Week 5 - Midterm Test and Proofs

First lets talk midterm test. Overall, I felt pretty good coming out of it.  I was wrestling with the last question (#3) for longer than I should have been. I noticed that one of them was an “exclusive-or” statement while the second one was a “not the intersection” statement and at first glance they seemed almost identical (negating one would also mean negating the other).  So I put away my truth tables and dusted off my Venn diagrams. Luckily, I noticed the rest of the universe “not (P or Q)” would also satisfy “not (P and Q)” but would negate the exclusive or “P and not Q or Q and not P”.  So I think I have this one pinned down. Venn diagrams: 1 v.s. Truth Tables: 0.

Prof. Heap posted solutions almost immediately and we even took it up the next class, which was great! I will note that for the first two questions I did not provide specific examples that listed elements like he did. He also mentioned that some students simply translated the statements into English. I hope I was not one of them but I don’t remember. Question 2 is where I’m uncertain because the proofs for the two statements about natural numbers are so basic that you could do a proof “by inspection” like I’ve seen in a lot of textbooks. So it is quite possible that my explanations for this question inadvertently became direct translations of the statements into English.  Lets hope not, but I guess what's done is done.

On the subject of proofs, moreover, simple proofs (like the one above), I find it tedious to have to prove things that are “by definition” true, or “by inspection” true. Prof. Heap had warned us that we were going to struggle with how far our proofs should go. But a perfect example would be in class where we'd get to a “simple proof” stage and I would be struggling to find a proof without saying this is “by definition” true, or “by inspection” true and then all of a sudden he would turn around and tell us, “Oh, you don’t have to prove that.”  Hopefully, our homework problems will be at least one step above a basic definition so we aren't running around in circles saying, “it’s true because I said it’s true!”