Week 9: Midterm and Assignment

As for the assignment, after getting to see the sample solutions I've been able to realize a few things that I didn't really understand or do correctly with how I wrote my proof. While in most cases I managed to get the general idea correct (whether it was to prove or to disprove, how to go through with that in a general sense) there were many areas that I may have overthought about how to prove it and did stuff that was unnecessary or poorly written and constructed. For example, I wasn't entirely familiar with the basics of how to handle multiple quantifiers in succession in a clear and concise way. In the assignment, I would write them out of order based on the type of quantifier I'm dealing with and what set they are in, rather than just take the quantifiers in the order they are presented. I also probably over-thought about how to represent an example of an existential variable by connecting every single one to an existing universal variable. It seems that in the solutions that isn't completely necessary and it really depends on what you're trying to prove and whether the variables are supposed to be based off one another or not. That definitely made things more complicated when, for example, in claim 1.2 I tried to base every existential variable off of d which was universal, which was both unnecessary, made the proof structure messy, and was possibly quite flawed. It seems that you only match an existential quantifier to a universal one if that existential comes after the universal in the order (Eg. 1.2: w is based off of d, but x and e are not).

That being said, I didn't repeat these mistakes for the midterm (as far as I can recall) since I learned about how to handle proofs more clearly and concisely beforehand. I was significantly more confident in my ability to write proofs during the midterm than the assignment, and I do believe I did a much better job in it too. The midterm went a lot more smoothly and I think my proofs are well ordered and don't have the same kind of gaps in transition that some of my assignment work did.