending, though it's not meant to be; it's also not meant to imply that the course is easy). When the course was first introduced, it was called "The Joy of Math," and I hope that you will look at it in that light.

The course begins with the foundation of mathematical discourse: formal logic and sets. The second part of the course covers important proof techniques, such as direct proof, proof by contradiction, and mathematical induction. In the third part, we will apply all of that to some important mathematical ideas such as functions, relations, and infinity.

The textbook for the course is Book of Proof, Third Edition, by Richard Hammack. It is available as a free PDF. If you would like a printed copy, an inexpensive paperback is available, but I have not asked the bookstore to stock copies. According the book's description, "This book is an introduction to the language and standard proof methods of mathematics. It is a bridge from the computational courses (such as calculus or differential equations) that students typically encounter in their first year of college to a more abstract outlook. It lays a foundation for more theoretical courses such as topology, analysis and abstract algebra."

Note that you are not required to bring the book to class. However, I will assign homework problems from the book, and I will expect you to do assigned readings.

We will cover most of the book, leaving out Chapters 3 and 13 and a few other sections. I will add a little extra material in the form of brief introductions to at least two mathematical fields, topology and group theory.

Although I have planned out the course in some detail, I don't want to be too rigid about sticking to the exact plan. We will evaluate how the course is going from time to time and discuss changes that we should make. In particular, the right balance of time among lectures, presentation, and group work is open to discussion.

Assignments and Presentation

Homework will be collected weekly, and sometimes more frequently. Many homework exercises will be taken from the textbook, but there will often be some additional problems. For most homework, you are allowed and encouraged to discuss the homework exercises with other people in the class. Some classes will be problem sessions where you get to work on homework with other people in the class. However, you should always write up your own solutions in your own words.

In many cases, I will allow you to resubmit proofs to improve your grade. However, I will only do that if you have made a reasonable effort to do the proof in the first place.

We will discuss how to use the LaTeX typesetting system for writing homework solutions. This might be required later in the course.

Sometimes, I will ask students to present proofs at the board. The proofs might be homework exercises, or they might be individual assignments. Everyone in the class should expect to do a couple of these presentations over the course of the semester. Presentations will be part of the homework grade for the course, but they will mostly get full credit as long as a reasonable effort is made.

In addition, I will ask every student to meet with me from time to time. At these meetings, we can go over the homework and discuss the course. I will sometimes ask you to present your homework solutions to me on the board. The first meeting will be in the first week of classes. I will ask you to set up an appointment at the first class meeting.

Tests and Final Exam

There will be two in-class tests and a final exam. The tests and exam will check your understanding of definitions and basic concepts, and they will include only a few simple proofs. Your main work with proofs will be in the homework assignments.

The in-class tests will be on Frday, September 27 and on Friday, November 1. The final exam will take place at the time scheduled by the Registrar, Saturday, December 14, at 1:30 PM. The final exam will be only a little longer than an in-class test.


Your numerical grade for the course will be determined as follows:

            First in-class test:      15%
            Second in-class test:     15%
            Final Exam:               20%
            Homework, etc.:           50%

Letter grades are assigned approximately as follows: 90-100: A; 80-89: B; 65-79: C; 55-64: D; 0-54: F. Grades near a cutoff get a + or -. (Note that I sometimes curve a numeric grade upwards when it seems that the grades do not reflect the level of ability of the class.)

Attendance Policy

It is important for you to be at every class and to arrive for class on time. I understand that that is not always possible. If exceptional circumstances force you to miss class, you should discuss with me your absence and the material that you missed. I do not count attendance explicitly in your grade for the course, but I do reserve the right to lower your grade because of excessive absences.

Statements from the Center for Teaching and Learning

At Hobart and William Smith Colleges, we encourage you to learn collaboratively and to seek the resources that will enable you to succeed. The Center for Teaching and Learning (CTL) is one of those resources: CTL programs and staff help you engage with your learning, accomplish the tasks before you, enhance your thinking and skills, and empower you to do your best. Resources at CTL are many: Teaching Fellows provide content support in 12 departments, Study Mentors help you manage your time and responsibilities, Writing Fellows help you think well on paper, Q Fellows support you in courses that require math, and professional staff help you assess academic needs.

Disability Accommodations: If you are a student with a disability for which you may need accommodations, you should self-identify, provide appropriate documentation of your disability, and register for services with Disability Services at the Center for Teaching and Learning (CTL). Disability related accommodations and services generally will not be provided until the registration and documentation process is complete. The guidelines for documenting disabilities can be found at the following website: Please direct questions about this process or Disability Services at HWS to Christen Davis, Coordinator of Disability Services, at or x 3351.

Office Hours, E-mail, WWW

My office is room 313 in Lansing Hall. My office phone extension is 3398. I am on campus most days, and you are welcome to come in anytime you can find me there. I will announce regular office hours and post them on my office door and on the course web page as soon as my schedule is determined, but note that your office visits are certainly not restricted to my regular office hours!

My e-mail address is The home page for this course on the web is located at This page will contain a weekly guide to the course, including homework assignments. You will want to bookmark this page! This course does not use Canvas.

Tentative Schedule

Here is a very tentative schedule for the course. I will try to keep approximately to this schedule, but the actual schedule will be posted weekly on the course web page. Tests will be given on the dates indicated, but might not cover exactly the material stated in this schedule.

Dates Topics and Readings
Aug. 26, 28, 30 Introduction to the course
Sets, Sections 1.1–1.6
Sept. 2, 4, 6 Sets, Sections 1.8–1.11
Extra: Topological Spaces
Sept. 9, 11, 13 Logic, Sections 2.1–2.7
Sept. 16, 18, 20 Logic, Sections 2.8–2.12
Direct Proof, Sections 4.1–4.3
Sept. 23, 25, 27 Direct Proof, Sections 4.4–4.5
Test on Friday, September 27, covering up to 4.3
Sept. 30; Oct. 2, 4 Proof by Contrapositive and Contradiction, Sections 5.1–6.3
Oct. 7, 9, 11 If-and-only-if and Existence Proofs, Sections 7.1–7.4
Advice about proofs, Section 6.4
Oct. 16, 18 Proofs with Sets, Sections 8.1–8.4
No class on Monday because of Fall Break
Oct. 21, 23, 25 Disproof, Sections 9.1–9.3
Mathematical Induction, Sections 10.1–10.3
Oct. 28, 30; Nov. 1 Mathematical Induction, Sections 10.4–10.5
Test on Friday, November 1, on Chapters 4–10
Nov. 4, 6, 8 Relations, Sections 11.1–11.4
Nov. 11, 13, 15 Relations, Sections 11.5–11.6
Extra: Some Group Theory
Nov. 18, 20, 22 Functions, Sections 12.1–12.6
Thanksgiving Break, November 25–29
Dec. 2, 4, 6 Cardinality and Infinity, Sections 14.1–14.4
December 9 Last Day of Class! Wrap up the course.
December 14 Final Exam: Saturday, December 14, 1:30 PM