On this page, there will be a record of daily lectures, generally posted after the lectures, with handouts/links etc.

**Monday, January 13, 2020**- First lecture!
- Polygon exploration activity: Your First Mission (investigating polyhedra) and the polyhedron pictures. See also this, for fun.

**Wednesday, January 15, 2020**- A first proof (if n is even, then n squared is even), and a video of me talking through it.

**Friday, January 17, 2020**- We talked about active reading of a textbook. Here’s a post explaining that. We used this textbook as an example.
- We took up proof writing and then critiqued some example student proofs.

**Wednesday, January 22nd, 2020**- We had our first proofs quiz!
- We talked about the basics of sets: element, equality, subset, empty set

**Friday, January 24th, 2020**- We covered set builder notation and had a little practice in the form of a game: set game instructions and cards.
- We had our first badges quiz, which was just Sets I and Sets III.
- Here’s a 2 minute video that addresses the issue of nested sets.

**Monday, January 27th, 2020**- We did some set builder notation examples.
- We discussed the notion of a Cartesian product and did a worksheet on Cartesian products.

**Wednesday, January 29th, 2020**- We introduced power set, and did a worksheet on Subsets vs. Elements.

**Friday, January 31st, 2020**- We introduced proof by contradiction, using the definition of rational and irrational numbers.

**Monday, February 3rd, 2020**- Info on studies about cognition and math:
- Worksheet on negation handed out in class.
- My video on if-then statements.
- My video on the Dystopian Perspective on negation of statements, particularly if-then statements.

**Wednesday, February 5th, 2020**- We took up the badges quiz, and homework worksheet on negation above.
- We had a proofs quiz.

**Friday, February 7th, 2020.**- We did the beginning of a worksheet on Setting up Contradiction.
- We did some Contradiction Puzzles. These are proofs by contradiction that are fun because they have a creative spark to them.
- We had a badges quiz.

**Monday, February 10th, 2020.**- You had a substitute teacher: you had a proofs quiz, and finished up material on sets.

**Wednesday, February 12th, 2020.**- We introduced the idea of Boolean algebra and did truth tables.
- Basic Truth Tables for filling in, and a filled version.
- SPQR Logic Game: cards and instructions.

**Friday, February 14th, 2020.**- We introduced Proof by Contrapositive and did examples, then a badges quiz.
- Proof by Contrapositive Setup Sheet and solutions.

**Monday, February 17th, 2020.****Wednesday, February 19th, 2020.**- Pigeonhole worksheet with hints included.

**Friday, February 21st, 2020.**- Solutions to pigeonhole.
- We discussed contrapositive, converse, tautology, contradiction.

**Monday, February 24th, 2020.**- We discussed existential proofs, and I gave examples, and then there was a Proof Quiz.

**Wednesday, February 26th, 2020.****Friday, February 28th, 2020.**- We did a worksheet on counting problems, most people finished the first page of this at least, then had a badges quiz.

**Monday, March 2nd, 2020.**- We took up solutions to Quiz 5, the Existential Proofs worksheet (solutions here), and took up problems 1-6 from the counting worksheet, including discussing the Multiplication Principle, which can be read about in Hammack, Section 3.2.
- I emphasized that in order to answer a counting problem, one needs to imagine the process as a time-ordered series of tasks (e.g. in making a burrito; first choose tortilla (2 possibilities), then choose bean type (2 possibilities), then choose topping (3 possibilities) in order to count 2x2x3 different possible burritos). Different ways to do a task will result in different ways to count, but the answer should always be the same!

**Wednesday, March 4th, 2020.**- We took up a solution to Quiz 6, then worked on a Second Counting Worksheet.
- We discovered (via the worksheet) and then gave a formula for the ways to arrange k objects chosen from n objects. (The answer is n!/(n-k)!; make sure you know why!)
- We discovered (via the worksheet) and then defined the binomial coefficient, “n choose k”, which counts the number of ways to choose k objects from n objects (order of the k objects doesn’t matter). (The answer is n!/(n-k)!k!; make sure you know why!)
- The worksheet led us to the notion of a “Combinatorial Proof”, that is, a way to prove a formula by coming up with a counting question that both sides answer correctly. The example was that 2^n is equal to a sum of binomial coefficients. This is exposited as Example 1.4.5 here, albeit with pizza instead of committees. We will revisit this.
- I started taking Blackboard Photos. You can now visit the Blackboard Photos Gallery to see what has been going on in class.

**Friday, March 6, 2020**- Today I talked about counting principles: Subtracting a Case, Breaking into Cases (Addition), and Uniform Overcounting (Division). Please see the blackboard notes gallery (check photo dates to find correct photos) for exposition and examples.
- I also took up problems from the counting worksheets we did.
- Then we had a badges quiz which had new badges: Counting and Synthesis badges are now available.

**Monday, March 9th, 2020**- We discussed coronavirus adjustments; see the post.
- I gave two examples of combinatorial proof, i.e. proving an equation by showing that both sides are correct ways to count the same thing. See blackboard gallery.
- We did a worksheet on combinatorial proof. Here are solutions to the first two problems, and then a longer solutions/study-sheet/hints-sheet to accompany it so you can work at home.

**Wednesday, March 11th, 2020**- Our first day meeting on Zoom!
- We will be testing zoom, learning etiquette, technology, etc.
- As possible, we will get started on these two worksheets in zoom “breakout rooms” as well as all together with a whiteboard feature: functions first, functions second.
- Pics from my whiteboard notes can be found on the Blackboard Pics Gallery as usual. In future I will record my lectures.

**Friday, March 13th, 2020**- I will lecture about functions and then we will try out some improved breakout functionality with this google doc version of the second worksheet. (You must be logged into your CU account to access the worksheet; please make a separate copy to edit for yourself.)

**Monday, March 16th, 2020**- Here are solutions to the two worksheets: first functions solutions and second functions solutions.
- Today I will take these solutions up, and practice definitions and proof-writing for injectivity and surjectivity.
- As usual, video is uploaded to canvas and screenshots are added to the Blackboard gallery.

**Wednesday, March 18th, 2020**- We discussed proofs for injectivity/surjectivity some more.
- We discussed cardinality as a motivation for defining composition and inverses.
- We introduced the definition of composition.
- We did examples and the first question on a worksheet: PDF version, google docs version (click “open with google docs” to see a “correctly” formatted version).

**Friday, March 20th, 2020.**- We will take up the first question on the worksheet above. Here are solutions.
- We will introduce inverses, do examples and finish the worksheet above.
- We will discuss the very important relationship between inverses and bijectivity using the following: worksheet with blanks, worksheet with answers.
- We will return to the idea of cardinality.

**Monday, March 30th, 2020.**- We will discuss the worksheet on inverses and bijectivity, then discuss cardinalities and some theorems relating to that, and then, time permitting, take up an interesting challenge.

**Wednesday, April 1st, 2020.**- We did a LaTeX tutorial. LaTeX is THE way to typeset homework. The link is “LaTeX Introduction” on the upper left on the course website, and a video is available on canvas. I’ve also posted it as a freely available blog post on proofofconcept.

**Friday, April 3rd, 2020.**- Here’s some notes covering the basics of cardinalities.
- Today we will do some exciting and interesting proofs about cardinalities — which are the same, which are different?

**Monday, April 6th, 2020.**- We will take up the three theorems in the notes above.
- Then we will play a game — you may find it useful to open a whiteboardfox.
- Then we will do a game-related worksheet.

**Wednesday, April 8th, 2020.**

- We discussed worksheet solutions, which is to say, how to write a proof of the winning/losing positions in the take-2-or-1 game.
- This introduced the notion of induction.
- Then we considered the following problem: which integer amounts of postage can be formed from 3 and 5 cent stamps? We answered this and wrote an inductive proof.

**Friday, April 10th, 2020.**- More practice with induction. Here’s an induction worksheet.

**Monday, April 13th, 2020.**- For more YouTube examples of induction, try ThinkWell vids.
- Today in class I will discuss the Well Ordering Principle and give an example proof of the Least Counterexample method, which is essentially equivalent to induction.
- Here are some erroneous induction proofs (outside link).
- Here are some axioms for the integers.

**Wednesday, April 15th, 2020.**- Here’s are modular addition and multiplication tables as PDF.
- Here’s a tool I made to make an addition or multiplication table online.
- Here’s a fun visualization (multiplication tables) and another fun visualization (modular dynamics) that I made.
- Here is a worksheet on modular arithmetic.

**Friday, April 17th, 2020.**- We talked more about modular arithmetic, including the notion of “dividing” mod n.
- We started on a Worksheet on Relations.

**Monday, April 20th, 2020.**- We finished the worksheet on relations, talked about equivalence relations and equivalence classes, and showed how the rationals are build from the integers with equivalence relations.

**Wednesday, April 22nd, 2020.**- We will cover constructing the real numbers from the rationals.
- We will do more examples of induction.
- Here’s a proof without words of the example we did from induction.