## Course Materials

- Online textbook Book of Proof (Hammack) — free PDF online
- Online textbook Proof and the Art of Mathematics (Hamkins) — free through CU libraries
- Grades are on canvas.
- The self-evaluation sheet.
- The proof-grading rubric.

### Quizzes

- First proof quiz solutions (Feb 1)
- Second proof quiz solutions (Feb 8)
- Counting quiz (Feb 15) and its solutions.
- Third proof quiz solutions (Feb 22)
- Sets quiz (Mar 1) and its solutions.
- Fourth proof quiz solutions (Mar 8).
- Logic quiz (Mar 15) and its solutions.
- Fifth proof quiz solutions (Mar 22)
- Sixth proof quiz solutions (Apr 5)
- Relations quiz (Apr 19) and its solutions.

**Practice proofs for Feedback**

- Sudoku (Jan 23)
- Set is Integers (Jan 25)
- Set containment ( Jan 27)
- Cups problem (Jan 30) + ChatGPT’s failures
- Number of subsets (Feb 1)
- Number of orderings (Feb 3)
- Combinatorial proof (Feb 6)
- Combinatorial proof (Feb 10)
- Proof about squares (Feb 13)
- Contradiction proof (Feb 17)
- Contradiction proof (Feb 20)
- Contradiction proof (Feb 22)
- Contradiction proof (Feb 24)
- Contradiction proof (Feb 27)
- disproof (Mar 3)
- disproof (Mar 6)
- induction (Mar 22)
- induction (Apr 5)
- modular arithmetic (Apr 7)
- induction (Apr 10)
- induction (Apr 12)
- relations (Apr 14)

### Practice problems for tests

- Counting problems and solution set.
- Sets problems and solutions.
- Existence proofs worksheet and its solutions.
- Logic problems and solution set.
- Relations questions and solution set.
- Functions questions and solution set.

### Other Daily Post And Worksheet Solutions

- Jan 27: PDF and video explanation (be aware these are example solutions, as there are many possible solutions — reach out if you want me to double-check yours!)
- Negation worksheet and solutions.

## Day-by-day Archive

The following calendar will be populated with links to material from each lecture, so you can catch up on anything you’ve missed and revisit important material.

**Wednesday, January 18th, 2022:**

- Snow day!

**Friday, January 20th, 2022:**

- The real first day of class. We did a group activity (see instructions and polyhedron sheet).

**Monday, January 23rd, 2022:**

- We discussed the solutions to the first proof assignment (sudoku), what a proof is, what purpose it serves and what we’re looking for in this class.
- We read the textbook together with an emphasis on active reading, and addressed some questions from the reading (Section 1.1 of Hammack).

**Wednesday, January 25th, 2022:**

- We looked over a sets/set-builder study sheet, introduced the notion of a subset, and played a bingo game to practice all these notions. We have essentially wrapped up the material in 1.1 and 1.3 of Hammack. Here’s the notes/study-sheet and the bingo sheet and card deck.

**Friday, January 27th, 2022:**

- We looked over the proof assignment (set containment)
- We discussed the sock and cups problems.
- We worked on a counting sheet.

**Monday, January 30th, 2022:**

- We looked over the cups problem solution.
- We discussed the Multiplication Principle (Textbook Section 3.2, phrased in terms of “lists” Section 3.1) for counting problems and did the first few problems of the counting sheet together.
- Class notes from our discussion.

**Wednesday, February 1st, 2022:**

- We had our first in class proof quiz. The schedule of quizzes is under the “Grading” tab.
- We worked on some more counting problems from our counting worksheet.
- Class notes from our discussion.

**Friday, February 3rd, 2022:**

- We discussed a variety of counting problems from our counting worksheet.
- These included
*overcounting*. - We recording the
*“Division principle”*for uniform overcounting. - We defined the
*binomial coefficient*and proved a formula for it. - We proved a
*combinatorial identity*(a sum of binomial coefficients comes out to a power of two) using*combinatorial proof*. - Here are class notes.

**Monday, February 6th, 2022:**

- We spent more than half the class discussing proof-writing with the first proof quiz as an example. Here’s the solutions/writing comments document.
- Then, we worked on the combinatorial proof worksheet.

**Wednesday, February 8th, 2023:**

- We worked on the combinatorial proof worksheet.
- We had the second proof test.
- Here are class notes.

**Friday, February 10th, 2023:**

- We reviewed the counting principles/overview
- I gave out a solution/study sheet for combinatorial proof.
- We defined and practiced Cartesian products and power sets
- Here are class notes.

**Monday, February 13th, 2023:**

- We covered intersection, union, set difference, complement, disjoint (definitions for sets).
- We proved that the square root of 2 is irrational.
- Here are class notes.

**Wednesday, February 15th, 2023:**

- We discussed Second proof quiz solutions
- We took up questions about practice counting problems and solution set.
- We took the Counting quiz; here are its solutions.

**Friday, February 17th, 2023:**

**Monday, February 20th, 2023:**

- Took up proof for feedback.
- Went over solutions to contradiction puzzlesheet.
- Started on Set Up Contradiction sheet.
- Some class notes (scribbles, really).
- Class was on Zoom because of illness (video is on canvas under “Media Gallery”).

**Wednesday, February 22nd, 2023:**

- Set Up Contradiction sheet (didn’t get to this)
- Negation worksheet (we started this)
- Proof Quiz

**Friday, February 24th, 2023:**

- Class note (summary of negation)
- negation worksheet with answers from class

**Monday, February 27th, 2023:**

- We looked over the solutions to the last proof quiz.
- We introduced Boolean algebra and truth tables as a way of verifying logical laws. We revisited some of the negations we had met previously and checked them as truth tables.
- Logical laws listing
- Notes from class

**Wednesday, March 1st, 2023:**

- Took up Sets problems and solutions.
- Sets quiz; here are its solutions.

**Friday, March 3rd, 2023:**

- We finished working through logical laws and did a few examples of boolean algebra using logical laws (simplifying/manipulating expressions; see Logical laws listing sheet
- We did the Wason Selection Task experiment on the class (and it worked very well) — here is PDF (in case you want to try it on your roommate)
- Then we introduced quantifiers (Hammack, Chapter 2.7), and handed out a Quantifiers Practice sheet.
- class notes
- Handed back Sets quiz ; here are its solutions.

**Monday, March 6th, 2023:**

- Quantifiers practice sheet and solutions
- solns to feedback proofs (1,2)
- class notes
- Existence proofs worksheet and its solutions.

**Wednesday, March 8th, 2023:**

- We did some example proofs of the existence variety, then we had the proof quiz.
- class notes

**Friday, March 10th, 2023:**

- We talked about negating quantifiers
- We did a fun non-constructive existence proof
- We took up quantifiers-negation practice sheet
- We discussed using quantifiers to define limits (justifying calculus)
- class notes

**Monday, March 13th, 2023:**

- We covered contrapositive and converse.
- We demonstrated proof by contrapositive.
- We demonstrated proof of an “if-and-only-if”
- We began Graph Theory by defining a graph.
- class notes

**Wednesday, March 15th, 2023:**

- Reviewed solutions to last proof test
- Reviewed solutions to practice problems for logic
- Took the logic quiz

**Friday, March 17th, 2023:**

- We discussed how many different math problems reveal themselves to be graph theory problems.
- Class notes

**Monday, March 20th, 2023:**

- We introduced induction with the Take-2-or-1 Game.
- Class notes and my writeup.

**Wednesday, March 22nd, 2023:**

- Class notes (we took up the proof for feedback, a variation on the take-2-or-1 game).
- Proof quiz #5 (contrapositive)

**Friday, March 24th, 2023:**

- There were videos to watch at home; see the daily post instructions.

**Monday, April 3rd, 2023:**

- We took up Fifth proof quiz solutions (Mar 22)
- We did two examples of proof by induction.
- The second proof (about graphs) comes with a formally-written version.
- Class notes.

**Wednesday, April 5th, 2023:**

- We introduced modular arithmetic.
- Class notes. Accompanying videos (user manual and under the hood).
- We had a proof quiz (induction).

**Friday, April 7th, 2023:**

- We took up the modular arithmetic practice sheet. (notes on this from class)
- We discussed number systems (“rings”), additive and multiplicative inverses and whether they exist in Z/nZ.
- We took up the proof for feedback.
- We defined the rational numbers via a notion of equivalence on pairs of integers.
- Class notes.
- A recording of class is available on canvas.

**Monday, April 10th, 2023:**

- We took up the last proof quiz (induction).
- We summarized some motivations for the notion of an “equivalence relation”
- We listed three properties it should have: reflexive, symmetric, transitive
- We verified that equivalence modulo n has these properties.
- We formally defined a relation, and discussed examples of relations (as sets of ordered pairs and as arrow diagrams).
- We formally defined the properties reflexive, symmetric and transitive, and gave examples and non-examples.
- Class notes.
- A recording of class is available on canvas.

**Wednesday, April 12th, 2023:**

- I was ill, so in class you just did the proof quiz, and the daily post was longer than usual, covering equivalence relations and partitions.

**Friday, April 14th, 2023:**

- Took up solutions to latest proof for feedback.
- Reviewed the basic ideas of equivalence relations and partitions.
- Introduced the definition of a function, both informally and formally (as a subset of a Cartesian product).
- How to represent a function: table, graph, arrow diagram, formula (sometimes), set of pairs.
- Injective, Surjective and Bijective
- Class notes.
- Class was recorded; video on canvas.

**Monday, April 17th, 2023:**

**Wednesday, April 19th, 2023:**

- We took up the review questions on relations.
- Class notes.
- We did the relations quiz.

**Friday, April 21st, 2023:**

- We did examples of proofs of injectivity and surjectivity.
- We discussed image and pre-image.
- We kept going on the second functions worksheet. Here are notes from class on that worksheet and complete solutions.
- Class notes.
- Video is on canvas.

**Monday, April 26th, 2023:**

- We discussed, for functions on finite sets, when injectivity and surjectivity are possible (together or alone).
- We proved several of these, including that for sets of the same size, injectivity implies surjectivity, and vice versa.
- We discussed pigeonhole principle and did examples.
- class notes
- Video is on canvas

**Wednesday, April 26th, 2023:**

- We did a few last pigeonhole problems
- We defined composition, identity function, and inverse functions, with examples
- class notes
- Video on canvas

**Friday, April 28th, 2023:**

- we covered inverse functions
- proved that a function is bijective if and only if it has an inverse
- class notes
- video on canvas

**Monday May 1st, 2023:**

- we talked about cardinalities
- class notes
- video on canvas