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.