CSC165 Lecture Week 3

I was always told that university would come with a lot of new challenges, and I think I faced my first one this week. In high school I grasped most topics very quickly and very easily. The only thing I really struggled with were proofs in math, but we didn't have to know them for tests, so I am ashamed to say that I pretty much just ignored them. Now that I actually have to learn this stuff, I have to figure out how to approach these problems. The tutorial exercise for this week took me two hours to do and I had to have a lot of outside help. Hopefully, in the future I will get better at this kind of problem solving.

Question 2 from the tutorial:

To help me solve answer this question, I first went to the CS help centre. I got two bits of information that helped me. First the x on the left and within each set of brackets is not necessarily the same x, but they could be the same x. I also figured out from looking at the online notes that (a) and (b) are actually laws and they are called quantifier distributive laws.

I then Googled quantifier distributive laws to see if I could get a different perspective that would help me. This website http://goose.ycp.edu/~dbabcock/PastCourses/mat235/lecture/lecture06.html, gave me concrete definitions of the predicates, which helped me to finally solve the problems. Let D be the set of people. Let P(x) mean x likes pie and Q(x) mean x likes cake.

Three things to keep in mind:
1. An implication can only be proven false if the antecedent is true while the consequent is false.
2. Or (V) is true when one statement is true and false when both statements are false.
3. And (^) is true when both statements are true and false when one statement is false.

And now my answers to the questions:
(a) is true in both directions. Translating it into English makes it obvious. Everybody likes pie and cake, therefore everybody likes pie and everybody likes cake. Everybody likes pie and everybody likes cake, therefore everybody likes pie and cake.

(b) is always true from left to right. When someone likes cake and pie (proving the right true), that same someone likes pie and that same someone likes cakes, so the left cannot be false. The other way however is sometimes false. Let's say person A likes pie and person B likes cake (proving the left true). There does not have to exist a single person C who like both (proving the left false). The antecedent can therefore be proven true when the consequent is false.

(c) is always true from right to left. Everyone likes pie (proving the left true), therefore everyone likes pie (proving the right true). The other way is false because everyone likes pie or cakes could mean that, for example, half of everyone likes pie while the other half only like cake (proving the right true), therefore there may exist some person A who does not like pie and some person B who does not like cake (proving the left false).

(d) is true in both directions. It's the same as saying: someone likes pie/cake therefore someone likes pie/cake, which makes both sides true.

These are, of course, not formal proofs and I can't Google around for answers on assignments and test, but I'm hoping that this was a decent first step. I learned some new strategies for understanding problems (rewording into something I understand, coming up with concrete examples) so I'm getting better at this.

Now I shall move onto my next challenge of learning to do laundry because I cannot take it home this weekend.

CSC165 Lecture Weeks 1 and 2

This is my first post for CSC165, so just a quick housekeeping note. This used to be a physics blog for physics class, but from this point forward all posts will be dedicated to CSC165.

Learning about mathematical expression and reasoning or logic is pretty much the same as learning a new language. Instead of learning new words we are leaning new symbols such as ∀ for universal quantification and ∃ for existential quantification. Instead of learning grammar, we are learning new, more precise ways to interpret the English language. Surprisingly, I am actually really enjoying this course, which is odd since I hated those years where I was forced to learn French.

The first week’s topic was very intuitive for me. Universal quantifiers like ‘all’ and ‘every’ can only be proven by looking at every element in the set, and can be disproven by one counter example. Existential quantifies like ‘some’ and ‘there exists’ can be proven by one example and disproven only by looking at all the elements of a set. I think this duality is really elegant and really makes sense in everyday English.

On the other hand, the topic from the past two weeks that I have found most interesting is the vacuous truth and it seems very counter intuitive. It’s pretty weird to think that a statement such as, ‘If the sun does not rise, then pigs will fly’, is true. In everyday English, this whole sentence is absurd. None of these things will ever happen, so it would seem as if this sentence where completely false. Although, once I think about it, the fact that this statement is true starts to make a lot more sense. If P, then Q implications can only be proven false if there is an example where P is true but Q is false. In the case of the statement above, P is never true, so this statement can never be proven false, and is therefore true. I really like how precisely this the rules for implication are defined. It definitely trips me up sometimes, but I do often find everyday English to be too loose and ambiguous, so maybe that’s why I’m liking this course.