Disprove (Week 7)

Last week, we learned how to prove a statement false. As it turns out, the simplest way to prove something false is to prove the negation true. Since if the negation is true then the negation of the negation which would be the original statement, has to be false.
If a statement begins with a for all quantifier, we can quickly assess to see if it can be potentially a false statement by looking for a counter example. If the statement is an implication then we would try to think up something that satisfies the antecedent but not the consequence. If there indeed does exist one such case then we can assume the statement is false as the for all quantification is not longer met. Following this, we would negate the statement and proceed to prove the negated statement as we would for any other proofs. In the case of our example with a false for all quantified statement, we would now go forward by proving there a there exists quantification and most likely pick the example as the same one we used to figure out that the original statement was false in the first place.

Existential vs Universal Proofs (Week 6)

So far in this course, there seems to be two categories of proofs in terms of structure. One where we prove that something holds for all cases. Another where we prove that there is at least one such case that holds. When we attempt to disprove a statement, we will negate it and prove that the negation is true. It's negation will generally consist of existential and/or universal quantifies also, simply with existential switched to universal and vice-versa. Therefore, negations do not really have a different proof structure than existential or universal.

When proving a statement that includes an existential quantifier, we need to pick a value  that satisfy the required set. If that value is currently unknown and we may be able to derive it after going through the proof a bit, we would leave a blank line to fill it in later. Then the rest of the proof for the existential portion will be to prove the statement inside the world where we picked the value. If the statement holds for that value then we can conclude that there exist a value which satisfy the statement and thus conclude our proof.

When proving a statement that includes a universal quantifier, we need to assume a variable is a typical member of the set. Then in the world, we need to assume that the antecedent is true. Using the antecedent, we need to derive the consequence, and if we can, then we conclude that the antecedent implies the consequence. Since we did it for a typical member of the set, we then extrapolate and conclude that for all members of the set, the statement holds and thus conclude our proof.

Simply looking at the two, I seem to see that proofs including universal quantifiers requires a lot more assumption and bigger assumptions too than that of ones with only existential. Sometimes it seems that we need to take a big leap and conclude that it will work for any members of the set.

On Proofs (Week 5)

The structure of a proof in this course actually makes a lot of sense. However, some of the rough work gets confusing at times, especially when the only viable option seems to be to use a "definition" of something. Although the definitions of most things makes sense after the professor writes it down, I believe that I would not have been able to conceive the definition as a way to proceed in a proof. What I mean is that I would not be sure what I can use and when I would use it. In the professor's examples, he would of course have already known before hand what he could use but if I were given a new proof of something I am completely unfamiliar with, I do not believe that I could proceed far into that proof before getting stuck.
Compared to delta-epsilon proofs of limit or mathematical proof by induction (from MAT137), I find plain proofs (the ones that just says "prove") to be the most difficult to approach. Mainly it is because it seems like things are somethings pulled out of thing air and used to proceed in the proof steps. I understand that we can assume things to be true and then we would indent and thus enter a world where said thing is true. I understand that we need to end by concluding the statement is true or that its false by contradiction. However, most of the initial steps of the "prove something" parts seems like pure magic. Perhaps I do not know enough laws or theorems to approach it the same way as the professor. I guess my major focus now in this course is to learn why something can be used in a proof.

LaTeX for Assignment 1 (Week 4)

Prior to this week, I didn't know how to use LaTeX. So when I figured out that Assignment 1 had to be typed and couldn't just have been scanned into a PDF file, I started trying to download LaTeX and getting it to work. It took me a while before I realized that it needed a separate compiler editor to let me type. Even after having gotten the editor, I didn't know how to start a typing. I tried using the only symbol I knew in LaTeX (\forall) but nothing showed up in the preview. I eventually looked at the online editors and saw that I needed to use \start{document} and the closing \end{document} to encompass all of my text. I found this to be a hassle.
Not knowing most of the symbols, I had to browse through online sources to figure how to compute them. Mostly using Piazza to start a new post and looking under the symbols tab for the right command and then not saving the post. Although it took me a while to learn to be comfortable with LaTeX (about a day), I realized how much easier it is to use it rather than copying and pasting symbols into Word.
Then one day, Professor Heap told us during a lecture that we are allowed to just type things like \exists and \forall in Word. I started to think that it wasn't necessary for me to learn LaTeX. However, while working on assignment 1 late at night, trying to correct any potential errors before submitting it before the deadline, it struck me how much more manageable it was to simply look at symbols that matched the ones I wrote down on paper for my draft. This was especially so because I was feeling tired by then and would have easily missed translated symbols and words with '\'. Thus it caused to be greatful that I learned LaTeX. Plus it's similarity to programming languages helped me connect back to programming, which I have not done a lot of since I entered University for Computer Science ironically.