Day 15, Mat385

Last time

Next time

Announcements
- Our exam is next time: it will cover everything we covered in the first three chapters.
  1. You're allowed a one-page sheet (front and back) of notes, and a writing utensil.
  2. You won't need a calculator.
  As I told you, you'll be allowed (in fact, required) to skip one question. There may be some that you may not skip, but you'll be able to skip one of the others.
- We'll be starting section 6.1 when we get back (after finishing up chapter 4); so you do have some reading and thinking to do over spring break.
- Also I'm hoping that you'll have figured out what your project will entail; be ready to make a decision when you come back from the break.
Last time:
- Let's go over our quiz
  1. key
  2. Fibonacci:
    1. If you index your Fibonacci numbers from 0, then this property is actually true from $n \ge 2$!
    2. The easiest way is directly, but the definition of the recurrence relation for Fibonacci numbers. (See "ta-da", in the key!:)
  3. You were to write a linear, first-order, constant coefficient recurrence relation. That involves both a base case, here $S(1)=1$, and the function relating $S(n)$ and $S(n+1)$.
    You hopefully recognized it as the sum of the first $n$ odd natural numbers, and then invoked a theorem you proved last week -- that the sum is $n^2$.
- We got pretty well through Section 4.1 (Sets).
- One focal point was the analogy between the binary operations of logic ($\land$ and $\lor$) and the binary operations of sets ($\cup$ and $\cap$).
- The power set was the primary tool we encountered, and we saw how it might be used in Venn diagrams to describe how many different subsets one can make with two sets in a universe.
  It is also the tool by which we come to the realization that infinity comes in various sizes....
  More on all of this after spring break.
Today:
1. We'll start with a little overview of where we've been so far.
2. Here is a link to all quizzes, and keys.
3. I'll have a couple of students do a few demos (if they're willing):
  - Brandon/Peyton/Ba: Section 3.1, #58, p. 177
  - Philip: Section 3.2, #12, p. 198
4. You might expect
  - some induction problem.
  - a contraposition/contradition problem.
5. Questions?
Links:
- Pi Day: How One Irrational Number Made Us Modern: The famous mathematical ratio, estimated to more than 22 trillion digits (and counting), is the perfect symbol for our species's long effort to tame infinity.

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