All posts by profstange

For Monday, April 10th, 2023

Proof for Feedback:

Another induction:  Hamkins (not Hammack), Chapter 4, exercise 4.4:  prove that for integers n \ge 2, we have f_n\text{\textless}2^n where f_n is the n-th Fibonacci number.


  1. I don’t get the sense that the class was feeling good about induction on the quiz.  I’m slightly changing the wednesday quiz schedule, so this coming wednesday will be another chance at an induction proof (grading will be updated so you get best 4 of 8 proofs instead of best 4 of 7).
  2. Read Hamkins (not Hammack), Chapter 4 (you may optionally skip the “Buckets of Fish” part), and do the proof for feedback above.
  3. Do some of Hamkins’ Chapter 4 exercises.
  4. Study induction.
  5. For those who missed Friday’s class, there’s a video on canvas, plus info as usual at “Materials Archive” above.

Due Friday April 7th

For Friday:

Proof for Feedback:

modular arithmetic (do this after tasks 1 & 2 below)


  1. Watch my video about Modular Arithmetic (9:22).  It repeats some of what we did in class Wednesday, but repetition is a tried-and-true method of learning. 🙂
  2. Do the associated practice-sheet (answer key at the back).
  3. If time remains, watch the next follow-up video (17:26).  This covers why addition and multiplication are well-defined (in other words, why they “work” mod n).
  4. Don’t forget that if you want to do a logic grade improvement, I’d like your study sheet by Friday.

Due Wednesday April 5th

Proof for feedback:

Hammack, Chapter 10, #4.


  1. Wednesday’s quiz is an inductive proof test.  See “Grading” tab above for the slightly modified quiz schedule.
  2. If you’d like a grade improvement for Logic, please contact me with a study sheet by this Friday.
  3. Read Hammack Chapter 10 up to end of 10.2 (he divides induction up into “weak” and “strong” which just refer to patterns of what depends on what; it’s all induction).
  4. Do exercises for Hammack Chapter 10.
  5. In class monday we did a graph theory induction proof.  Here’s a formally-written version.
  6. If you contacted me with a sets study sheet for a grade improvement, and I didn’t write you an email today (Monday), then contact me again to make sure I didn’t miss you.

Spring Break! What to do.

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 Wednesday, March 22nd, 2023

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.


  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 Wednesday, March 15th

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. 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, March 13th, 2023

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.


  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. Video 1:
        2. Video 2:
        3. Video 3:
  3. See if you can finish the rest of limit part (first two pages) of the worksheet from Friday.

For Friday, March 10th, 2023

Proof for Feedback:



  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.