On each homework assignment, please write (i) your name, (ii) name of course, and (iii) homework number. You are allowed and encouraged to work together on homework. Yet, each student is expected to turn in their own work. In general, late homework will not be accepted. However, you are allowed to turn in **up to three late homework assignments with no questions asked**. Unless you have made arrangements in advance with me, homework turned in after class will be considered late. When doing your homework, I encourage you to use the Elements of Style for Proofs (see Appendix B of the course notes as a reference.

The following assignments are due at the beginning of the indicated class meeting. However, most assignments will be collected at the end of the class meeting. I reserve the right to modify the assignment if the need arises. These exercises will form the basis of the student-led presentations. Daily assignments will be graded on a $\checkmark$-system. During class, **you are only allowed and encouraged to annotate your homework using the colored marker pens that I provide**.

**Daily Homework 1:**Read the syllabus and write down 5 important items.*Note:*All of the exam dates only count as a single item. Turn in on your own paper at the beginning of class. (Due Friday, January 19)**Daily Homework 2:**Read and sign the Student Contract. (Due Friday, January 19)**Daily Homework 3:**Stop by my office (AMB 176) and say hello. If I’m not there, just slide note under my door saying you stopped by. (Due by 5PM on Friday, January 19)**Daily Homework 4:**Read all of Chapter 1: Introduction of the course notes. In addition, complete 2.1-2.11 in Chapter 2: An Introduction to Groups and digest the surrounding text along the way. (Due Friday, January 19)**Daily Homework 5:**Skim through Appendix A: Elements of Style of Proof of the textbook. In addition, complete 2.12-2.14, 2.17-2.19 in Chapter 2: An Introduction to Groups and digest the surrounding text along the way. (Due Monday, January 22)**Daily Homework 6:**Skim through Appendix B: Fancy Mathematical Terms of the textbook. In addition, complete 2.20-2.24, 2.26 in Chapter 2: An Introduction to Groups and digest the surrounding text along the way. (Due Wednesday, January 24)**Daily Homework 7:**Skim through Appendix C: Definitions in Mathematics of the textbook. In addition, complete 2.27-2.29, 2.31, 2.32, 2.34-2.38 in Chapter 2: An Introduction to Groups and digest the surrounding text along the way. (Due Friday, January 26)**Daily Homework 8:**Complete 2.39-2.43 in Chapter 2: An Introduction to Groups and digest the surrounding text along the way. (Due Monday, January 29)**Daily Homework 9:**Complete 2.44, 2.45, 2.47, 2.48, 2.50, 2.53-2.55 in Chapter 2: An Introduction to Groups and digest the surrounding text along the way. (Due Wednesday, January 31)**Daily Homework 10:**Complete 2.56-2.64 in Chapter 2: An Introduction to Groups and digest the surrounding text along the way. (Due Friday, February 2)**Daily Homework 11:**Complete 2.65-2.67, 2.69, 270(abc) in Chapter 2: An Introduction to Groups and digest the surrounding text along the way. (Due Monday, February 5)**Daily Homework 12:**Complete 2.70(defg) and 2.71 in Chapter 2: An Introduction to Groups and digest the surrounding text along the way. (Due Wednesday, February 7)**Daily Homework 13:**Complete 3.13-3.20 in Chapter 3: Subgroups and Isomorphisms and digest the surrounding text along the way. (Due Friday, February 16)**Daily Homework 14:**Complete 3.21 and 3.22 in Chapter 3: Subgroups and Isomorphisms and digest the surrounding text along the way. (Due Monday, February 19)**Daily Homework 15:**Complete 3.23, 3.25-3.32 in Chapter 3: Subgroups and Isomorphisms and digest the surrounding text along the way. (Due Wednesday, February 21)**Daily Homework 16:**Complete 3.33-3.40 in Chapter 3: Subgroups and Isomorphisms and digest the surrounding text along the way. (Due Friday, February 23)**Daily Homework 17:**Complete 3.41-3.45, 3.47-3.50 in Chapter 3: Subgroups and Isomorphisms and digest the surrounding text along the way. (Due Monday, February 26)**Daily Homework 18:**Complete 3.51-3.55 in Chapter 3: Subgroups and Isomorphisms and digest the surrounding text along the way. (Due Wednesday, February 28)**Daily Homework 19:**Complete 3.56-3.61 in Chapter 3: Subgroups and Isomorphisms and digest the surrounding text along the way. (Due Friday, March 2)**Daily Homework 20:**Complete 3.62-3.65 in Chapter 3: Subgroups and Isomorphisms and 4.1-4.4, 4.6-4.8, 4.11-4.15 in Chapter 4: Families of Groups. (Due Monday, March 5)**Daily Homework 21:**Complete 4.17-4.20 in Chapter 4: Families of Groups. (Due Wednesday, March 7)**Daily Homework 22:**Complete 4.44, 4.45, one of 4.48-4.49, one of 4.51-4.53, and think about 4.55-4.57 in Chapter 4: Families of Groups. (Due Wednesday, March 28)**Daily Homework 23:**Complete 4.58-4.60, 4.62, 4.63, 4.65-4.70 in Chapter 4: Families of Groups. (Due Friday, March 30)**Daily Homework 24:**Complete 4.71-4.84 in Chapter 4: Families of Groups. (Due Monday, April 2)**Daily Homework 25:**Complete 4.85-4.90, 4.94, 4.95 in Chapter 4: Families of Groups. (Due Wednesday, April 4)**Daily Homework 26:**Complete 4.96-4.98, 4.100-4.104 in Chapter 4: Families of Groups. (Due Friday, April 6)**Daily Homework 27:**Complete 4.106, 4.107, 4.110, 4.111 in Chapter 4: Families of Groups and then complete 5.2-5.9, 5.17, 5.20 in Chapter 5: Cosets, Lagrange’s Theorem, and Normal Subgroups. Also, review 5.10-5.14, 5.16, 5.19, which should all look familiar from the take-home portion of Exam 2. (Due Monday, April 9)**Daily Homework 28:**Complete 5.21, 5.22, 5.24, 5.25, 5.28-5.33, 5.35 in Chapter 5: Cosets, Lagrange’s Theorem, and Normal Subgroups. (Due Wednesday, April 11)**Daily Homework 29:**Skim Section 6.1 and complete 6.7, 6.8, 6.10, 6.11, 6.12 in Chapter 6: Products and Quotients of Groups. (Due Friday, April 13)**Daily Homework 30:**Complete 6.13, 6.14, 6.16, 6.17, 6.20, 6.24, 6.25 in Chapter 6: Products and Quotients of Groups. (Due Monday, April 16)**Daily Homework 31:**Complete 7.12, 7.15, any two of 7.16-7.18, any two of 7.21-7.24 in Chapter 7: Homomorphisms and the Isomorphism Theorems. For 7.21-7.24, you should use the First Isomorphism, which we will wrap up on Friday. (Due Friday, April 27)**Daily Homework 32:**Complete 8.8, 8.9, 8.15-8.18, 8.20, 8.27, 8.31-8.33 in Chapter 8: An Introduction to Rings. (Due Monday, April 30)**Daily Homework 33:**Complete 8.36-8.38 in Chapter 8: An Introduction to Rings. (Due Wednesday, May 2)**Daily Homework 34:**Complete 8.39, 8.42, 8.43 (prove that the kernel is an ideal and prove that the map $\psi:R/\ker(\phi)\to \phi(r)$ defined via $\psi(r+\ker(\phi))=\phi(r)$ preserves multiplication), 8.48 in Chapter 8: An Introduction to Rings. (Due Friday, May 4)

In addition to the Daily Homework, we will also have Weekly Homework assignments. For most of these assignments, you will be required to submit 2-3 formally written proofs. Typically, two of the problems will come directly from the Daily Homework from the previous week. Any additional problems will likely be new. You will be required to type your submission using LaTeX (see below for more on this). You can either submit a hardcopy of your assignment or email me the PDF of your completed work. If you email me the PDF, please name your file as `WeeklyX-Lastname.pdf`

, where `X`

is the number of the assignment and `Lastname`

is your last name. Notice there are no spaces in the filename.

**Weekly Homework 1:**On the Course Materials page there is a list of videos about growth mindset and productive failure under “Miscellaneous Materials”. Watch “Grit: the power of passion and perseverance” and any other 4 from the list and then write a reflection that is at least 10 sentences long. You should comment on each of the videos you watched, but rather than reflecting on each video separately, try to reflect on growth mindset, productive failure, and grit, in general. You are required to type your reflection in LaTeX. For this assignment, I suggest you use the template on Overleaf found here instead of using the “Start your homework in Overleaf” link below. (Due Thursday, January 25 by 8PM)**Weekly Homework 2:**Prove**two**of Theorems 2.29, 2.37, 2.39. You must type up your proofs using LaTeX. I suggest you use my Overleaf template, which you can access by clicking the “Start your homework in Overleaf” link below. (Due Thursday, February 1 by 8PM)**Weekly Homework 3:**Prove**two**of Corollary 2.41, Theorem 2.42, Theorem 2.44, Theorem 2.45, Theorem 2.47(a), Theorem 2.47(b), Theorem 2.50, Theorem 2.63. You must type up your proofs using LaTeX. (Due Thursday, February 8 by 8PM)**Weekly Homework 4:**Complete each of the following tasks. (Due Thursday, February 22 by 8PM)- Prove
**one**of Theorems 3.19 and 3.21. - Determine whether each of the following statements is true or false. If a statement is true, write a short proof. If a statement is false, justify your reasoning. In each case, the context should make it clear what each letter represents. In particular, in Items 1, 3, and 5, $r$ represents rotation of a square by a quarter turn clockwise. But in Item 4, $r$ represents rotating a triangle by a third of a turn clockwise.
- $\{s, r, sr, rs\}\leq D_4$
- $\{1, -1, i, -i, j, -j\}\leq Q_8$
- $\{e, sr, rs, r^2\}\leq D_4$
- $\{e, r, r^2\} \leq D_3$
- $\{e, r, r^2\} \leq D_4$

- Prove
**Weekly Homework 5:**Prove**two**of Theorem 3.23, Theorem 3.24, Theorem 3.51, Theorem 3.52, Theorem 3.53, Theorem 3.54. (Due Thursday, March 8 by 8PM)**Weekly Homework 6:**Prove**two**of Theorem 4.10, Theorem 4.17, Theorem 4.19, Theorem 4.27, Theorem 4.39. (Due Thursday, March 29 by 8PM)**Weekly Homework 7:**Prove**two**of Theorem 4.41, Theorem 4.44, Theorem 4.45, Problem 4.51, Problem 4.52. (Due Thursday, April 5 by 8PM)

You are required to use LaTeX to type up your Weekly Homework assignments. To do this, I suggest that you use my LaTeX Homework Template. The easiest way to get started with LaTeX is to use an online editor. I recommend using Overleaf, but there are other options. The good folks over at Overleaf have preloaded my template, so to get started, all you need to do is click the link below.

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MAT 226: Discrete Math

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Flagstaff and NAU sit at the base of the San Francisco Peaks, on homelands sacred to Native Americans throughout the region. The Peaks, which includes Humphreys Peak (12,633 feet), the highest point in Arizona, have religious significance to several Native American tribes. In particular, the Peaks form the Diné (Navajo) sacred mountain of the west, called Dook'o'oosłííd, which means "the summit that never melts". The Hopi name for the Peaks is Nuva'tukya'ovi, which translates to "place-of-snow-on-the-very-top". The land in the area surrounding Flagstaff is the ancestral homeland of the Hopi, Ndee/Nnēē (Western Apache), Yavapai, A:shiwi (Zuni Pueblo), and Diné (Navajo). We honor their past, present, and future generations, who have lived here for millennia and will forever call this place home.