This course ended on
December 13, 2019

Math 135: First Steps into Advanced Mathematics

       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

Fifteenth Week: December 9 and Final Exam

Monday, December 9, is the last day of class. The final exam will be given in our usual classroom on Saturday, December 14, at 1:30 PM.

Study Guide for the Exam

You might want to review the Test 1 Study Guide and the Test 2 Study Guide.

Here are my sample solutions to Homework 10.

I will have the following office hours during reading period and exams, and I might well be in my office at other times as well:

           Monday, December 9:     10:15 to 10:50 AM, 12:15 to 1:15 PM
           Wednesday, December 11: 12:00 to 3:00 PM
           Friday, December 13:    12:00 to 3:00 PM
           Saturday, December 14:  11:00 AM to 1:20 PM

Homework 11 is due in class on Monday, December 9. If you want to do rewrites on Homework 11, you can pick up the graded homework during my office hours on Wednesday and turn in your rewrites during my office hours on Friday, December 13.

Here are my sample solutions to Homework 11.

Fourteenth Week: December 2, 4, and 6

In this last full week of classes, we will finish Chapter 14, and we will look briefly at some basic group theory (unless we decide on some other short topic).

The following homework is due in class on Monday, December 9:

Homework #11

Thirteenth Week: November 18, 20, and 22

We will finish Chapter 12 this week and then move on to Chapter 14, which covers cardinality, especially infinite cardinalities. We will also be discussing what we should do during the last week of classes.

The following homework is due in class on Wednesday, November 20:

Homework #10

Here are my sample answers to Homework 9.

There are no classes for the week of November 25 because of Thanksgiving. Happy Thanksgiving! Enjoy the break!

Twelfth Week: November 11, 13, and 15

Having finished Chapter 11 last week, we will move on to Chapter 12, which covers functions and their properties. You should read all of Chapter 12, although we might not finish it until next week.

The following homework, #9, from Chapter 11 is due in class on Wednesday, November 13:

           Section 11.1,  Exercises  2, 4, 10
           Section 11.2,  Exercise   2
           Section 11.3,  Exercises  2, 4, 10, 12
           Section 11.4,  Exercises  2, 6

Eleventh Week: November 4, 6, and 8

This week we begin the fourth part of the textbook, moving a general consideration of proof techniques to a look at some specific mathematical ideas. The topic for the week is relations, especially equivalence relations. The reading is Chapter 11. You should read the entire chapter, although we will probably still be working on it at the beginning of next week.

Here are my sample answers to Homework 8.

Tenth Week: October 28 and 30; November 1

There is a test this week, on Friday, November 1, covering Chapters 4 through 10. A study guide will be handed out in class on Monday.

We will continue with Chapter 10 on Monday. On Wednesday, we will review for the test and will finish up any part of Chapter 10 that we have not yet covered. If there is extra time, we might start work on Chapter 11.

Homework 8 is due in class on Wednesday. Sample solutions tot he homework will be available at that time.

Ninth Week: October 21, 23, and 25

The topic for the week is mathematical induction, and the reading is Chapter 10. There is a test next week, on Friday, November 1, covering Chapters 4 through 10.

The following homework is due in class next Wednesday. Because of the test next Friday, I will make my solutions to this homework available when I collect the homework, so no rewrites will be possible on this homework.

           Homework #8, due Wednesday October 30:
                Chapter 10, Exercise # 2, 8, 10, 18, 34
                Two additional problems from this handout

Here are my Answers to Homeworks 5 and 6.

And here are my Answers to Homework 7.

Eighth Week: October 16 and 18

There is no class on Monday this week because of Fall break.

You should read Chapter 8, Sections 1 to 3, and Chapter 9. I will not cover Section 8.4, but you are certainly encouraged to read it for your own interest. The following problems from the textbook are due in class on Friday, October 18:

       Homework #7:
            Chapter 7 exercises, # 2, 8, 12, 16, 32
            Chapter 8 exercises, # 4, 10, 16, 22

Once we get through Chapter 9, we will spend a little time reviewing all the very general proof techniques from Chapters 4 through 9 before going on to Chapter 10.

Seventh Week: October 7, 9, and 11

The reading for the week is Chapter 7 and starting Chapter 8. We will continue talking about proof techniques. Chapter 7 covers "if and only if" proofs and existence proofs. Chapter 8 discusses how to prove things about sets.

Your LaTeX homework is due on Monday. There is also a written homework assignment due on Wednesday:

Homework #6

Sixth Week: September 30; October 2 and 4

We will continue studying proof techniques for the next few weeks. The reading for the week is Chapters 5 and 6 in the textbook, which cover proof by contrapositive and proof by contradiction.

But we will also be doing something different at the beginning of this week. We will look at the LaTeX typesetting system, which is used to produce high-quality mathematical texts such as textbooks and journal articles, as well as many of the handouts for this course. I will ask every student to get a free account on the on-line LaTeX web site,, and to use it to write some mathematics, probably including your next homework assignment.

The next homework assignment is due next Monday, October 7. It will be written in LaTeX on The assignment was handed out in class, and you can get a copy here:

A Guide to LaTeX, written by Professor Alden Gassert, will also be handed out in class. Note that the features that are documented as requiring Prof. Gassert's TeX file are not available for the first homework assignment. Also, note that there is a lot of information in the guide that you won't need yet.

Fifth Week: September 23, 25, and 27

There is a test on Friday, September 27. A study guide for the test will be handed out in class on Monday. The following homework is due on Wednesday:

Homework #4

We will spend Monday working on proofs and Wednesday doing review for the test. You can read the rest of Chapter 4, Sections 4 and 5. We might or might get to those Sections or Wednesday but in any case, they are not on the test.

My answers for Homework #1 and my answers for Homework #3 are available here.

Fourth Week: September 16, 18, and 20

The homework problems from the textbook that were assigned last week [see below] are due on Wednesday.

The reading for the week is to finish Chapter 2 and to read Sections 1 through 3 in Chapter 4. Note that we will not cover Chapter 3. We will finish up the general introduction to formal logic from Chapter 2. We will then move on to Chapter 4, which begins the discussion of proof and methods of proof. In particular, Section 4.3 covers the most important single proof technique: direct proof.

There is a test coming up at the end of next week, on Friday, September 27. It will cover everything up through Section 4.3, but will not cover any material beyond that that we do next week.

Homework for next Wednesday will be posted soon.

Third Week: September 9, 11, and 13

Homework #2 is due on Wednesday.

The reading for the week is Chapter 2, Sections 1 through 7. This week, we begin our study of mathematical logic. We will look at propositional logic, which concerns statements that can be either true or false, and logical operations such as AND, OR, NOT, and IMPLIES. Hopefully, by the end of the week, we will get to Section 7, which introduces the quantifiers FOR ALL and THERE EXISTS.

The following homework from the textbook is due in class next Wednesday:

       Section 2.1, Exercises 2, 4, 8, 10  — if it's not a statement, explain why
       Section 2.3, Exercises 2, 4, 10  — no explanation necessary
       Section 2.4, Exercises 2, 4   — no explanation necessary
       Section 2.6, Exercises 2, 8, 10, 12, 14  — some explanation necessary!
       Section 2.7, Exercises 2, 4, 6  — give English translation and why it is true or false

(Note: There won't be any rewrites for this homework, since there are no "proofs".)

Second Week: September 2, 4, and 6

We will finish Chapter 1 early in the week. For the rest of the week, we will work on a handout about topology. In particular, we will be looking at the basic definitions in a branch of topology known as point-set topology, which will allow us to apply some of the things that we have been learning about sets.

You should finish reading Chapter 1 by Monday. For Wednesday, you should read the handout about topology, which will be distributed in class and is also available through this link:

Handout 1: Topology

This homework on the topology handout is due in class next Wednesday (changed from the original due date):

Homework #2

I will need to leave my Thursday office hours early this week, at 1:50. However, I will be available earlier in the day from 10:00 to 12:00.

First Week: August 26, 28, and 30

Welcome to the course!

As a reading assignment for the week, you should read Chapter 1 in the textbook, although we might not finish the entire chapter this week in class. You should also read the course syllabus. You should download the free PDF version of the textbook, available here:

Book Of Proof (PDF)

(There is also a print version if you would prefer that.)

The first homework assignment will be handed out in class on Wednesday.