- 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.
- There will be a Proofs Quiz, using the method of contrapositive proof.
- Complete the contrapositive setup worksheet and compare to solutions. Here: Proof by Contrapositive Setup Sheet and solutions.
- Read Hammack’s Chapter 5.
- Do the two contrapositive proofs I assigned in class:
- If xy is greater than 100, then at least one of x or y is greater than 10.
- If 3 divides n^2 then 3 divides n.
- Do Exercises from Chapter 5.
- Happy Valentine’s!
- 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.
- We will begin boolean algebra on Wednesday.
- 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.
- Watch Video of an informal explanation of why the square root of 2 is irrational (Numberphile) (8 or 9 minutes long). He rambles interestingly for a while about paper and peeing toward the sun, but he does eventually give a proof.
- 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.
Professor Katherine Stange, Spring 2018