QUIZ TODAY: Badges (all). Last chance on sets & logic in class. If you still miss them, there’s an opportunity to earn the badge by visiting me in office hours to demonstrate mastery. More on that after the quiz.

Quiz coming up on Friday: a proof by contradiction. Prepare by doing examples from your book and working on the suggested examples posted here.

To do for class:

Prepare for the quizzes coming up.

Also, suggested exercises of Chapter 7 covering many different methods of proof (so choosing the correct method is part of it), which you should do sometime this week: 2, 8, 10, 12, 17, 21, 28.

Also, when time allows this week: reading on relations: Section 11 up to end of 11.1

Happy Halloween! Please feel welcome to wear a math costume to class (non-math costumes are alright too, but who would do that?).

Notes:

On Wed we’ll do a badges quiz with the last sets/logic; I’ll return this week’s one on Monday. If you are still missing sets/logic badges after that you can demonstrate mastery one-on-one in my office hours.

It was pointed out it is time for new groups. I’m going to put this off for one week; so we’ll make new groups on Friday.

To do for class:

Read your textbook, Section 7. This covers some variations on the theme of proof: if-and-only-if (just two if-thens), equivalent statements (piles of if-thens), existence proofs (show the relevant example), and the concept of constructive or non-constructive proof (we encountered this with CHOMP).

I will pick groups to present as usual, but if you haven’t had a chance to present, and would like to make sure you get a chance, you can request to present, either for this week’s group-work, or you can choose something you’d like to present (e.g. from Wednesday’s worksheet or a proof we didn’t cover another time), and email me ahead of time that you’d like to do this.

Also, next week will probably be the last in-class chance for sets and logic badges, but you will be welcome to come to my office during office hours to demonstrate your understanding of a badge you have missed before the end of term. I’ll set details after the last in-class opportunity.

Quiz: Badges quiz. This will have proof setups and counting. I will give a *second last chance* on all sets and logic badges.

To do for class:

Tabulate for yourself which badges you have earned, and decide which ones you need to work on. Study those topics.

Compare the badges you believe you have earned with my record on D2L, and bring your past quiz as evidence to report the discrepancy to me. I believe I missed recording one badges quiz, so this is crucial!

Please bring your contradiction quiz for me to input grades also, if you haven’t yet. Sorry for the mixup on this — I forgot to record the fourth proof quiz (contradiction) quiz grades and need you to show them to me again.

Here are the worksheets for counting from class last week: first and second.

Please try to complete the worksheets as far as possible. On the first, make sure you are confident of your answers on questions 1-7 at least. On the second, make sure you are confident of your answers on sections 1 and 2, at least.

You may wish to read some or all of Hammack, Chapter 3. We do counting a little differently (with an emphasis on the process, not on the formalism of lists, sets, factorials and subsets), so if you find Hammack is too much of a departure, then just focus on understanding the worksheets by discussing them with me and with your group.

Sorry that I forgot to give out the quiz on Monday, I really have trouble reading the clock. Worse than that, I forgot to record the grades on the quiz for contradiction.

PLEASE BRING YOUR GRADED QUIZ ON CONTRADICTION TO CLASS WED

Quiz today: Induction. See Monday’s daily post for review problems to do in preparation. This will be a weak induction involving a formula.

To do for today:

Prepare for your induction quiz! (See Monday’s post for suggestions)

Quiz today: Induction. It will be an induction involving some equations, not game theory.

To do for class:

Prepare for the induction quiz. Hammack has lots of good examples to work through. Read pages 154-164 and do some example problems (remember, solutions in the back for odds), suggested: 1,3,5,9,11,19,33. Chapter 10.3 is also a nice example.

ADDED NOTE: Please keep in mind group-works are graded on participation more than final product, and if one of the problems gets your group stuck in a serious way, please feel you have permission to cut your losses after an honest effort, and go study for whatever midterm or whatever you need to do!

To do for today:

Read in Hammack about Induction (Chapter 10). That is, read pages 154 — 160.

Please read actively, as always. In particular, pick one of the induction proofs to really study in detail. Study it until it is in your bones. Study it until you feel confident you could present it at the board without notes. Don’t memorize words, but instead, you should understand steps and structure. Techniques for doing this: write a point-form summary of the structure; practice the structure out-loud with your eyes closed; practice paraphrasing the main idea (like an abstract for the proof) to your dog. When you’re done, close the book and write it up from your understanding from start to finish. Open the book back up and compare yours with the book. If there are differences, decide if they are important or not. This is considered homework, and you should compare these “rewritten” proofs at the next group-work meeting.

Note: learning a proof like this should eventually feel like learning the plot to a movie. If you really got into an episode of your favourite show, then afterward you should be able to recount the plotlines, and correctly answer questions about motivation of the characters etc. A proof should feel like this!

Please note: there are two types of induction discussed in the book. The first, in the reading above, is “regular induction” where to prove the theorem works for n, you need only use the assumption that it works for n-1 (the previous step). In class, in our motivating example, we were really using “strong induction” where to prove it for n, you use the fact that it’s true for all of the previous steps. The notion of strong induction is discussed in section 10.1, if you want a further comparison. I don’t like to distinguish these two types with special terms (they are all examples of the same thing, and you should learn to be flexible!) but the book does this.