Week 5: More proofs

I'm glad to have assignment 1 and term test 1 behind me. I had felt very unsure about everything in assignment 1, but one trip to Larry's office hours turned out to be very helpful and productive. Question 5 had made no sense to me, but I walked out of there feeling confident about my answers. I felt that I did decently on the test as well. Thank goodness for the practice tests, they were excellent practice and prepared me well so that there were no surprises on the test.

This week, we learned more proof techniques. In general, the material seemed straightforward, as it always does once the answer is in front already of you. Proofs are meant to be read easily, but I'm starting to realize that actually coming up with a proof requires a substantial amount of creativity. I will need to do many practice problems if I hope to do well on the next test. For the next assignment, I anticipate a lot of staring blankly at the questions, some useless scribbling, then giving up and binge watching The Walking Dead (zombies are really gross).

Week 4: You can't prove that I didn't write this in October

This week we talked about transitivity, limits, and structured proofs.

The definition of a limit was a little hard for me to understand. For example, take the function we saw in lecture, f(x) = x^2:

When I had first seen something similar to this in a calculus textbook, it made no sense to me. But in the context of this lecture, one way that I think of it is: "If my enemy chooses any e, then I can choose a d such that for all x-values that differ from 2 by less than d, the corresponding f(x) values will differ from 4 by less than e." So for some function g(x):

From this, I can see how as d gets smaller (x → xo), e can also get smaller (y → yo), so that:

We also started learning about proofs, and looked at some examples for direct proofs of universally quantified implications. I appreciate how rigorous proofs are, and how clearly and throughly one step must lead to the next, so that ultimately P leads to Q, and the implication is undeniably true.

Then we came across the "unexpected hanging paradox", and for a second I felt like I had just unlearned everything from the last twenty minutes!