Math 299 – Spring 2024: Training Workouts

Welcome to Math 299, where mathematicians are born!  I will post assignments and announcements here throughout the semester.  Check back frequently.  Below are links to some resources we will be using in the course.

• Download and install Dropbox on your laptop and send me the email address you used for your Dropbox account if you haven’t done so already.
• Read the course syllabus (see below).  Let me know if you have any questions.
• Read Problem #1 on the 2021 Prove it! admissions test and then play the Scrambler! Product Catalog game until you figure out how to reliably beat any goal on all three machines (Juggler, Frogger, Whirligig). You do not have to answer parts (a)-(f) of the question or hand in anything, but you should try figure out a strategy that is guaranteed to beat all three machines no matter what goal comes up when you select “New Goal”. I will ask you in class how to solve each machine. Prize cookies may be awarded. Optional: If you figure out all three levels, you can try the more advanced Scrambler game.

• 0. For each assignment that you hand in in Dropbox, make a folder called “Assignement #n”, where n is the Assignment number I have posted here. So for this assignment, put it in a subfolder of our shared Dropbox folder called “Assignment #1” and put your document in there. All assignments must be handed in prior to class on the day they are due. Be sure to name your files as explained in the syllabus below.
• 1. Answer Problem #1.4 at the end of Chapter 1 in the lecture notes. Type up your answer (you can use any editor for this assignment) and save your file to Dropbox as described above.
• 2. Try to figure out how to consistently beat Trix Game.  This is another example of a Toy Proof system. I’ll ask for volunteers in class. Write up your solution and put it in the same document as the provious problem. Cookies may be awarded.

Reminder: For each assignment that you hand in in Dropbox, make a folder called “Assignement #n”, where n is the Assignment number I have posted here. So for this assignment, put it in a subfolder of our shared Dropbox folder called “Assignment #2” and put your documents in there. All assignments must be handed in prior to class on the day they are due. Be sure to name your files as explained in the syllabus below.

• 1. Prove each of the following Circle-Dot theorems. You must use the Toy Proof software to ensure that your proofs are correct. Prove them using the software, and then print that web page to a PDF document and place that pdf document in your Dropbox homework folder with the appropriate file name. You can prove all four parts in one proof if you want, rather than proving them separately (but its ok if you prove then separately too). If you put one pdf for each problem below, name the files so I know which one is which. You can also just insert the printout into a larger document so that you can type comments or mark up the pdf with a stylus (for my iPad users).
  • Thm H: $\bullet\bigcirc\bullet$
  • Thm J: $\bigcirc\bigcirc\bigcirc$
  • Thm M: $\bigcirc\bigcirc\bullet\bullet\bullet$
  • Thm R:  $\bullet\bigcirc\bullet\bigcirc\bullet\bigcirc$
• 2. (Bonus) Can you explain why every circle-dot expression can be proven in the Circle-Dot toy system, or if not, determine with certainty exactly those expressions which can? Write up your answer in a pdf document (using whatever editor you like for now) and put it in the same Dropbox folder with this assignment.
• 3. Answer question 2.1 in the lecture notes below (all parts). Do not simplify your answers (other than inserting parentheses where needed).
• 4. Bring your laptops to class from now on.

• 1. Answer questions 2.2 and 2.3 in the lecture notes. You can type up your answers using any editor or write them by hand on paper. Either way scan or print your document to a pdf or save your Lurch file and put it in the appropriate folder in Dropbox. (Don’t put a .doc or .docx or .tex file or photos or images.)

• 1. Choose any five theorems that contain at least two different variables from Problems 4.12 thru 4.29 in the Lecture notes, and make a truth table for those to show that they are tautologies. You don’t have to type this in Lurch, but put your answer in Dropbox along with the next part even if they are separate files.

• 2. Click on this link and answer the questions in that Lurch document. Save frequently to your Dropbox folder using the Lurch File> Save menu option. Hand it in the same folder in Dropbox as the previous problem.

• Click on this link and answer the questions in that Lurch document. Save frequently to your Dropbox folder using the Lurch File> Save menu option

• Click on this link and answer the questions in that Lurch document. Save frequently to your Dropbox folder using the Lurch File > Save menu option.

• Click on this link and answer the questions in that Lurch document. Save frequently to your Dropbox folder using the Lurch File > Save menu option.

1. Memorize the Rules of Logic for a short quiz at the start of class on Tuesday. Memorize the template form of the twelve rules for Propositional logic in the table on pages 15-16 in the lecture notes, the four rules for quantifiers in the second table in section 5.4 of the lecture notes, and the two rules for equality in the second table in section 5.5 of the lecture notes. You will just have to write them all down from memory.
2. Click on this link and answer the questions in that Lurch document. Save frequently to your Dropbox folder using the Lurch File > Save menu option.

• Click on this link and answer the questions in that Lurch document. Save frequently to your Dropbox folder using the Lurch File > Save menu option.

Midterm Exam – Part 1

This assignment is Part 1 of your Midterm Exam, so every problem is worth triple what it would normally be worth on the homework.  No googling or discussing these problems with others, but it is open notes, so you can look at the lecture notes or previous examples or homework problems. Enjoy! 

• Click on this link and answer the questions in that Lurch document. Save frequently to your Dropbox folder using the Lurch File > Save menu option.

Midterm Exam – Part 2

This assignment is Part 2 of your Midterm Exam, so every problem is worth triple what it would normally be worth on the homework.  No googling or discussing these problems with others, but it is open notes, so you can look at the lecture notes or previous examples or homework problems. Enjoy! 

• Click on this link and answer the questions in that Lurch document. Save frequently to your Dropbox folder using the Lurch File > Save menu option.

1. Memorize the Peano Axioms for a short quiz at the start of class on Tuesday. You should memorize the rules derived from the axioms given in template form in the table on pages 38-39, not the original axioms on page 37. For the derived rules you can either write them in template form or in the more compact turnstile notation. For example, you can write N2 as $\sigma(m)=\sigma(n) \vdash m=n$ and M0 as simply $\vdash n\cdot 0=0$. You will have to write them all down from memory.
2. Click on this link and answer the questions in that Lurch document. Save frequently to your Dropbox folder using the Lurch File > Save menu option.

1. Memorize the Definitions of Number Theory for a short quiz at the start of class on Thursday. You should memorize the rules derived from the axioms given in template form in the table on pages 39-40, not the original axioms on page 38.
2. Click on this link and answer the questions in that Lurch document. Save frequently to your Dropbox folder using the Lurch File > Save menu option.

1. Click on this link and answer the questions in that Lurch document. Save frequently to your Dropbox folder using the Lurch File > Save menu option.

1. Click on this link and answer the questions in that Lurch document. Save frequently to your Dropbox folder using the Lurch File > Save menu option.

1. Memorize the Axioms for Real Numbers for a short quiz at the start of class on Thursday. You should memorize the rules in the Table on page 52 of the Lecture Notes.
2. Click on this link and answer the questions in that Lurch document. Save frequently to your Dropbox folder using the Lurch File > Save menu option.

1. Click on this link and answer the questions in that Lurch document. Save frequently to your Dropbox folder using the Lurch File > Save menu option.

1. Click on this link and answer the questions in that Lurch document. Save frequently to your Dropbox folder using the Lurch File > Save menu option.

1. Memorize the Axioms for Set Theory for a short quiz at the start of class on Thursday (yes, Thursday, not Tuesday). You should memorize just the compact abbreviated rules in the Table on page 60 of the Lecture Notes.
2. Click on this link and answer the questions in that Lurch document. Save frequently to your Dropbox folder using the Lurch File > Save menu option.

1. Memorize the Axioms for Set Theory for a short quiz at the start of class on Thursday. You should memorize just the compact abbreviated rules in the Table on page 60 of the Lecture Notes.
2. Click on this link and answer the questions in that Lurch document. Save frequently to your Dropbox folder using the Lurch File > Save menu option.

1. Click on this link and answer the questions in that Lurch document. Save frequently to your Dropbox folder using the Lurch File > Save menu option.

1. Memorize the definitions for functions for a short quiz at the start of class. Memorize the first table in section 10.4 of the lecture notes. I will provide you with the names of the definitions.
2. Click on this link and answer the questions in that Lurch document. Save frequently to your Dropbox folder using the Lurch File > Save menu option.

1. Add all of the reasons to the second half of the proof that we went over quickly at the end of class on Thursday. The proof is in your Dropbox>handouts>in class folder named Apr 23 b.lurch. Add the reasons and then save your resulting Lurch file in the same Assignment #23 folder with the homework Lurch file below. Thus, your folder should contain two Lurch files for this assignment, not just one. You can save it with the same name, but put it in the Assignment #23 folder, otherwise it will get overwritten the next time I share class files with you.
2. Click on this link and answer the questions in that Lurch document. Save frequently to your Dropbox folder using the Lurch File > Save menu option.

1. First Encounters with $\LaTeX$. Choose any nontrivial proof that you got correct on Assignments 15 thru 23. Write up a short LaTeX article that explains that proof to a reader in full expository style, including a title, abstract, introduction, the theorem statement, and the proof itself. Your target audience should be students in an Introduction to Proof course at another university. You should cite at least one reference (our lecture notes are a good reference if you are referring to the theorem numbers or definitions in the notes). Your article should follow all of the conventions of mathematical writing explained in class and in Chapter 11 of the Lecture notes. Download just the pdf from Overleaf and put it in your Dropbox folder for this assignment. Here is an example document (don’t do this proof)
2. Click on this link and answer the questions in that Lurch document. Save frequently to your Dropbox folder using the Lurch File > Save menu option.

• Write up and hand in a combinatorial proof for each of the following. Write your proofs in LaTeX using the Homework Article style and put your pdf file in Dropbox as usual.  You cannot use any algebra, arithmetic, substitution, or previous formulas derived in or out of class. Just 100% pure counting! Bonus points will be awarded for nontrivial figures, but they aren’t required. Put all of the problems in a single document with a title and abstract, and restate the theorem you are trying to prove along with it’s problem number. You can cite this web page and the Lecture Notes as a reference. There is an example of a combinatorial proof here.
  • 1. Thm 12.10 (counting over computing) $$\binom{6}{2}\cdot\binom{4}{3}=\binom{6}{3}\cdot\binom{3}{2}$$
  • 2. Thm 12.13 (binomial complement) $$\binom{m+n}{m}=\binom{m+n}{n}$$
  • 3. Thm 12.15 (choose vs. permute) $$(n)_k=\binom{n}{k}\cdot k!$$
  • 4. Thm 12.17 (good things come in pairs) $$\binom{2n+2}{k} = \binom{2n}{k} + 2\cdot\binom{2n}{k-1} + \binom{2n}{k-2}$$
  • 5. Thm 12.26 (combination recursion) $$\binom{n+1}{k+1}=\binom{n}{k}+\binom{n}{k+1}$$

• Write up and hand in a combinatorial proof for each of the following. Write your proofs in LaTeX using the Homework Article style and put your pdf file in Dropbox as usual.  You cannot use any algebra, arithmetic, substitution, or previous formulas derived in or out of class. Just 100% pure counting! Bonus points will be awarded for nontrivial asymptote or original figures, but they aren’t required. Put all of the problems in a single document with a title, abstract, and you can cite this web page and the Lecture Notes as a reference.
  • 1. Thm 12.16 (Ordered choice) $$\binom{n}{k}\cdot\binom{n-k}{j}=\binom{n}{k+j}\cdot\binom{k+j}{k}$$
  • 2. Thm 12.11 (more counting over computing) $$\left(\!\!\!\middle(\genfrac{}{}{0pt}{}{8}{3}\middle)\!\!\!\right)=\left(\!\!\!\middle(\genfrac{}{}{0pt}{}{4}{7}\middle)\!\!\!\right)$$
  • 3. Variant of 12.17 (good things come in triples) $$\binom{n+3}{k}=\binom{n}{k}+3\cdot\binom{n}{k-1}+3\cdot\binom{n}{k-2}+\binom{n}{k-3}$$
  • 4. Thm 12.30 (Gauss Redux) $$1+2+\cdots+n=\binom{n+1}{2}$$

• Our Final Exam will be at 5:15pm on Thursday, May 16 in our classroom. Bring your laptops. You will be asked to do as many proofs correctly as you can in two hours. I will email you with any additional details. Game on!