Your quiz will be a constructive existence proof, and will involve rational/irrational numbers. Study appropriately (see your groupwork, and read Hammack 7.3-7.4). Here are a couple similar types of proofs appropriate to study by:

For every irrational x, there exists an irrational y such that xy = 1.

For every two rational numbers x and y, there is an irrational between x and y (we did this one before).

Every rational number can be written as a product of two rational numbers.

Please compare your Quiz #5 to the solutions and contact me if you want to discuss.

Please think about how to write an inductive proof that all trees on n vertices have n-1 edges. Here is a handout with the relevant definitions and the question. Please attempt to write a proof and bring it to class on Wednesday. Even if you aren’t sure you are doing it right, having made the attempt will prepare you to get the most out of class.

Announcement: There’s a Math Club Talk on Wednesday night. This week it is about computational complexity: how hard is it to compute certain things, like factoring.

Also, some people asked about how to include a picture in latex. The short story is to make sure your preamble where you include packages includes the package:

\usepackage{graphicx}

And then include a picture where you want it in your document by putting (in text mode, on its own line, surrounded by empty lines):

\includegraphics[width=3in]{mypicture.jpg}

You can find more info about this online, for example here. You can adjust the width to make your picture bigger or smaller. You can include jpg, png or pdf files, among others. You can put it inside a “figure” environment if you want it to “float” and position itself with a caption (see link above).

Next week (Monday Feb 26) will be the last opportunity for Sets badges on the quiz. There may be an opportunity to earn one you are missing in office hour later (but you can’t do them all in office hour), so buckle down and study Sets.

To do for Friday:

Prepare for group presentation day.

Prepare for your proof quiz. It will be a reasoning/counting proof similar to the proof we did in class on Wednesday. See the Wednesday class notes on Graph Definition and Handshake Lemma. Study this proof so you understand it very well, and you will be prepared for the quiz.

Spend some time thinking about the problems on our Graph Theory Warmup handout. That is, solve the Warmup problems, and give some thought to the Exploration problems while you are eating dinner, having a shower, or falling asleep (all of which are excellent times to think about math).

Study for sets badges for Monday if applicable (see above).

In class on Monday, we covered truth tables and logical implication, including the notions of contradiction and tautology. We used a handout of truth-tables and played a logic game (instructions and cards).

For Wednesday: (links below are coming shortly)

Fill in the truth table handout. Lines 3 and 4 (each a single table) are each a demonstration (proof, actually!) of a certain logical equivalence. Write down what the logical equivalence is, and then compare to the solutions.

Read Hammack, Chapter 2 up to the end of 2.5, doing the exercises. Keep going if you like. This formalizes the logic we have done intuitively. I will do a mini lecture in class on this material, and then take it a bit further.

Make sure you have sorted out your new groups.

The badges quiz will cover all the previous badges PLUS Logic I and Logic V. The Sets I-IV badges will appear perhaps two more times (including today). So get on them this week if you don’t have them yet! (Canvas is now up to date — note 2=full, 1=partial 0=no credit.) Otherwise focus on getting Logic IV and Proofs I. Then with your spare time, if available, try the new ones.

I have created a 7-minute recap of my advice on how to negate sentences. Watch this whenever you feel you need it for studying. It will be listed under resources. In particular, I suggest you watch it before tackling the Logic IV badge.

Complete the Negation II worksheet, and then compare with the solutions. Bring any questions you might have to class.

There will be a proof quiz (#4). It will be a proof by contradiction, which can also be proved using pigeonhole principle. My main advice for you is to study negation and setting up proof by contradiction. If you can set up proof by contradiction correctly, the proof will not be otherwise long.

Make sure your group is organized for meeting next week. Contact me if you are having trouble.

In the Canvas spreadsheet, grades for badges are recorded as

0 (no credit)

1 (partial)

2 (full)

This differs from the very first quiz, where they were marked on your paper 0, 1/2, 1. But it became apparent that having integers instead of fractions in the spreadsheet on canvas was preferable. Your papers from Badges Quiz #2 and onwards are marked 0, 1, 2.

There is no extra credit for badges.

For Wednesday:

Make absolutely sure you are in contact with your new group-mates and have a plan for Friday. Go to Canvas, click on “People”, then “Groups”. You can contact your group via Canvas.

Please complete the worksheetNegation I that we began in class. (Note: there was a typo on the sheet handed out in #7, 8, 9, where “shoes” at the end should have been “hat”. This is now corrected.)

I realized when administering the quiz on Friday that it was confusing what I was after, in terms of what you can and cannot use. That’s poor quiz design on my part. Therefore we’ll have a “re-do”.

You can download a copy of the quiz, print it out, and write up your best proof for handing in on Monday. This will replace your in-class quiz for grading. This is optional (if you don’t hand something in, I’ll use the quiz you handed in in class). The new copy of the quiz (at the link above) has better instructions on what you can use or not use. You can also email me for clarifications.

For this re-do, it is a violation of the honor code to work together, or use the help of a tutor or friend or any outside source, including the internet. You can, however, use your course notes and textbook. There is no time limit besides the deadline for hand-in, which is in class Monday.

GROUPS ANNOUCEMENT:

We are reshuffling groups. I am forming new groups using the survey you completed on Canvas about group preferences. (If you didn’t complete the survey, you must have forgotten to read the website before Friday’s class. You’ll be assigned to groups randomly in that case.) The new groups will be formed on canvas and you can check them by logging in. You can now log in to canvas to find your new groups. You should also be able to contact one another there.

For Monday’s class:

Re-do your Proof Quiz #3 as described above if you desire (optional).

Contact your new groups to schedule a time for this week’s groupwork.

Please read Hammack, Sections 4.4 and 4.5, and do exercises 14-17 (compare with solutions to 15, 17 in the back). This deals with the topic of “cases” and the use of the phrase “without loss of generality” (also known as WLOG). These will come up naturally in future, but for now I’ll consider them covered by your reading in the book, at least in the sense that you are familiar with them, if not a master of them.

In class, we will work on negation and introduce truth tables.

There will be a badges quiz, and the available badges will be the same as last time. A good goal for this quiz is to finish off any Sets badges you have not yet earned, and then focus on Proofs I if you have room for more. For example, 70% of students have earned Sets I, and so I’ll probably stop including it in badges quizzes in a couple weeks (I will warn you before the last chance). The Logic IV badge is material we will cover soon, but is still a bit “ahead” of where we are.

NOTE: Thursday’s office hour has to end slightly early, as I mentioned before. It will end by 1:45 but possibly I have to go at 1:40.

For class friday:

It is group presentation day, don’t forget.

We’ll make new groups! Please fill out this form (to get to it, go to Quizzes on Canvas) on your groupwork preferences. I will try to do some “roommate matching” based on your feedback for creating the next groups.

The proof quiz will be a proof by contradiction. It will be in the style of Hammack Chapter 6, exercises 2, 3, 9, 11, and the proofs by contradiction you have seen in class or your groupwork assignments.

Study for your proof quiz. Here are some tips on how. I suggest you study by examining the proofs just mentioned above. The study goal is to understand the structure of the proofs so that you can reproduce them without aids. It’s important not to memorize, however, as a memorized proof is of limited value compared to one you can reproduce from understanding its structure and principles. One tip here is to focus on giving an explanation of how to build the proof, instead of giving the proof itself. Don’t memorize sentence-by-sentence but instead focus on how you would explain the process of discovering the proof (for example, the process of working backwards from your goal, or unravelling a definition). (I do my best to model this behaviour in class, to give you some examples.)