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.

We’ve essentially finished counting. The test will be Wednesday. Just some general advice: you can go back to the exercises suggested in the last daily post as a study aid, as well as the course notes & materials, and reach out to me for help if needed.

Don’t forget the proof for feedback.

Today we covered Power Sets and Cartesian Products. These are chapters 1.2, 1.3, 1.4 in Hammack. Actively read these chapters (remember the reading strategies we discussed, and consider making a study sheet).

Let . This is the set of sets which do not contain themselves. Is ? Discuss on discord!

Do Exercises from Hammack 1.2, 1.3, 1.4 (comparing to answers in the back) as needed to solidify knowledge, if time remains (I’ll also remind you next daily post to come back to these exercises, so if time does not remain, that’s ok too).

Remember to put in a solid hour of useful work with these daily posts for full credit.

Look at solutions to old proofs (Materials Archive tab above).

Practice basic counting problems. At this point (with all our various principles: multiplication, addition, subtraction, division), you should be able to do all the exercises in Sections 3.1 through 3.5 of Hammack. Please do as many of these as you have time for (answer to odds are in the back). This is the type of problem we’ll see on the next wednesday quiz (which is on counting).

Obtain your proof 1 quiz (handed back Monday; if you missed class ask me and I’ll send you a photograph of it), and check out my feedback. Also look over this document about model solutions which we spent a while discussing in class. It’s got lots of important feedback/discussion on writing.

Then, when you feel you’ve absorbed the feedback, close all your resources/notes and do the proof on a blank page. Hand this in for feedback. (Don’t just copy one of the model solutions I’ve given you; that would have very little learning value.)

We have another proof quiz on Wednesday! You can see the schedule of proofs by clicking on the Grading tab at the top of the page. (Note: at some point I reordered the 2nd and 3rd topics.)

In fact, here’s a fun counting question: looking at the schedule of quizzes, how many ways could I reorder them? Two proof quizzes are not considered different.

In class we started a worksheet on combinatorial proof puzzles. Spend some more time on this, up to problem 4 ideally, more if you want. We will tackle it next day in class. Here are Friday’s class notes which are relevant.

Another counting question: how many different burritos can you actually make at Chipotle? Discuss on discord (is this problem well-defined/precise?)