WEEK 4

Hello everyone, welcome to week 4 post for csc165 course. That week was terrific, proving anything is the basic of any science or non-science subject.

CLASS

Before start writing my thoughts about the class, just a reminder that the first midterm is next week, as usual, I am completely crazy, but let’s see how we can manage that, good luck to everyone, and don’t forget the aid sheet, if you need some reference, check the link with old exams here (update: The old midterm can be found here). If I have some time int he future and before the exam, I will try to summarize information about the previous final exams.

On week 4 we had, Bi-implication, Transitivity, Mix quantifiers, and the star of the week Proofs.

Just to clarify, all writing on those posts are explanations or thoughts that helped me to memorize or learn concepts, even if you don’t find what I think in any book, it doesn’t mean it is wrong or that I am 100% right.

So, let’s begin with bi-implication, that I think to be only a special case of the implication itself, or the “if and only if” thought, and it is represented by P <=> Q, and is the same of (P => Q) /\ (Q => P), and using implication we find that it is the same of (¬P \/ Q) /\ (¬Q \/ P). Following some laws, it can be found the identity, and also the exclusive OR with the negation of the bi-implication.

Transitivity is a basic property on set theory, and says that if A is related to B, and B is related to C, then A is related to C. Basically, if (A => B) /\ (B => C), we can assume that A => C.

Mixed quantifiers is the “evolution” of what we’ve been using so far, instead of one set X, now we can use set X and set Y, and as many sets as we wish to express what we want to say. Also, it increases the complexity of the predicates we can use.

Now, let’s focus on Proofs. It does not matter if is one way or another, if you believe something when someone proves it is wrong or true. Every one need to prove something on the daily life. So, you use an argument that is precise enough for the situation. In the science case, it is necessary to be precise enough, but if you’re trying to convince your friend that the new restaurant is awesome you don’t need to be as precise as a mathematician.

To get the proof, first you need to completely understand the problem and spend time to discover the truth of the proof. After, you just need to write the proof. Believe me, you’re going to spend the most part of the time thinking (or discovering) a way to prove, so, be imaginative.

In other to write a proof in a precise way, the scientific community uses some well know structures, and for now, on week 4, we are going to learn the direct proof of universally quantified implications. Don’t be scared, the name is huge, but if you break it, then it is not so scary, just think that as parts:

1. Direct proof…
2. Universally quantified…
3. implications…

The only unknown part that we didn’t studied before is the number 1, and it is self explanatory.

In order, to prove that R => T, we should remember something that I said at the beginning about the bi-implication (if A=>B and B=>C, then A=>C). In that case, if we put our values inside the bi-implication, R should be A, and T should be C, and to prove that implication we need to think of a B, where B is know to be TRUE. In fact, we need to find a chain, and that slide from class is perfect to show a chain, and also, show the fully sentence:

But no proof is equal to another, it could have the same structure, so the only thing that can make you truly learn is practicing.

TUTORIAL

The tutorial was great, we made some progress toward learning proofs and the questions were some how challenging, and the quiz was just one question which could be solved with laws explained during the tutorial, I had no trouble to go through the tutorial this time.

#I hope you liked, and don’t forget to check the week 5 (soon),
#I am finally up to date with my posts, I am so happy :D
#Questions, comments, suggestion?! Write it below, I will answer as soon as possible.
#Thank you.