# Lectures

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

• Monday, January 13, 2020
• Wednesday, January 15, 2020
• Friday, January 17, 2020
• 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
• Monday, January 27th, 2020
• Wednesday, January 29th, 2020
• Friday, January 31st, 2020
• We introduced proof by contradiction, using the definition of rational and irrational numbers.
• Monday, February 3rd, 2020
• 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.
• Monday, February 10th, 2020.
• You had a substitute teacher:  you had a proofs quiz, and finished up material on sets.
• Wednesday, February 12th, 2020.
• Friday, February 14th, 2020.
• Monday, February 17th, 2020.
• Wednesday, February 19th, 2020.
• Friday, February 21st, 2020.
• 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.
• 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.
• Monday, March 9th, 2020
• 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
• 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.
• 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.
• 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.
• 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.
• 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.