As per the second to last Monday of the semester, I’m thinking over the past 12 or so weeks, and wondering if I’m any smarter at this math stuff. Casey did say we’d learn how to prove things in his class, but am I there yet?
I’m not going to lie, I really thought I would take this class, and somewhere right in the middle of the semester we’d come up with this universal proof writing formula that could be used for all mathy things, everywhere.
What I’m realizing ties back to the quotes we blogged about at the beginning of the semester (corny I know). Math isn’t all formulas. It requires common sense, and the sense to throw that common sense out the window when necessary (ie thinking about infinity). Math is what you make it, and for me it’s still fun. Here’s what I’ve learned so far:
- What a hypothesis is. I may have learned about this in all science classes since 3rd grade, but now I finally get it. Really all you need to know about the hypothesis, is that you always assume it true. And if it’s false, your work there is done. (ie if a unicorn runs through the library, I’ll ace college. neither is going to happen, so why worry about it. in math terms, we call it a vacuously true statement).
- What a conclusion is, and how easy it is to make a proof into proving a very, very small part of a statement. Like with implication statements. You basically assume 75% of the statement is true and just prove the last conclusion. The key here is distinguishing identifying the conclusion and having something be easy to prove. Just because you only have to prove 25% of a statement, doesn’t mean it’s easy.
- Counterexamples are the holy grail of FOM. Very rare, hard to find, but freaking awesome when found.
- Implications are straightforward, sometimes. At least you know when you see a little ⇒, you know what you’re dealing with.
- Crazy letters can help too. Knowing what ∀ and ∃ mean is usually about 33% of the battle.
- Proving sets are equal requires two things: patience, and proving that each is a subset of the other one.
- Functions are not just functions. If you’re a fan of the good old calculus function, run far way from FOM right now.
- Injectivity, surjectivity, and bijectivity will help you in ways unimaginable at the beginning of the semester. Just learn what they mean and never forget it.
- Sometimes, when you can’t figure out a statement, it helps to turn it around, a lot. (ie converse, contrapositive, etc.)
- Induction is probably the easiest concept (which isn’t really even that simple). Just think in terms of k, and prove that every other k after that acts the same way. Or, start with k, and prove that all k’s before that are true. Sound good?
- In the words of Casey Douglas “And when it all comes down to it, all you’re doing is wiggling”. Wiggles are equivalence classes, which are partitions. There is two much notation for all three of these things, so once you understand any of it, stick with one and chug through with it.
- When you’re proving an equivalence relation, just prove three things: transivity, reflexivity, and symmetry.
- There are so many different kinds of infinity. Knowing this should make you feel very small (so eat the extra donut if you want to).
- Just because there are different kinds of infinity, does not mean you can leave infinity be. Match infinite sets up to each other using functions and keep on provin’.
- Modulo is not spanish for anything, at least not in FOM.
- And finally, math is only fun when you give yourself time to think about how really awesome it is.
Maybe FOM is making us smarter, even though it feels pretty confusing mostly all of the time.