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”).

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.)

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.

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.)

In class friday we talked about quantifiers. Read Hammack, Section 2.7 and do exercises.

I handed back the Sets test. Check out the solutions. and make sure you understand anything that caused you trouble.

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.

You can also do exercises from Sections 1.1 – 1.7 of Hammack.

The example sheet doesn’t contain too many with pictures. Look at Hammack, Section 1.2 Exercises part B, Figures 1.2, 1.4, 1.5, 1.6. In class, for example, we drew some pictures in the real Cartesian plane (a square, a disc, a line), and asked whether they were Cartesian products or not. Don’t forget about these types of sets.

Here’s a reminder of the material:

basic definitions including set, element, equality, empty set, cardinality, subset

Set-builder notation and interval notation

ordered pairs, Cartesian products and powers, subset and powersets, including cardinality of such

operations (union, intersection, difference), universe, complement, Venn diagrams to visualize these

Don’t forget to get me your study sheet if you want to improve your counting grade (optional grade improvement activity; see Grading > Content Modules > Grade Improvement from top bar). Monday is best, Wednesday is the last opportunity.

Check out solutions to the last proof. Did you notice the statement was false?

For more negation practice, Hammack, Exercises 2.10 (odds have answers in the back, as usual). Keep in mind there are many correct answers to a single negation problem (but they all mean the same thing).

Don’t forget the sets test is coming up. Very soon I will put up some practice problems (these will go up on the Materials Archive page). The material covered is:

basic definitions including set, element, equality, empty set, cardinality, subset

Set-builder notation and interval notation

ordered pairs, Cartesian products and powers, subset and powersets, including cardinality of such

operations (union, intersection, difference), universe, complement, Venn diagrams to visualize these

You have a proof test on Wednesday. Review the proofs by contradiction that we have covered (hint: Materials Archive tab), in preparation for the proof test. Focus on understanding the structure of a proof by contradiction and the negation necessary.

In class on Wednesday we will take up this worksheet on setting up proofs by contradiction (which is really about negating). Take a stab at it before class, as practice with the structure of proof by contradiction.

One week to our sets test, so please keep working on solidifying knowledge there. Read Hammack, Section 1.6 and do exercises as needed.

Click the “Grading” tab above and read the part about “Content Modules” > “Grade Improvement”; this is info on how to improve your grade on the counting test if you want to. The first step is to send me a study sheet to look over; once I approve of that we can do an oral retake together. There’s a bit of a deadline: you two weeks to get your study sheet approved and retake scheduled.

We have basically finished the material on sets (to be tested in a week and a half). For review/solidification, read Hammack, Section 1.5 and do exercises as needed.

Ponder the rest of the contradiction puzzles, which we will take up together next class.

Read Hamkins (not Hammack), Chapter 1, A Classical Beginning (pages 1 through 8). This is about the proof of square root of 2 being irrational. Note: Hamkins can be read through CU’s library system online! This link(or searching “Hamkins Proof and the Art of Mathematics” at CU Library) should get you to the library’s listing, then click through for online access; may require being on campus, using CU VPN (library instructions) or using library proxy to prove you are a student.

Hamkins, Chapter 1, Exercise 1.9 and 1.10. (Both proofs are wrong — what’s wrong?)

I have still noticed some confusion on “element of” vs. “subset of”. Do this worksheet on the topic. The answers, if correct, spell a little message (ask on discord if it isn’t clear).

Study for your test on Wednesday. You can hand in some practice problems on canvas. Resources/info:

Test will be 20 minutes at the end of class, on counting problems. The problems will be short-answer problems (not proofs), grades mainly based on getting the answer correct.

Materials from class are all available under “Materials Archive” tab. In particular, the main topics are:

Multiplication, addition, subtraction, and division principles

How to count all sorts of things by use of the principles above

How to count the number of ways to order k elements chosen from n elements (and why it works)

How to count the number of subsets of a set of size n (and why it works)

How to count the number of subsets of size k chosen from a set of size n (binomial coefficient and its formula and why it works)

Here are some example test problems I’ve given in the past. For hints, discuss on discord (use the “spoiler” feature on discord if hinting/spoiling anything). For full solutions, see this solution set. BUT I suggest you use the full solutions only when you are confident you have solved it yourself — asking for a hint is a better way to proceed for more effective learning.

Hammack chapters 3.1-3.4 exercises are excellent practice.