We've made it to the end of the course! "FINALLY," you say.
Well hold your horses there, buster brown. We're not out of the
woods yet. We have, of course... the DREADED FINAL EXAM to worry about!!!
No, there's no use running.
Or, perhaps you may be saying, "Alex, I spent an entire semester in this
class and I don't remember anything!" or "I don't feel like I learned anything."
Like flames... at the side of your face.
Well I'm here to say: POPPYCOCK.
You remember plenty, and you've probably
learned lots of things! Here, I'll guide you by the hand and list some of the stuff
that you probably know and don't even know you know!
Don't worry, I'll STEP you through it. Hahahahahahahaha
Look at all the stuff we learned!
Chapter 1 was on logic:
- Natural Deduction in Propositional Logic
- Existencial and Universal Quantifiers
- Natural Deduction in First-Order Logic
- Building Proof Trees
- The difference between intuitionistic and classical logic
- The law of the excluded middle
- Proof by contradiction
- Variables, propositions, atoms, constants
- Homework: DeMorgan's Laws
- Homework: Double Negation
- The Project: The Gentzen System
Building proof trees: somewhat painful!
Chapter 2 was on set theory
- What's a set?
- Set equivalence, subsets, and elements
- Russell's paradox
- Power sets
- Unions, Intersections
- The definition of the Natural Numbers
- Countable/uncountable infinities
- The Axiom of Choice?!?
- Complete (strong) induction, weak induction
- Homework: Big Unions/Intersections
Chapter 3 was all about functions
- What is a function?
- Ordered pairs, cartesian products
- Function composition
- Recursion and the Fibonnaci/Lucas numbers
- Inverses of functions
- Images and inverse images
- Injections, Surjections, Bijections
- The Pigeonhole Principle
- Hilbert's Space Filling Curve
- How do we REALLY prove things?
- Homework: The Triangular Numbers
- Homework: Hilbert's Hotel!
Chapter 4 was an introduction to graphs
- What are graphs?
- Verticies and Edges
- Directed, Undirected
- Paths and Strongly Connected Components
- Node degrees, in degree and out degree
- Cycles, Connectivity
- Trees and Forests
- Spanning Trees
- Graph Isomorphism
And that's it! If you know all of that, you're surely set for the
final. And look how much you know! Maybe you knew some of it before, maybe some
of it will fall out of your head soon, but that's OK! You've surely gained
something from the class.
I'll be having office hours next week for any questions you have.
It's been delightful being your TA this semester, you guys were
great, even if you did bombard me with 100000000 emails.
I guess it's time to say goodbye...
No no, don't worry about me, I'll be fine.
Haha, just kidding! It is time for RIDICULOUS DANCING.
Enjoy your winter break!