Hi all!

Have a happy and restful spring break.  Video links for Friday’s video lecture are below.  Your homework over break will be to read Hamkins (not Hammack), Chapter 4 up to the end of Section 4.3 (pages 27-32) about induction, and to watch the Friday lecture videos.  Just spend some time thinking and absorbing, nothing to hand in.

Video 1 (19 mins) is hosted on google drive, and I made it for you today, showing a standard inductive proof:  view it here.

Video 2 (22 mins) is hosted on YouTube and I made it in 2018 showing a graph-based inductive proof:  view it here.  (I apologize that I can’t find my original file from 2018 and have to give a YouTube link, which may serve you an ad that Youtube makes money from. 🙂

For Wed:

Proof for feedback:

State and prove a variation on today’s Take-2-or-1 Theorem, but where players can take 1 or 3 stones.  Use as a model the proof from class notes and/or my writeup.

Tasks:

1. Note:  Friday, March 24th will be a recorded lecture (so no need to attend class); you will find it on this website and view it over spring break at your leisure.
2. Wednesday is a proof quiz.  It will be a proof by contrapositive.  Examples can be found in Chapter 5 of Hammack.  Do some exercises from there as practice.
3. Because of spring break, you will have until April 7th to do a logic study sheet in order to request a logic grade improvement.  Solutions to the quiz can be found in the “Materials Archive” page as usual.

For Monday:

Proof for feedback

graphs

Tasks:

1. Using the theorem we proved in class, prove that if n is the number of vertices of a graph having odd degree, then n is even.
2. For an extra resource on graph theory, a nice book is Introduction to graph theory by Robin J. Wilson.  Here’s a website that may be helpful too.

For Friday:

Proof for feedback:

big-oh-inspired

Tasks:

1. Please hand in your self-evaluation sheet again this time, so I can see how the semester is going for you on that front.
2. You’ve got a proof for feedback above.

For Wednesday:

1. no proof for feedback because there’s a content module quiz
2. if you wish to do a grade improvement for the sets module, please get me a study sheet in the next few days.  Further info is above in Grading > Content Modules > Grade Improvement and the Study Sheet tab has info on making a nice study sheet.
3. Wednesday is the Logic quiz.  Here are the topics:
   1. 1. boolean expressions and truth tables
      2. converse and contrapositive
      3. quantifiers (for all, there exists)
      4. negating statements
      5. logical equivalence, logical laws and algebraic manipulations of boolean expressions
4. Practice problems for the logic quiz (solutions here).

For Monday:

Proof for feedback:  Revisit any of your previous proofs for feedback and resubmit a new draft based on your previous feedback.  Pick one that you struggled with the first time around.

Tasks:

1. Your logic test is on Wednesday.  I will post some example/practice problems shortly on here (watch this spot).
2. Watch this description of the limit definition of a sequence; this should help clarify some of what we were doing in class:
   1. 1. 1. Video 1: https://www.youtube.com/watch?v=PRTjvMA2nCY
         2. Video 2: https://www.youtube.com/watch?v=0UCRZAsIkXM
         3. Video 3: https://www.youtube.com/watch?v=lCW8BBBQRyc
3. See if you can finish the rest of limit part (first two pages) of the worksheet from Friday.

Proof for Feedback:

epsilon-delta

Tasks:

1. Read Hammack, Section 2.10, about Negating.  Do exercises as needed.  Pay special attention to the negation of quantifiers.
2. Negating quantifiers is the last topic for “Logic” which will be tested on Wednesday.  Here’s a worksheet on practice with quantifiers.  Give it a go.  We will take this up in class.

Proof for Feedback:

choose #3, 4 or 5 (your choice) from the Existence Proofs Worksheet

Tasks:

1. Wednesday is a proof test!  It will be an existence proof.  For more practice, see Hammack, Chapter 7, exercises 12, 17, 18, 20 (uses the symbol “divides”; it says “11 divides 2^n-1”).
2. Finish your existence proofs worksheet; choose one of them to request feedback on from the grader (put it first in what you hand in).  In class we will take up the solutions.  (Of course, don’t look at this before you do your proof for feedback, that would be kind of silly.)
3. With what time remains, read Hamkins, Chapter 3 up to end of 3.2 (Number Theory, Prime Numbers and Fundamental Theory of Arithmetic).  This involves an existence proof.

Proof for feedback:

feedback proof

Tasks:

1. Friday attendance has been slipping.  You will need to reflect this in your self-eval sheets, so if this is you, please find time to re-engage with class, for the sake of your grade.  (And yes, I’ve been keeping track.)
2. In class friday we talked about quantifiers.  Read Hammack, Section 2.7 and do exercises.
3. Then, try to finish out the Quantifiers Practice sheet.
4. I handed back the Sets test.  Check out the solutions. and make sure you understand anything that caused you trouble.
5. If you want to do a retake for Sets, info is above in Grading > Content Modules > Grade Improvement and the Study Sheet tab has info on making a nice study sheet.

For Friday:

it’s not a long proof

Tasks:

1. Read Hammack, Chapter 2, up to the end of Section 2.3.
2. Do exercises for Hammack, Section 2.3.
3. Do exercises for Hammack, Section 2.2.