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.)

Quizzes are generally handed back next lecture day. Please frequently check your grades on canvas for correctness and completeness.

Tips for coming to office hour: bring your past work. If you have a question about a quiz problem, have a suggested correct solution or re-written proof to look at. Be prepared with specific questions.

Toward the end of semester I will give you some extra opportunities to demonstrate missed badges, but this opportunity is not unlimited — it will be an extra opportunity for 2-3 badges by doing extra work. The only way to guarantee success is to consistently study and master badges each week throughout semester.

The undergraduate math club holds talks you may be interested in: Undergrad Math Club.

To do for Wednesday’s class:

Please finish the worksheet on Setting up Contradiction that we started in class today. We will take it up together.

Please write a new, good copy of a correct proof for the last proof quiz, as applicable (i.e. if you didn’t score perfectly). If you scored (0, 1, or 2) out of 4 on the Reasoning Score of Quiz #2, be aware that this likely indicates that you need to work extra to keep up with the class. Everyone can master the material in this class, but the class does go at a certain fixed pace, and you are in a “catch up” position at this point. I’m happy to talk to you in office hour. Please bring your good copy of the proof to office hour so we have something to work with.

Sets IV: ordered pairs, Cartesian products and powers, subset and powersets, including cardinality

Logic IV: negating statements

Proofs I: setting up a proof by contradiction

The new groupwork has been posted. I will go one more week before remixing groups, but as always, contact me if you have trouble scheduling 2 hours or any other problem. If you have 6 people in one group, consider breaking into two groups of 3.

For today’s class:

We will mostly focus on proofs by contradiction today. (The Logic IV and Proofs I badges are relevant to the study of proof by contradiction, so they will begin to be available on the badges quizzes.)

Bring your homework about “threeven” that was assigned in the previous daily post.

Read Hammack, page 111 (the first page of Chapter 6), and from page 113 to the end of Section 6.2 on page 116. In a couple spots, there’s a bit of notation from Boolean Logic in this chapter, because Hammack assumes we have covered Chapter 2 before getting to Chapter 6. Consider it an exercise in adaptive reading — just work around the notations you don’t know. We will be working on Boolean Logic soon, and for now skipping over his reference to it still leaves a very useful reading.

Study for the Badges Quiz. My advice: look your scores up on canvas, look over your returned quiz, and decide on 2 or 3 badges you’d like to focus on getting full credit for this week. (Study a few excellently, instead of all of them passably.) To study, it is helpful to find the relevant material in Hammack and do exercises (odd answers are in the back) to brush up on the concepts. If you have questions about the badges quiz material, I can take them up at the beginning of class Monday before we do proof by contradiction.

It is crucial that you are checking the website between classes and keeping up with homework tasks between classes, whether graded or not. (Hint: this Friday I may check for completeness.)

General announcements:

THIS THURSDAY OFFICE HOUR IS CANCELLED due to an unavoidable conflict. Please email if you’d like to arrange to meet.

Friday is another group presentation day. Please put your group member names and section number on your PDF.

Friday will also have a proof quiz. It will be a “direct proof” (in Hammack’s parlance), similar to the examples of Section 4.3, and to exercises 1-7, 10-12, 19-20 of Chapter 4, and to the proofs from the first groupwork assignment.

Note: office hours will end 15 minutes early Feb 8th.

Groupwork submitted on canvas must from now on be typset, by LaTeX or Word or any other method. But not handwritten.

I hope you all enjoyed the LaTeX tutorial on Wednesday. You can access a LaTeX overview and links, including the sample file, on the navigation bar at left. Learning LaTeX will be very helpful for your undergraduate career!

If the badges quiz confused you, or worried you, remember that you have multiple attempts at each badge. You can check your current badge earnings in canvas. Please read Grading again for details.

Checklist for today’s lecture:

Prepare for group presentation (see above).

Study for proof quiz (see above).

We have spent some time on boring old even and odd integers. Your task for Friday is to invent a notion of “threeven” (i.e. divisible by 3) and two types of non-threeven-ness (i.e. two distinct ways an integer can fail to be threeven). In other words,

The numbers 0,3,6,9,… are threeven

The numbers 1,4,7,10,… are one type of non-threeven

The numbers 2,5,8,11,… are the other type of non-threeven

You can get creative in giving them names. What I’d like you to do is:

Give formal definitions of the three types of integers

Give a formal proof that if is threeven, then is threeven. Hints:

You may find it helpful to read some of Hammack, section 6, namely the first page and the example at the bottom of page 115.

You may find it helpful to first prove that if is not threeven, then is not threeven. You can then use this as a lemma if you like (you can revisit the notion of a ‘lemma’ on page 88 of Hammack).

If you are still unsure, but have given it a good effort, write up what you have as incomplete ideas.