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:

• Proof by contradiction puzzlesheet.

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:

• Functions worksheet the first
• Functions worksheet the second
• Class notes.

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