Do you have a question or comment about the course? The answer might be in the course policies. If your question isn't answered in the course policies, please send me an email.

My office is **Room 204** in the Math Department. My office hours often change, so I maintain a calendar showing the times I will be available. I am often in my office outside of those hours, and I'll be happy to answer questions if you drop by outside of office hours, provided I am not busy with something else. I am also happy to make an appointment if my office hours are not convenient for you.

Goals: become familiar with the course policies

Reading: course policies

**Due Monday, August 22:** Please fill out this survey. This is not graded, but it is required!

Goals: make precise mathematical statements; start learning to read mathematics

Reading: Book of proof,
§2.1—2.4 (pp. 33-46), §2.7. There are occasional mentions of sets in
the reading; for the most part, you can ignore these right now, but you
may find it possible to understand some of them if you reinterpret *x ∈ ℤ* to mean "*x* is an integer" and *x ∈ ℕ* to mean "*x* is a positive integer".

Recommended exercises (important exercises in **bold**): §2.1, #1-3, 7, **9**, 11-13; §2.2, 1-5, **6**, 7, 8, 14; §2.3, **1-13**; §2.4, **1-5**.
The exercises in §2.3 and §2.4 are particularly important. They ask
you to rephrase mathematical statements, which will be a fundamental
skill when we need to find the right form of a statement to prove.

**Activity, Monday, August 22:** Exploration 1

Suggestions for successful communication. These are broad techniques that you can try to apply in discussions, in lecture, when reader, when writing, or in any other situation where you need to communicate mathematically (and maybe also non-mathematically).

**Due Wednesday, August 24:** Pick a game or sport that you enjoy
and carefully explain its rules. Your explanation should be a page or
less in length. The goal is not to explain every rule, especially in a
complicated game, but to explain yourself precisely. *You may not use the name of the game or any terminology that gives away what the game is!*

**Activity, Wednesday, August 24:** Exploration 2

**LaTeX workshop: Friday, August 26**, in a special location, **ECES 107**.
You may bring a laptop if you would prefer to use it instead of the
computers in the lab. Here is the document we will work with: [pdf] [tex] [overleaf]

**Due Monday, August 29:** In class on Wednesday, you received
one of your classmates' descriptions of a game. Please read what your
classmate wrote and comment on it. Look for places where the game
procedure is unclear. Are there any situations where the rules do not
explain what to do? Are there any situations where the rules tell you
to do two (or more) different things? Describe any such situations as
clearly as you can. Suggestion: If you want to refer to a specific
place on your classmate's paper, make a small symbol at that place on
their paper, and then use that symbol when referring to that place in
your comments.

**Quiz: Wednesday, August 31**, in the last 10 minutes of class.

**Independent submission suggestions:**

- Diagram some of the sentences from exercises in Chapter 2, §§ 2.1-2.3, and especially §2.9.
- Diagram some of the sentences from the sample proof in Exploration 4 or from the proofs you are submitting for Wednesday.
- Chapter 4 contains many mathematical sentences in its theorems and proofs. Try diagramming some of these.
- Make up some mathematical statements of your own and diagram them.

Goals: typeset mathematics using Latex; write direct proofs; identify assumed common knowledge

Reading: Book of proof, Chapter 4

Suggested exercises:

- Chapter 4 (pp. 100-101), #1-7,10,11,18,19 are good exercises in
setting up direct proofs. Figure out how to phrase these statements in
the form "∀
*x*,*P(x)*⇒*Q(x)*" and use direct proof. Remember that*x*∈ ℤ means "*x*is an integer." Can you find a way to use Exercises 4 and 5 to give a quick proof of 3? Can you use Exercise 6 and a proposition from the chapter to do Exercise 10 quickly? - You should get some practice setting up proofs by cases (Exercises 14-17), but once you get the idea, you don't gain much more by setting up more proofs by cases. Just do one or two of these exercises.
- Exercises 8, 9, and 26 are good, and I recommend them, but they may require some clever ideas in addition to proofwriting skills.
- I strongly recommend Exercises
**20**and**27**. These should not be the first exercises you try, because they are challenging, but I encourage you to work up to them. You will need to find a good formulation of these statements as implications to prove them.

Activity: Exploration 4

**Due Wednesday, September 7:** Complete the exercises on the second page of Exploration 4 [pdf] [tex] [overleaf]

Activity: Exploration 6

A **quiz** is likely on **Monday, September 12**. Expect to be asked to diagram sentences, set up direct proofs, and identify assumptions in mathematical writing.

**Due Wednesday, September 14:**

- Make a list of fundamental properties of, operations on, and relationships between integers. Your list should have about 10 to 20 entries. To find ideas, try looking for facts about the integers that we use without proof in class and in the textbook.
- Do one of Exercises 2, 4, 6, 12, 18 from Chapter 4 (p. 100) of
*Book of proof*. Write a clear and careful proof and make sure to indicate which facts about the integers from your list you used in your proof.

Goals: write proofs using induction

Reading: Book of proof, Chapter 10

Suggested exercises: In order to get practice with the form of induction proofs, you should do exercises like Chapter 10, #1-7, 15, 20, 34 (#8 is similar, but uses facts about the factorial that may not be familiar). These require you to prove *formulas*, which can sometimes be easier than proving more complex mathematical statements. Once you are comfortable with the form of induction proofs, you should do some of Chapter 10, #9-13, 16, 19, 22. Problems #25-28, 30 will ask you to use a recursive definition in your induction; this is a new skill to develop, so make sure to do at least one of these (#29 is similar, but requires you to know about binomial coefficients). I recommend taking a look at #30—you may find the formula surprising. Problems #32 and 33 are important because they use induction to prove things that are not easily expressed using formulas; please attempt these!

Activity for **Monday, September 19**: Exploration 8 [pdf] [tex] [overleaf]

**Due Friday, September 23**: Complete Question 5 of Exploration 8. Working in groups is strongly encouraged.

**Due Monday, September 26**: Complete Question 6 of Exploration 8. Working in groups is strongly encouraged.

Activity: Spherical graphs [pdf] [tex] [overleaf]

Reading for **Wednesday, October 12**: Make sure to read the discussions of strong induction and proof by smallest counterexample in §10.1 and §10.2 of Book of Proof.

**Due Friday, October 14**: Prove that if a spherical graph has *V* vertices, *E* edges, and *F* faces then *V - E + F = 2*.

Activity: The game of Nim

Activity: Euclid's algorithm

**Due someday:** Using Latex, write a paper that includes (i) an
explanation of the rules of the game of Nim, (ii) a careful statement of
who will win the game under optimal play, in terms of the initial
number of tokens in the heap, (iii) an explanation of the winning
strategy.

**Due someday:** Revise your paper about Nim, this
time including (iv) a convincing explanation of why the winning strategy
actually is a winning strategy.

Goals: become comfortable with the language of sets (including functions and relations); use sets to express mathematical ideas; write proofs about sets; use sets to prove statements about other mathematical objects

Reading for **Monday, October 31**: Book of proof, Chapters 1 and 8. You need to read all of both chapters, but focus at first on the concepts and operations (equality of sets, subset, union, intersection, difference, well ordering) introducted in §§1.1, 1.3, 1.5, and 1.9 and proofs involving those concepts in §§8.1—8.3.

Suggested homework for **Monday, November 7**: complete Exploration 20. Expect a quiz soon with a similar format.

Suggested exercises (as of **Friday, November 4**) for practice with set operations: You should be able to do all of the exercises in §§1.1, 1.3, although you don't need to do all of these exercises. Do enough to feel confident about doing the others. In §1.5, you should be able to do 1, 2, 7, and 8.

Activity for **Monday, November 7**: Exploration 18

Suggested homework for **Friday, November 11**: Practice writing proofs about sets. I strongly suggest doing §8, #1-5, 19-20, 26-28 (problems #26-28 are closely related to Exploration 18). Some of these can be tricky to do completely, but the most important part is to set up the proofs and figure out exactly what you would have to do to write a complete proof. Please also practice writing formal proofs about sets by doing a selection of 6, 8-9, 12-15, 21-22 (we did #6 in class on November 7). On these problems in particular, try to make your proofs as formally complete as possible, showing every step of the reasoning and referencing the relevant definitions.

Reading for **Monday, November 14**: Book of proof, Chapter 12. The text makes referece to relations, so you may wish to refer back to the beginning of Chapter 11, especially Definition 11.1. You don't need to read all of Chapter 11, yet.

**Due Friday, November 18**: Prove the following statement:

**Theorem:** If *a* and *b* are integers with greatest common divisor *d*, the sets *{ax + by : x,y ∈ ℤ}* and *{ n ∈ ℤ : d|n}* coincide.

Suggested exercises from §12.1 for **Monday, November 28**: You should be able to do all of the exercises in §12.1; it is very helpful to think carefully about what needs to be done in #9—12. Problem #11 uses the notation ℤ_{5}, which we have not introduced in class. You may replace ℤ_{5} with the set *{0,1,2,3,4}* for this problem.

Suggested exercises from §12.2 for **Monday, November 28**: It is essential to be able to do problems like #1—10 in §12.2. Problem #10 is particularly interesting and it (or a problem like it) may be discussed in class. Problems #11—13 and #18 are a little more challenging, but problems like these are likely on the final exam, so I encourage you to practice with them now.

Reading for **Monday, November 28**: Book of proof, §1.2, §1.3, §12.4—12.5 (pp. 208—214). Some of this this has been assigned earlier, but we have not used these topics much in class yet. We will make use of these ideas for the rest of the semester, though.

Suggested exercises from §12.4 and §12.5: The exercises in these sections are mostly practice with definitions and setting up proofs about functions. You should understand *what needs to be done* in all of these exercises, even if you get stuck on some of them. The textbook does not have many conceptual exercises in these sections, so make sure also to practice with problems that come from class. I suggest thinking about Problem 8 from §12.5, though (note that *X* means *ℤ - X* in this exercise).

Suggested exercises from §12.5:

Reading for **Wednesday, November 30**: Book of proof, §11.1—11.2 (pp. 175—183)

**Due Monday, December 5**: Solve the question in Exploration 23 [pdf] [tex] [overleaf]. Note: this will be accepted late.

**Due Friday, December 9**: Either revise your proof of the following statement:

**Theorem.** If *a* and *b* are integers with greatest common divisor *d*, the sets *{ax + by : x,y ∈ ℤ}* and *{ n ∈ ℤ : d|n}* coincide.

**Definition.** For each integer *n*, let *ℤ/nℤ* be the set of equivalence classes of the equivalence relation *R = { (x,y) ∈ ℤ × ℤ : n|x - y}*.

**Theorem.** Suppose that *n* is an integer. The functions *a : ℤ/nℤ × ℤ/nℤ → ℤ/nℤ* and *m : ℤ/nℤ × ℤ/nℤ → ℤ/nℤ* defined by *a([x],[y]) = [x + y]* and *m([x],[y]) = [xy]* are well-defined.

Extra question (not required): We usually write *[x][y]* instead of *m([x],[y])* and call it *multiplication modulo n*. For which values of *n ∈ ℤ* is it possible to *divide* modulo *n*? That is, for which *n ∈ ℤ* is it possible to solve the equation *[x][y] = [1]* for every *[x] ∈ ℤ / n ℤ* except *[x] = [0]*?

Goals: relate set operations to numbers

Reading: Book of proof, Chapters 3 and 13

Activity: The game of set

Activity: Hilbert's hotel

Quiz 1

Quiz 2

Quiz 3

Quiz 4

Quiz 5

Quiz 6

Quiz 7

Quiz 8

Quiz 9

Quiz 10

Quiz 11

Midterm - Part I

Midterm - Part II

Exploration 1

Exploration 2

Exploration 3

Exploration 4

Exploration 6

Exploration 7

Exploration 8

Exploration 9

Exploration 10

Exploration 11

Exploration 12

Exploration 13

Exploration 14

Exploration 15

Exploration 16

Exploration 17

Exploration 18

Exploration 19

Exploration 20

Exploration 21

Exploration 22

Exploration 23

There will be one midterm exam and one final exam, taken outside of class. These will all involve a group work component.

Midterm: Tuesday, Oct. 25, 6—8pm in DUAN G125

Final: Sunday, Dec. 11, 7:30—10pm in ECCR 151

A reference discussing most of the important aspects of LaTeX you will use.

A very quick introduction to LaTeX.

If you need to know the command for some symbol, try using Detexify.