Here’s a few existence proofs. Give these a go, and bring your attempts to class. I encourage you to work with friends, as always, as long as you write up your own work.

On Monday, the Proof quiz will be an existence proof. It may involve a novel definition, or the definition of convergence, or things you have already met and studied.

Next week the Badges quiz will no longer have Sets badges on it. You can begin to earn Sets badges in office hour (one per week), if you are still missing them. To do that, you must first prepare a study sheet. That is, a page of notes/information/examples describing the key concepts of that badge. Bring this to office hour. I will go over this with you, and if it demonstrates you have studied well on your own, then I will give you an oral exam on the badge, meaning, I will ask you to work some problems at the board (or at my table) while I watch and interact with you. If you complete the problems, then you will have earned the badge. If you come in unprepared (scribbled insufficient study sheet), then you will not earn the badge. If my regular hours don’t work, email to set up another time. I’m available. Note: you can only earn one badge per week, so this doesn’t replace the in-class quizzes, but it can help if you are getting stuck on one of the badges for some reason.

Next week will be the last week for the Proofs I and II badges. So please study these for the quiz. I will also begin adding Counting badges.

Feel free, for the purposes of our course, to focus on the formal definition of convergence, and not continuity. Complete the worksheet’s first two pages if you didn’t in class. Make sure you are very confident and comfortable with those two examples.

Friday is the last day for Sets badges in badges quizzes. As I mentioned before, I will follow up next week with info on how to earn them by oral exam in my office hour after the in-class quizzes. Please study as needed.

Using the video, write a formal proof that between any two rational numbers, there exists an irrational number. Be careful: the video isn’t careful about some details, and it’s just an informal discussion, so to do this right, you have to formalize and fill in some details that are missing.

With any remaining time, spend quality time with Hammack, Sections 7.3 and 7.4. There are useful exercises here.

Note: Sets Badges will be available in class for the last time Friday. I will then allow you to attempt them as an oral exam in office hour (more details next week).

An announcement follows for a special event you might be interested in. Enjoy math? Maybe your future includes math research as an undergraduate! There’s also a cool poster.

Math Research : DemystifiedWhat’s undergraduate math research anyways? Who is it for? It might not be what you think it’s like and it might be for you, even if you haven’t realized it yet! So come to MATH 350 on Wednesday Feb 26 from 5-6 pm to learn more about undergraduate math research, hear about past projects done by students at CU and learn more about the opportunities for math research experiences available to the undergraduate students on campus. There will also be be free soft drinks and pizza. Everyone is welcome!

When: Wednesday Feb 26 from 5-6 pm

Where: MATH 350 (Mathematics Building)

Who: Organized by the Math Club and Diversity Committee

Monday we will have a Proofs Quiz. It will be something that can be solved by pigeonhole, but it won’t be a “gotcha” — it will either be fairly clear what the pigeons/holes need to be, or I will provide a hint to avoid you just getting stuck. I’ll be looking for your being able to correctly structure the logical argument once that is decided, and communicate well.

Work on Logic this weekend: especially quantifiers, which take practice! Read Section 2.7 again if needed, and do the exercises. Then, take the english sentences that result, cover the original exercises, and see if you can reverse-engineer the symbolic expressions with quantifiers.

Here’s another good resource on quantifiers. In particular, do exercises from that webpage; those are good ones!

Just a look-ahead: the Sets badges will be on the Badges quiz for the last time next week. So you will want to study up on any that you still need to earn.

An announcement follows for a special event you might be interested in. Enjoy math? Maybe your future includes math research as an undergraduate! There’s also a cool poster.

Math Research : Demystified

What’s undergraduate math research anyways? Who is it for? It might not be what you think it’s like and it might be for you, even if you haven’t realized it yet! So come to MATH 350 on Wednesday Feb 26 from 5-6 pm to learn more about undergraduate math research, hear about past projects done by students at CU and learn more about the opportunities for math research experiences available to the undergraduate students on campus. There will also be be free soft drinks and pizza. Everyone is welcome!

When: Wednesday Feb 26 from 5-6 pm

Where: MATH 350 (Mathematics Building)

Who: Organized by the Math Club and Diversity Committee

Finish up what you can of the worksheet from class today. You’ll find the worksheet online including a hint sheet — with that you should be able to make further progress. This is the topic of Pigeonhole Principle and I will now assume you can use this as a proof method (in simpler circumstances). Some examples with worked solutions can be found in Hammack, Section 12.3.

Badges on Friday will now include all Logic badges. There’s one we haven’t discussed yet: Converse and Contrapositive. This is a quick one, and I’ll discuss it briefly in class, but you can also look this up in your book. The quiz will also include Sets badges for the second-to-last time, so please study any of those that you are missing.

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.