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.

WEDNESDAY WE WILL HAVE A LATEX WORKSHOP IN DUAN G116. NOT OUR REGULAR ROOM!

The room is a computer lab but I’m concerned it won’t have enough computers to accommodate everyone (the school couldn’t get me anything better). Therefore please bring a laptop if you have one. The only thing you need is web access, no need to prepare. I will give an introduction and overview and then I will debug and walk around to help as you work through the worksheet. If you are a True Master of LaTeX already, you may skip this class, or come with your LaTeX questions, but please email me to let me know if you plan to skip.

General announcements:

Group homework will be weekly. Make time to meet for two hours. I won’t exceed this. If you don’t finish in two hours, but made a decent effort during the two hours, then call it a day and pack up — that’s ok. If you have any trouble with groupwork, contact me privately if you wish and I will try to help. This includes scheduling issues, personality conflicts, or anything else. Your next group work is due Friday. See Grading.

Proof quizzes. Our first one was Friday. It was returned on Monday. Please pull it out of your bag now and take a look at my comments. The comments explain how to improve your grade for next time, so make sure you understand. If you don’t, please ask me before/after class or in office hour. See Grading.

Badges quizzes. Our first one was Monday. Remember that you don’t need to do everything on the quiz. Each section is a “badge” which you will have many chances to earn. You may wish to choose two per week to study for and work hard at, instead of always attempting everything. Your grades will be updated in canvas so you can check them before the next quiz and know which badges to work on. See Grading.

Daily posts. By now you have realized that you must check this website for daily posts and that I will rely on you to have done the homework between each class.

Monday in class we proved that the square root of 2 is irrational. Here’s an enjoyable video from Numberphile that recaps the proof. (If you’ve never seen Numberphile, you are missing out.)

I will post all the handouts from class etc. under Resources, so if you miss something, you can find it there.

Checklist for Wednesday:

Read the comments on your quiz which was returned Monday, and write out a proof that you believe addresses those comments. Suggestion: staple this new proof to the old quiz and keep for your records. Depending on your confidence in this exercise, see me in office hour (Wed/Thur 1-2 or by appt) if you feel unsure. I’m happy to look it over.

Complete the handout from class today, including the part on powersets which was rushed in class. To check your answers in the table, the correct answers read, from bottom to top, “NUMBERS RULE THE UNIVERSE.” If you don’t understand why one of your answers is wrong, ask me!

Read Hammack, 1.5, 1.6 and 1.7. Section 1.7 is about Venn Diagrams. These are a sort of “visual aid” to the ideas of 1.5 and 1.6. Read actively, as always. Make an outline only if you’ve found that a useful exercise in past.

Do as many exercises from 1.5, 1.6 and 1.7 as you need to feel confident.

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

We have not covered all this information in class yet, so by working ahead you can get a jump-start, if you feel like it. The idea is that you can choose which questions to attempt. You need not attempt them all, but I like to give you many options. You will get many (but not infinitely many) attempts at each badge, so this is not your only chance.

Please read Sections 1.3 and 1.4 of Hammack. Remember: read actively!

As usual, do as many exercises as you need for Section 1.3 and 1.4 to feel comfortable.

Announcement: I have set regular office hours WED/THUR 1-2 pm. The third hour is “floating” because with 50+ students, no set hours will work for everyone. That means as needed, I will schedule the third to meet the needs of those who ask, and announce the time to everyone. Therefore, please email me if the time above won’t work for you and you’d like to meet that week.

Reminder: groupwork is posted and due Friday. This one may require 2 hours.

If you have any scheduling issues with groups, please contact me.

Friday will be a group presentation day and a FIRST PROOF QUIZ. Your only assigned task is to prepare for these.

A little more info about group presentations and what to expect:

Your group will have handed in (by Friday morning 8 am), a PDF (scan of handwriting is ok as long as it is legible) via canvas of at least the three main proofs assigned in the Groupwork Assignment (and possibly the extra one too), or whatever you have accomplished. I will assign a small grade for groupwork but it is assessed on a complete/partial/incomplete rubric. Errors are fine. In fact, I hope you make interesting errors, because that’s where learning happens.

In class, you will hand in a paper copy (handwritten is fine) of the Groupwork Report (you can find a blank copy under Resources). This is just a brief accounting of what happened in your group. It is very helpful for me to know how things went, whether there were sticking points and confusions, how long it took, etc. This should only take a few minutes to complete.

In class, I will use my computer to load up the PDFs from canvas. I will load up a proof and ask for the presenter from that group to come to the front and explain what their group wrote. The class will then discuss writing and reasoning and offer constructive criticism and praise. This is meant to be a safe environment for errors, and I ask you to be thoughtful in your comments. In fact, I will pick proofs that have interesting errors so we can all learn.

Some more information about the first proof quiz:

This is a bit of a dry run. Remember (from Grading), that we drop just less than half of the proof quizzes. But we need to get started so I can see where the class is.

The quiz will ask you to prove something similar to the proofs we have seen. In particular, it will ask for a proof like:

If n is even, then n^2 is even.

If n is odd, then 2n is even.

If n and m are odd, then n + m is even.

etc.

You may wish to examine my proof grading rubric, available under Resources.

The group presentations will be a great way of studying for the quiz, as we will discuss writing and reasoning for similar proofs.

Other comments:

I meant to get a bit farther today than we did; apologies we didn’t get to talk proofs. We will on Friday.

The back of today’s handout on Cartesian products is very interesting material, and I encourage you to wrestle with it on your own time; we may get a chance to come back to it in class in future. Meanwhile, Chapter 1.10 in Hammack talks about some of this.

I will set office hours as soon as my schedule settles down, but please just email me to set a time to meet as needed.