Studying for Test 2

Since test 2 is tomorrow, I will be spending the night reviewing the material that we need to know (aka proofing), and piecing together a 'cheat sheet' that will (hopefully) help me during the test just in case I somehow blank out and forget natural and real numbers are (sure, laugh at me now, but things like this are more likely to happen than you's think).

I feel that through the large amount proofing we had to do in assignment 2, I've got a pretty solid fundamental understanding of the structuring and execution of a good proof, although delta epsilons have been haunting me since the day we introduced it in calculus, so I'll definitely be paying very close attention to those.

Some other helpful things I decided to include:

Ceiling(x): the smallest integer no smaller than x
Prove ]x[  + 1 > x:

Since ]x[ + 1 > ]x[ and we know  ]x[ >  x (by definition of ceiling)
Then ]x[ + 1 > ]x[ >= x
So ]x[ + 1 > x


Strategies for proofs:
1. Sometimes changing the direction and proving the contrapositive is easier.
    Example: For all whole numbers x, if x^ 2  , then x is even. 
    Proving directly would be difficult, but proving that if x is not even, then x^2 is not even
    is fairly simple to do.

2. Disprove a statement by negation (I find it simpler and cleaner than disproving by contradiction, but maybe           
     it's a personal preference.) 


I spent at least an hour just staring at delta epsilons, and I think I can actually do them now (I've completely forgotten what I learned last semester in calculus). Looking at examples really help!


I finished my 'cheat sheet' (I just added a bunch of things onto the back of the one for test 1 since those things are still relevant), and I think I'll head to bed now.

♩ヽ(・ω・ヽ*)♬ Good luck everyone! ♪(*ﾉ・ω・)ﾉ♫