I hope everyone has/had a wonderful thanksgiving, whether remote, take-out, whatever it is/was. 🙂

To Know: Because of the Thanksgiving break, badges and proofs due dates are pushed back. Here’s the remaining due dates:

Badges will be due Monday, November 30th and Friday, December 4th.

Proof Quizzes will be due Wednesday, December 2nd and Monday, December 7th. They will both be on induction.

Practice setting up induction using this induction worksheet. This is your first practice for the Proofs III badge, and we’ll take it up in class, so you may wish to leave this badge for last and do it after class (it’s due Monday at midnight).

Please know that online FCQs are open. FCQs are used to evaluate your instructors for promotion and tenure, and to inform the department about their teaching effectiveness. I, personally, greatly appreciate feedback and work to improve my teaching using your feedback.

To hand in as the daily exercise, try your hand at a few proofs (whatever the 1 hour allows):

Complete the proof I ended class with on Monday. I’ll complete it in class Wednesday.

Prove that if G is a tree, then every edge of G is a cut edge. An edge is a “cut edge” if removing that edge produces a disconnected graph.

Prove that if every edge of a connected graph G is a cut edge, then G is a tree.

To Know: There is a pigeonhole proof quiz on canvas. Check canvas for available oral exams to schedule.

To Know: On Monday we’ll start talking about graph theory. We’ll use it to practice methods of proof, including induction, and return to the problem we started Day 1 of class with.

To Do:

Read Section 3.9 of Hammack, where he discusses the pigeonhole principle and its related “division principle” and does a couple of nice problems with it.

To Know: I have opened up the modular arithmetic badge and continued the open badges (due Friday). The only badge missing is Induction, which is our next big topic.

To Do: Read the Solutions to Proof Quiz #9. Read them actively, as always. There are three proofs presented, and you only submitted one, so there’s something to learn there. And please read the second page with common errors, where I explain the most common error with surjectivity! (I decided to do solutions just text, not video this time.)

To Do: Take a look at the multiplication tables here, and collect as many patterns as you can. Just try to describe them.

To Do (IMPORTANT; NEW CONTENT): Watch my follow-up YouTube video on defining the integers modulo n (17:26). This is more practice with the formalism, and the work one needs to do to make sure integers mod n are well-defined.

To Do: Complete the exercise mentioned at 12:48 in the video. Hand in on canvas.

IMPORTANT TO DO: Watch this YouTube video I made: Modular Arithmetic User’s Manual (9:23). This is part of the course lecture material, in flipped form.

To Do: Here’s a follow-up worksheet to do and hand in on canvas’s daily dropbox.

To Do (time permitting): Think about how to finish off the proof from Friday’s class; I’ll finish it up on Monday.

To Know: There’s a proof quiz due Monday on canvas.

To Know: I’ve posted more badges for Friday. I’ve opened the Relations I & II badges. Make arrangements for any oral exams you may need. Please please review the videos/material for functions and figure out what you got wrong before attempting them again (there are standard errors you can overcome with a little attention to your first few attempts). The fastest way to reach me is via discord, and I’m always happy to help.

To Know: It’s getting close to end of semester. The last two badges may end up compressed (either more frequent deadlines or fewer attempts). That’s basically just Modular Arithmetic and Proofs III.

To Do: Look over your Proof Quiz 7 and 8 and compare to Proof Quiz 7 Solutions and Proof Quiz 8 Solutions. I’ve also posted a video solution for Proof Quiz 8 because I had some things to say about it. Please watch the video (available on canvas).

To Do: Read Section 14.4 of your text. This has the detailed proof that I only sketched in class today. Do exercises 1 and 2 to hand in.

To Know: I extended all the badges due dates to Monday, to give everyone more breathing room.

To Know: A proof quiz has been posted, due Monday.

To Know: There was some problem with the playlists on canvas; the other tab (instead of the default Playlists view) is a full directory of videos. They are all there. I’ve tried to fix the playlists by breaking them up by month. If a video is missing try the other tab.

To Do: In class on Wednesday we gave three “properties” of the definition of “same cardinality” that I labelled “Reflexivity”, “Symmetry” and “Transitivity”. Review the video/notes and then attempt a proof of Symmetry and a proof of Transitivity (I did Reflexivity in class). Hand in on canvas.

To know: I’ll be putting up the other functions badges tonight, as well as some repeats on older ones (notably the two proofs badges); make sure you take a crack at all of them. I’ll also open up some oral exams, so please schedule those with me if you’d like.

To Do: I ended Wednesday’s class with some sets you are familiar with, and I’d like you to try to come up with bijections between some of them, to show they are the same cardinality. Bring your bright ideas to class. To hand in on canvas: whatever thoughts you have on this, e.g. some bijections or some ideas that didn’t work, etc. (It’s a gently-graded daily post, since I want to give a bit of a break this week, but do hand in something thoughtful.)