Look at your Quiz #3 which was returned and make sure you understand my comments. If your proof wasn’t correct, write a correct proof and check it with me at office hours or after class.

Finish the two worksheets from class today (if you missed them, check the Lectures page).

Prove or disprove: There exists a real number x such that x^2 > 0 but x^3 < 0.

Prove or disprove: All pairs of integers a and b have gcd(a,b) > 1.

Spend time as needed (active reading + exercises) with Hammack, Sections 2.6 and 2.7. This is the material of Badges Logic III and Logic V which will now be on badges quizzes.

Please compare your truth table worksheet today to the filled worksheet, to make sure you have correct and understand every T and F there. Reminder: the first group of tables are *definitions* and the later groups further down the page can be calculated using them.

Some students wanted to take home the SPQR logic game — check the Lectures page for links.

Read Hammack 2.2-2.5 for a review of what was covered in class today.

Do Exercises for Section 2.3 and 2.4. This is a little practice in the use of language for conditionals. (You know your text has answers to all odd exercises at the back, right?)

Read Section 2.6, which is a look-ahead to our next topic.

Friday we will have a Badges quiz. It will now include Sets I, II, III, IV, Logic I, IV, Proofs I.

Use this hour to study and solidify any material from Sets (Hammack 1.1-1.7), Direct Proof (Hammack Chapter 4) or to study Proof by Contradiction (Hammack Chapter 6, which we will spend more time on). In particular, note that I haven’t done a ton of “cases” examples in class (Hammack 4.4), but you should get comfortable with those too.

Some advice on studying: making notes is good, but my rule of thumb is to write notes with the book closed. For example, don’t copy definitions out of the book. Instead, read and understand the definition (including inventing examples and non-examples of your own, etc.), then close the book and write it in your own words. Then compare to the book and see if you’ve forgotten any important hypotheses or anything like that. The over-arching principle is: knowledge that you reorganize/reinvent with your brain is knowledge you have actually owned.

Stay healthy! Thanks for the smiling picture of you guys I received from Prof. Green in email, that was great. 🙂

In class, we started two worksheets, one on setting up proofs by contradiction (about formal writing of such proofs), and contradiction puzzles (fun proofs that require a creative idea). At home, please complete both of these worksheets and bring them to class on Monday. If you aren’t sure of the secret key to one of these, discuss with your roommates, other math majors, or peers in the class to brainstorm ideas.

With remaining time, study for the proofs quiz, which will be a proof by contradiction on Monday. Hammack has exercises and examples.

Now, write up a nicely written mathematical style proof that the square root of 2 is irrational. You can and should follow the logic in the video (back up and replay as much as you like), but you need to write it formally, not just copy down his informal ramblings. The script of the video is not a nicely written mathematical proof. There are some details that need to be dealt with, so be careful and fill in any little holes as carefully as you can. (Please *don’t* look up write-ups of this proof online or in your text; use only the video and have the writing part be your own work.)

What you did above is a proof by contradiction. Answer these two:

What did you assume for contradiction?

What contradiction did you reach?

Did you notice the other tiny proof by contradiction contained in the bigger one (about even squares)? Do the same here:

What was the statement of this little fact?

What did you assume for contradiction to prove it?

What contradiction did you reach?

Please bring your work to class, as always, in case I check homework.

There will be a badges quiz on Friday. We will have Sets I, II, III and IV, as well as Proofs I.

We will do examples of Proofs I in class before the quiz, but basically the idea with that is to be able to figure out what you have to assume to set up proof by contradiction (the first sentence of the proof, typically). This is really practice in negation, more than anything else, and in formal writing.

If you have mastered your other badges, and you wish to read ahead, Sets II is very easy to do by self-study and is covered in Hammack, 1.5-1.7. Otherwise, focus on the badges we have covered in class and we’ll get to this in due time.

Remember, you can attempt as many or few of the badges as you like on the quiz each time. Your score for each badge can only go up, not down, so you are welcome to attempt badges you’ve already earned, to get feedback and practice, without any penalty. But you may wish to focus on badges you have not yet earned.

With remaining time, study for quizzes, catch up on reading the text and do practice problems, as always.

Please make an attempt to fill out the worksheet on negation handed out in class. We haven’t studied this yet, so some of it may be tricky, but just dive in and try to give it a go. We’ll take them up in class.

Budget 10 minutes to watch these two videos I made on the topic of today’s lecture:

There will be a proof quiz on Wednesday (we are catching up on quizzes). It will be another direct proof (Chapter 4, Hammack), but it will be more challenging than the first proof quiz, in one or several of the following ways:

It may require you to read and understand a novel definition and apply it.

It may require more steps, or breaking into cases.

It may require more creativity.

Note: I will shortly (but not up yet) post solutions to a few recent homework problems, so you have more things to compare to.

Quizzes: I somehow completely forgot to give the badges quiz on Friday, for which I apologize (I really try to avoid surprises). We will take it on Monday instead. That will also push the proof quiz back to Wednesday next week.

Today we did a warmup direct proof and I left you with two proofs to write at home:

Let a,b,c be integers. Then .

If a,b,c are integers such that a|b and a|c then a|(b+c) and a|(b-c).

In class we used the fact that the equation xy=1 has only two solutions in the integers, namely x=y=1 and x=y=-1. Can you prove this? Try contradiction!

Review the proofs by contradiction from class and also the first Proposition in Chapter 6 of Hammack (first page of the chapter), which is an example. Pay careful attention to the negation of the theorem, i.e. the supposition you make for contradiction.

With your remaining time, try a few at home: Chapter 6 exercises 1,2,6,7,8,9 as time allows.

Complete the worksheet we did today if you didn’t finish it in class. The circled letters spell a message (read from bottom to top) as a way to check your work. If you don’t have a copy, you can always find handouts under “Lectures”.

Grades: Quiz grades should stay basically up-to-date in canvas, so please log in regularly to check that there are no administrative errors! Please keep your quizzes until the end of semester, for reference.

Friday we will have a badges quiz! It will cover Sets I, III and IV. A little advice: choose one to focus on in your studying, and earn that one for sure, then move on to the next. Don’t spread yourself too thin and only do a halfway job on all of them. Better to do fewer problems well.

Reading: Hammack 1.4 on powersets, and exercises as needed.

Look over your returned quizzes so far (proof and badges) and make sure you understand the scoring and can complete the problems correctly. If you have any questions, email me or see me during office hour! I’m happy to help.

Active reading (it’s always active!): Hammack, Section 1.2 on Cartesian products.

Finish the Worksheet on Cartesian products we did in class (you can always find worksheets on the Lectures page), at least to the end of the first page. Consider the open-ended problems on the second page, as far as your brain takes them, but I won’t check those for completeness. We’ll take up the worksheet in class on Wednesday.

Do exercises from Hammack, Section 1.2 for what time remains.

Note: There’s a math club talk on Wednesday! (Speaker: Elizabeth (Boo) Grulke (CU Boulder); Title: A MIXED-METHOD STUDY OF ONLINE DISCUSSIONS IN MATHEMATICS; Time: Wednesday, January 29 at 5-6pm; Place: Math 350)