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:
Goals: typeset mathematics using Latex; write direct proofs; identify assumed common knowledge
Reading: Book of proof, Chapter 4
Suggested exercises:
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:
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 ℤ5, which we have not introduced in class. You may replace ℤ5 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.
or prove the theorem below: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.