# Materials Archive

## 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:

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:

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:

Wednesday, February 8th, 2023:

Friday, February 10th, 2023:

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:

Friday, February 17th, 2023:

Monday, February 20th, 2023:

Wednesday, February 22nd, 2023:

Friday, February 24th, 2023:

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:

Friday, March 3rd, 2023:

Monday, March 6th, 2023:

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:

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:

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:

Monday, April 3rd, 2023:

Wednesday, April 5th, 2023:

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:

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: