Third week: Proofs

This tutorial we did proofs. At first I was confused about proofs because I didn't know whether to negate them or use the rules; I only understood how to negate statements, not prove them.
So I didn't really understand the first tutorial question, but I got the second one. I was also confused about the third one because I thought it involved vacuous truth and I got the opposite answer than the answers the TA gave.

Then today I went to office hours and it cleared it up for me. 
I still find the logical notation very confusing; just when I think I have understood it and grasped the concepts, I forget it and have to relearn it and look at it over and over. Hopefully I'll soon be familiar enough with it for it not to be confusing anymore.

I put:

(A ^ B) => C
But the answer was:
(A V B) => C
And then I subbed in the statement:
If the temperature is hot, or the sun is shining, then it's swimming weather.
When looking at this statement, I didn't know if both A and B were needed for the statement to be true or not. That's why I was confused. I didn't know which one was right.
After office hours today, I saw how (A V B) => C could change to (A => C) ^ (B => C) by using the identities, distribution, and DeMorgan's law.

(A V B) => C
¬(A V B) V C
(¬A ^ ¬B) V C
(¬A V C) ^ (¬B V C)
(A => C) ^ (B => C)

As for the proofs, I now realize that I have to negate them, and I DO have to use DeMorgan's law and the other laws.

I still have trouble with them, but hopefully I can get them down by the test next week.

Week Two: Vacuous Truth & symbols

The tutorial exercises this week were even more confusing, so I decided to get some help. I went to office hours and asked about the material.

Despite the fact that English to symbols is supposed to be unambiguous, and despite of the point of it being to write it in a straightforward way which cannot mean something else, I found that the symbols were very unclear in the way they could be written. After trying it many different ways, my answers were, even by the end, all very different than what the actual answers were.

I finally grasped the concept of vacuous truth, which was really helpful in understanding some of the other concepts.

An implication can only be false if the antecedent is true and the consequent is false.
If the antecedent is false, it doesn't matter whether the consequent is false, the statement will still be true.

I also learned about inclusive vs. exclusive or; it's interesting how when we use 'or', we always use the inclusive 'or' (meaning 'A or B or both'), and we have to add more to the statement for it to be an exclusive 'or'.

Universality & Existentiality: The First Week

                Logic and expression. And reasoning. That’s what this course is about. How to express yourself and to practice your reasoning skills in solving problems logically. It’s an interesting combination, in my opinion: math combined with writing and expressing yourself. I never would have thought that you could express yourself in math. I have always leaned more toward expressing myself in writing, so this should be interesting.

I thought that learning about the different steps to solving a problem was very useful. I have always had trouble with how to tackle logical questions which require thought and reasoning, so I will always try to apply this method to all the problems I want to solve from now on. In fact, the other day when I started on a programming exercise for another computer science course (CSC148), I tried to use that approach to tackle the problem. As a result, I successfully solved the algorithm and coded it in even less time than I would usually take.

We started this week on logic. Universality and existentiality. It reminded me of grade 12 Philosophy, in which we also learned about these things. Lots of proofs with p’s and q’s, deductive and inductive reasoning...and other forms of logical comprehension.

The first problem the Prof showed us, the streetcar drama, was an interesting one. He said no-one had solved it since the 1900s. When I looked at the problem, it seemed sort of simple, because the explanation was simple, but when looking deeper into it and thinking about it in depth, I realized it was a lot harder than it seemed.

A: Haven't seen you in a long time! How old are your three kids now?

B: The product of their ages (rounded down to the nearest year) is 36.

A: That doesn't really answer my question.

B: Well, them sum of their ages is [fire engine goes by]

A: That still doesn't tell me how they are.

B: Well, the eldest plays piano...

A: Okay, I see: their ages are [you have to get off the streetcar]

In order to solve this problem, you must first understand the problem, the inputs and the outputs. Then you must think of different ways/methods to solve it. Only after you have done these first two steps can you officially start carrying out the plan you chose. If it doesn't work, figure out exactly what part is not working and why.

I admit I had problems with the first exercise we were asked to do in class, and it made me a little nervous about how the course would play out in the end. I have heard from a lot of people how they thought this course to be very rigorous and challenging, and I could see from the beginning how much thought was needed to be put into this in order to understand it.

I worked on the tutorial exercises as best I could, trying to apply the information from the lecture into the problems. Again, it reminded me very much of what we did in the logic unit of grade 12 Philosophy, which was good because I enjoyed that unit. We learned about universality (when we state that ‘all’ of p is q) and existentiality (when we state that ‘some’ or at least one of p is q).

The tutorial exercise made me think very hard because it was a bit confusing, and I came out with a number of different results, but by the end of the tutorial I had mostly understood it.