A little introduction to cardinality.

Lectured on the rest of inverse functions.

Great day with great discussions.

• At the board: 7.20, 7.23(a,b,c), 7.25, 7.26

Best. Day. Ever. Tons of good conversation that culminated with a question, a conjecture, and a proof. And that, is what math is all about! Extra special thanks to HF and NH!!

• At the board: 7.8, 7.16(e,f), 7.19(a,c), Additional Problem #1

• To turn in:
  • Proofs/solutions (or your best efforts) for: 7.20, 7.21, 7.23, 7.25, 7.26

More one-to-one and onto

• At the board: 7.13(c,d), 7.16(c,d)

• To turn in:
  • Proofs/solutions (or your best efforts) for: 7.19 and Additional Problem #1: find a bijection from $[1,3]$ to $[2,10]$, where $[1,3]$ and $[2,10]$ are intervals of real numbers.
  • Redo and turn in problems from last time: 7.8, 7.13(c,d), 7.15(a,b), 7.16(c,d,e,f).
• Hint:
  1. on the additional problem, your function should be something you can graph—think of lines.

Functions: one-to-one and onto

• At the board: 7.7, 7.13(a)(b), 7.15(c)(d), 7.16(a)(b)

• To turn in: Proofs/solutions (or your best efforts) for: 7.8, 7.13(c,d), 7.15(a,b), 7.16(c,d,e,f).
• Clarification:
  1. in Exercise 7.16, make sure to either prove or provide a counterexample to support your answer. If you need to prove a property, please make it a formal (but perhaps short) proof.

Exam 2. Finished. Happy Thanksgiving!

• To turn in: Proofs/solutions (or your best efforts) for: 7.7, 7.13(a)(b), 7.15(c)(d), 7.16(a)(b).
• Clarification:
  1. in Exercise 7.16, make sure to either prove or provide a counterexample to support your answer. If you need to prove a property, please make it a formal (but perhaps short) proof.

Functions

• At the board: 7.3, 7.5

• Info: See the Other Handouts section for an outline.

Even more modular arithmetic and divisibility by $3$.

• At the board: 6.80(a), 6.82, 6.83, 6.84

• To turn in: Proofs/solutions (or your best efforts) for: 7.3, 7.5.
• Review: Also bring any questions you have about the material that will be on the exam. Concrete problems (from our notes or from the review sheet) are usually the best.

More modular Arithmetic.

• At the board: 6.70, 6.71, 6.72, 6.78, 6.79

• To turn in: Proofs/solutions (or your best efforts) for: 6.80(a), 6.82, 6.83.
• Clarification: The point of 6.80 is that Definition 6.78 only tells you how to add or multiply two elements of $\mathbb{Z}/m\mathbb{Z}$ at a time. Now you want to add or multiply several at at time. Try using induction.

Modular Arithmetic.

• At the board: 6.52, 6.63, 6.64, 6.66, 6.69

• To turn in: Proofs/solutions (or your best efforts) for: 6.70, 6.71, 6.72, 6.78. (You need to download the notes for Chapter 6 again to get the section on modular arithmetic.)
• Hint: On 6.78, first practice multiplying in $\mathbb{Z}/6\mathbb{Z}$. For example, $[2]_6\cdot [5]_6 = [10]_6$. Also, $[10]_6 = [4]_6$ since $10-4$ is divisible by $6$. Thus, $[2]_6\cdot [5]_6 = [4]_6$. Similarly, $[5]_6\cdot [5]_6 = [1]_6$; do you see why?

Partitions.

• At the board: 6.40(b), 6.44(g), 6.48, 6.49, 6.50

• To turn in: Proofs/solutions (or your best efforts) for: 6.52, 6.63, 6.64, 6.66, 6.69. (You need to download the notes for Chapter 6 again to get the section on modular arithmetic.)
• Hint: On 6.64, translate what you want to prove into the language of divisibility, and then try to make use of the results you proved in Chapter 2.

Even more equivalence relations.

• At the board: I lectured over parts of 6.37, 6.41, 6.44, and 6.39

• To turn in: Proofs/solutions (or your best efforts) for: 6.40(b), 6.44 - only part (g) of 6.32, 6.48, 6.49, 6.50.
• Hint: In Theorem 6.40(b), you want to prove an "or" statement. A typical way to prove "$P$ or $Q$" is as follows: if $Q$ is true, then the statement "$P$ or $Q$" is automatically true, so we may assume that $Q$ is false and then use that to show that $P$ is true. Thus, you may assume that $[x]\cap[y] \neq \emptyset$ and use that to prove that $[x]=[y]$. Now, if $[x]\cap[y] \neq \emptyset$, then there exists some $c\in A$ such that $c \in [x]$ and $c \in [y]$. What does that tell you? (Theorem 6.39 may be helpful.)

More Equivalence Relations.

• At the board: parts of parts of 6.37, 6.38, parts of 6.39

• To turn in: Nothing!
• Reading: please (re)read from 6.33 through 6.44. You are high encouraged to work on 6.39, 6.40, and 6.44. I will use most of Monday's class to review these problems and introduce section 6.3.

Equivalence Relations.

• At the board: parts of 6.27, 6.30, parts of 6.32, parts of 6.37

• To turn in: Proofs/solutions (or your best efforts) for: 6.36, 6.37, 6.38, 6.39.
• Clarification:
  1. Please redo 6.37 even if you completed it before. Also a hint: the answer to "How many different sets of relatives are there?" is a number between 1 and 10.
  2. I will only grade your effort on 6.39, so please give it your all.

Relations.

• At the board: 6.20, 6.25, 6.26, parts of 6.27

• To turn in: Proofs/solutions (or your best efforts) for: 6.30, 6.32, 6.37. Also, you may redo Homework 24 (or just turn in what you had from last time)
• Hint:
  1. Read the directions for Exercise 6.32; make sure to use (and reference) your answers to 6.27.

Irrationality of $\sqrt{p}$ and relations.

• At the board: 5.17, 6.8, 6.9, 6.13, 6.19

• To turn in: Proofs/solutions (or your best efforts) for: 6.20, 6.25, 6.26, 6.27
• Hint:
  1. Read and reread Definition 6.23. Does Example 6.24 make sense?
  2. Here are more details Example 6.24(a). You know $1 \lt 2$ but $2 \nless 1$, so this example proves (via a counterexample) that $\lt$ is not symmetric. You also know that $1 \nless 1$, so $\lt$ is not reflexive. However, it is true that for all $x,y,z\in \mathbb{R}$, if $x\lt y$ and $y\lt z$ then $x \lt z$; this proves that $\lt$ is transitive.

Irrationality of $\sqrt{2}$ and the Well-Ordering Property.

• At the board:5.16, 5.4

• To turn in: Proofs/solutions (or your best efforts) for: 5.17, 6.8, 6.9, 6.13, 6.19
• Clarification:
  1. In proving Theorem 5.17, you are free to use theorems from section 5.1, even if we didn't prove them. Theorem 5.12 will probably be useful.
  2. For 6.13, you want to imagine the sets lying in $\mathbb{R} \times \mathbb{R}$, where the first coordinate is the $x$-coordinate and the second is the $y$-coordinate. Your answers could be given in words and/or pictures.

More complete induction

• At the board:4.20, 4.27, 4.29

• To turn in: Proofs/solutions (or your best efforts) for: 5.16, 6.5, 6.6
• Reading : Please read Theorem 4.3.4 about the Well-Ordering Principle. Also, make sure to read the beginning of Section 5.2 before starting 5.16.
• Clarification:
  1. For Theorem 5.16, I will only grade whether or not you thought about it. Make sure to write down your ideas! Also, make sure to read the footnote for the theorem.
  2. I will be giving a mini-lecture on the Well-Ordering Principle after we discuss the problems to turn in.

More Induction, and intro. to complete induction

• At the board:4.16, 4.19, 4.22

• To turn in: Proofs/solutions (or your best efforts) for: 4.20, 4.27, 4.29
• Hint:
  1. If you get stuck on any of these, make sure to clearly outline your proof by induction and carefully check the base case (called the base step in the notes).

More Induction.

- Rainbow induction. -

• At the board:4.4, 4.5, 4.6, 4.15

• To turn in: Proofs/solutions (or your best efforts) for: 4.16, 4.19, 4.22
• Hint:
  1. If you get stuck on any of these, make sure to clearly outline your proof by induction and carefully check the base case (called the base step in the notes).

Induction.

• At the board:Extra Problem #1, 4.13

• To turn in: Proofs/solutions (or your best efforts) for: 4.4, 4.5, 4.6, 4.15
• Hint:
  1. If you get stuck on any of these, make sure to clearly outline your proof by induction and carefully check the base case (called the base step in the notes).

Unions and intersections and the beginning of induction.

• At the board:3.26, 3.39
• Mini-lecture:Introduction to induction (to be continued...)

• To turn in: Proofs/solutions (or your best efforts) for the following problem:

  Extra Problem #1: Define $A_n = \{x\in \mathbb{R}\mid -n < x \le 2+\frac{1}{n}\}$. In other words, $A_n$ is the interval of real numbers $(-n,2+\frac{1}{n}]$.

  1. Find $A_1 \cap A_2$
  2. Find $\bigcap_{n\in \mathbb{N}}A_n$
  3. Find $A_1 \cup A_2$
  4. Find $\bigcup_{n\in \mathbb{N}}A_n$

• Reading : Make sure to read and reread section 4.1 up through Theorem 4.5. Try hard to get Theorems 4.4 and 4.5 started. This is really important!

Big unions and intersections.

• At the board:3.27, 3.37, 3.38, 3.41, 3.42

• To turn in: Proofs/solutions (or your best efforts) for: 3.26, 3.39, meditations on 4.4
• Reading : Make sure to read and reread section 4.1 up through problem 4.4. This is really important!
• Clarification:
  1. For Theorem 4.4, I will only grade whether or not you thought about it. Please make sure to come with lots of questions!
  2. I will be giving a mini-lecture on induction after we deal with 3.26 and 3.39.

Back to set theory.

• At the board:3.21(b), 3.25, 3.28, 3.35, 3.36

• To turn in: Proofs/solutions (or your best efforts) for: 3.27, 3.37, 3.38, 3.41, 3.42
• Hint:
  1. Make sure to (re)read all of the text at the beginning of Section 3.3.
• More Reading (same as last time): The text between Problem 3.28 and Problem 3.31. Then meditate on the following problem: if $U$ is the collection of all sets and $S = \{A\in U\mid A\notin A\}$, then which is true: $S\in S$ or $S\notin S$?

Exam 1. Done.

• The exam.

• To turn in: Proofs/solutions (or your best efforts) for: 3.21(b), 3.28, 3.35, 3.36
• Hint:
  1. Make sure to read all of the text at the beginning of Section 3.3 before starting 3.35.
• More Reading: The text between Problem 3.28 and Problem 3.31. Then meditate on the following problem: if $U$ is the collection of all sets and $S = \{A\in U\mid A\notin A\}$, then which is true: $S\in S$ or $S\notin S$?

The power set.

• At the board:3.20(a), 3.21(a), 3.25 (half of it)

• Info: See the Other Handouts section for an outline.

The power set.

• At the board:3.20(a) - started

• To turn in: Proofs/solutions (or your best efforts) for: 3.20(a), 3.21(a), 3.23, 3.25
• Clarification:
  1. This is an opportunity to completely redo the problem set from last time. Don't for get to start preparing for the exam too!

More set theory.

- Proving theorems. -

• At the board: 3.16, 3.17, 3.19

• To turn in: Proofs/solutions (or your best efforts) for: 3.20(a), 3.21(a), 3.23, 3.25
• Clarification:
  1. For Problem 3.25, I will only grade you on effort (for now). Please make sure to try hard on it, come with lots of questions, and be ready to present what you have.

Starting basic set theory.

• At the board: 3.4, 3.5, 3.9, 3.10, 3.15

• To turn in: Proofs/solutions (or your best efforts) for: 3.16, 3.17, 3.19
• Clarification:
  1. The sets in 3.16, are intervals of real numbers. Try to write your answers to this exercise in interval notation or using set-builder notation.
  2. Your goal in 3.17 is to prove $B^c\subseteq A^c$. So, using the hypothesis, you want to prove that for all $x\in B^c$, it follows that $x\in A^c$. As such, your proof should begin with "Let $x\in B^c$" and end with "Thus, $x\in A^c$."
  3. Before proving Theorem 3.19, make sure to read the text right before it. Your proof should have two parts (visually, two paragraphs). In one part you prove $A\setminus B \subseteq A \cap B^c$, and in the other part you prove $A \cap B^c \subseteq A\setminus B$.

Evaluating the truth of quantified statements.

• In groups: 2.69-2.80
• At the board: 2.69, 2.79, 2.80

• To turn in: Proofs/solutions (or your best efforts) for: 3.4, 3.5, 3.9, 3.10, 3.15
• Clarification:
  1. To clarify the directions for 3.4, here are two possible answers for part (a). Answer 1: $A = \{3,6,9,12,\ldots\}$. Answer 2: $A$ is the set of all natural numbers that are divisible by $3$.
  2. For 3.9, you should find that there are 8 different subsets.
  3. Before proving Theorem 3.10, make sure to read the text before Problem 3.7. Your goal in 3.10 is to prove $A\subseteq C$. So, using the additional hypotheses, you want to prove that for all $x\in A$, it follows that $x\in C$. As such, your proof should begin with "Let $x\in A$" and end with "Thus, $x\in C$."

Negating quantifiers.

• In groups: 2.68-2.74
• At the board: 2.68, 2.73, 2.74

• To turn in: Proofs/solutions (or your best efforts) for: 2.79, 2.80
• Clarification:
  1. There is a small something missing in the statement of Theorem 2.80. It should read "For all integers $n$, $3n^2 + n +14$ is even."
  2. The sentence before Theorem 2.80 suggests considering two cases. So, if you are having trouble proving the theorem all at once, try splitting your proof into two case: when $n$ is even and when $n$ is odd.

More quantifiers.

• In groups: 2.61-2.67
• At the board: 2.65, 2.66, 2.67

• To turn in: Proofs/solutions (or your best efforts) for: 2.68, 2.69, 2.73, 2.74
• Hint:
  1. Make sure to read (and reread) all of the words between the exercises. The ones between 2.73 and 2.74 are quite important.
  2. Again and again, quantifiers can be tricky; we will talk even more about these in class.

Remember, if you use any resources other than our notes and each other, you need to write these on the top of your homework. (See the syllabus for more details.)

Contradiction and quantifiers.

- Rainbow math -

• In groups: 2.55-2.62
• At the board: 2.55, 2.57, 2.58, 2.60

• To turn in: Proofs/solutions (or your best efforts) for: 2.63, 2.65, 2.66, 2.67
• Hint:
  1. Make sure to read (and reread) all of the words between the exercises.
  2. Again, quantifiers can be tricky; we will talk more about these in class.

Remember, if you use any resources other than our notes and each other, you need to write these on the top of your homework. (See the syllabus for more details.)

Contrapositive, Vader, and Palin.

• In groups: 2.43-2.52
• At the board: 2.43, 2.44, 2.45, 2.46, 2.48, 2.51

• To turn in: Proofs/solutions (or your best efforts) for: 2.55, 2.57, 2.58, 2.60, 2.61, 2.62,
• Hint:
  1. Remember to read the sentence right before 2.55. This is likely your first exposure to proof by contradiction. Read 2.53 and 2.54 carefully, but definitely come with questions on Monday.
  2. Also, be aware that quantifiers can be tricky, and the order in which they occur is extremely important.

Remember, if you use any resources other than our notes and each other, you need to write these on the top of your homework. (See the syllabus for more details.)

More Logic. Had great discussion about the contrapositive and negating conjunctions.

• In groups: 2.25-2.32
• At the board: 2.35, 2.36, 2.38, 2.41

• To turn in: Proofs/solutions (or your best efforts) for: 2.43, 2.44, 2.45, 2.46, 2.48, 2.51,
• Hint:
  1. Remember that the negation of "$x$ and $y$ are odd" is "$x$ or $y$ is even." Similarly, the negation of "$x$ or $y$ is even" is "$x$ and $y$ are odd."
  2. To do Exercises 2.48(b), you can use Corollary 2.47 (even though you are not ask to prove it).

Remember, if you use any resources other than our notes and each other, you need to write these on the top of your homework. (See the syllabus for more details.)

Logic.

• In groups: 2.25-2.32
• At the board: 2.28, 2.31

• To turn in: Proofs/solutions (or your best efforts) for: 2.34, 2.35, 2.36, 2.38, 2.40, 2.41, 2.43
• Hint: Read the sentence before Theorem 2.34. You can prove that two propositions are logically equivalent by constructing truth tables for each one and showing that they are the same. But, make sure to use your words like "consider the following truth tables" or "since the columns for $\neg(A\wedge B)$ and $\neg A \vee \neg B$ are identical..."

Remember, if you use any resources other than our notes and each other, you need to write these on the top of your homework. (See the syllabus for more details.)

You all. Are. Awesome.

• In groups: 2.21-2.25
• At the board: 2.18, 2.19, 2.22, 2.23

• Reading: Appendix C: Definitions in Mathematics
• To turn in: Proofs/solutions (or your best efforts) for: 2.25, 2.27, 2.28, 2.31, 2.32
• Challenge (but not for credit): Look at Theorem 2.23 again. Let's try to generalize it. Is it true that the for all $n\ge 2$, the sum of $n$ integers is divisible by $n$? If so, prove it. If not, can you determine for which values for $n$ it is true? (For example, Theorem 2.23 says that the statement is true when $n=3$, and it's also not hard to prove that it's true for $n=2$.)

Remember, if you use any resources other than our notes and each other, you need to write these on the top of your homework. (See the syllabus for more details.)

Another complete-sentences day with several good discussion surrounding this. Four stellar presentations by PA, JE$_1$, SK, and VT, and many comments from the rest. I'm loving it! ...hope you are too ☺

- So much beautiful math! -

• In groups: 2.14, 2.15, 2.16, 2.17, 2.18, 2.19, 2.20
• At the board: 2.14, 2.15, 2.16, 2.17

• Reading: Appendix B: Fancy Mathematical Terms
• To turn in: Proofs/solutions (or your best efforts) for: 2.21, 2.22, 2.23

Ramped up the math today, and had the first (of many) complete-sentences day. We were treated to five outstanding presentations by AL, AK, DL, PC, and AF, and quite a few people chimed in to create some nice discussions. Great job all around! And my apologies to AF for the rush at the end! Here are a few photos—sorry AL, the picture of your proof was super blurry ☹

- We do math here! -

• In groups: 2.7, 2.8, 2.9, 2.12, 2.13
• At the board: 2.7, 2.8, 2.9, 2.12, 2.13

• Reading: Appendix A: Elements of Style for Proofs: Item 11 through the end
• To turn in:
  1. Proofs/solutions (or your best efforts) for: 2.14, 2.15, 2.16, 2.17, 2.18, 2.19, 2.20

Yay math! Three great presentations today by HF, NS, and CE! There were lots of excellent talking points—many thanks for their patience and bravery as we talked through the details. And many thanks to the rest of the class for such a high level of participation on just the second day of class.

• In groups: 2.3-2.7
• At the board: 2.3, 2.4, 2.5

• Reading: Appendix A: Elements of Style for Proofs: the beginning through item 10 (i.e. the first two pages)
• To turn in:
  1. Short paragraph addressing one item from your reading from Appendix A that you had not though about before and/or did not think was important.
  2. Proofs of (or your best effort to prove) Problems 2.7, 2.8, 2.9, 2.12, 2.13 (from Chapter 2 of the course book)

Great to meet you all! Make sure to sign up for a ShareLaTeX account and get started on Writing Assignment 01. It's due this coming Friday!

• Reading: Sections 1.1-1.3 of the course book
• To turn in:
  1. Short paragraph about one aspect of an IBL approach that you view as positive and one concern that you have.
     (This is based off of the 1.1-1.3 reading.)
  2. Proofs of (or your best effort to prove) Theorem 2.3 and Theorem 2.4 (from Chapter 2 of the course book)