WEEK 3

Hello everyone, welcome to week 3 post, a continuation of week 2 material about the language of math.

CLASS

This week we covered Conjunction, Disjunction, Negation, Truth table and manipulation laws.

Conjunction and Disjunction are used to put together two or more predicates (for more about predicates, check week 2). Conjunction is the “and” and Disjunction is the “or”, and their symbol are “^” and “v” respectively. Again, sets has a really important role understanding those two. First, “and” can be intersection, and “or” can be union. Nothing more to talk about, just basic set concept.

We already talk about how to negate a predicate on week 2, but now we are talking about a whole statement, and instead of negating all the statement, we should negate just particular predicates inside the statement, for example:

How not to do:

How it should be done:

while negating a statement, always remember the quantifier table from week 2.

Another element that can be used is parenthesis, and we should be careful, it works similar to basic equations, but we should remember that what happens inside parenthesis does not affect or can be reflected to another parenthesis, and what is inside parenthesis is called “scope”. Also, parenthesis help us to organize the idea of the statement, during the class the following slide was presented:

Furthermore, the truth table was presented as a solution where we should deal with more than two predicates, since the Venn Diagram does not help us to understand properly when a large number of predicates is showed, for example, below we have more than two predicates, and it is quite confusing.

To help us with that and show all possible outputs, we use the truth table:

But we should focus on the special cases, that can be used to say interesting things about the statement, and this slide showed us a good picture of those special cases:

Moreover, we have some laws, and they are De Morgan’s Law, table example below. Also, commutative, Associative, and distributive laws are applicable.

TUTORIAL

The tutorial had some trouble, some people in the class had difficult to understand and solve the questions, but I need to say that is not easy to know how to apply what we learned in class to another problem that is completely different from the example that we saw during the class, but we need to think bigger and know that the language math can be applied almost to everything that we say on our daily life.
I had some problems to solve some part of question 2, but the truth table can be a great ally to solve most of the questions without the need to think in an example and try to deduce something from that. So, just go to the truth table, and let it do the job. After that you can think a good example to present your answer.

#I hope you liked, and don’t forget to check the week 4.
#Questions, comments, suggestion?! Write it below, I will answer as soon as possible.
#Thank you.

WEEK 2

Hello everyone, this post will be dedicated to the second week of class, a lot of material was covered. Also, we had our first tutorial followed by the quiz. So, let’s start with the class.

CLASS

The material covered was:
- Quantifiers, verify/falsify
- Sentences vs statements
- Predicates
- Implications and equivalence

In my opinion, the most relevant content was Quantifiers, Predicates and Implications, those were the keys to understand the content of week 3 (…shush, no spoilers).

At the beginning I was really scared about that math language, all those symbols and I never thought I could make sense of what it was trying to tell me, but now it is some how easy, but don’t be fooled, it is not easy, and also is really helpful if you have some understanding about set notation. The image below was used to demonstrate how the math language looks like.

The quantifiers are divided into Universal and Existential, and they were putted together in a really helpful table with how to verify and falsify, and the table is some how self explained.

A really amazing tool to not get lost when trying to verify or falsify is the Venn Diagram, where you can represent the sets and graphically shows what is empty or not.

Right after, the battle between what is a sentence and what is statement begun, it is quite easy to mix both because a statement is a sentence, but the opposite is not always true, but in the end it is very simple. In a statement, the object of the sentence is quantified, that is all. For example, if I say “The planet is beautiful” is planet quantified?! yes…no?! If you compare to that another sentence it gets easier to know which one is sentence or statement, and the another sentence that is a statement can be “All birds can fly”, just remember that what is being analyzed is which one is a statement and not if it is true.

The Predicate is something a little bit complex, but can be explained using very simple examples, using the bird example I can say the P = “The birds that can fly”, so “Bird x can fly”, then P(x), and “P(x)” is a predicate. So it is a full sentence compressed in two letter, it can save a lot of paper, can’t it?!
If I would like to negate it, just use that symbol “¬”. Now, you can easily know what I want to say with that “¬P(x)”. It can get a lot more complex, and sometimes more information about the sets should be provided.

Implication is the base to construct more complex thoughts mathematically speaking, and to anyone familiar with any computer language it should be easy peasy. Basically, “IF P, THEN Q” or “P => Q”, so P implies Q. But, be careful, since it does not mean that P causes Q, and some weird sentences can come out, like the one presented in the class “If it rains yesterday, then the sun rose today.”, and it is true. Also, some interesting rules came out, converse and contrapositive. So, tons of everyday sentences can be translated just with the symbol “=>”.

Following implication we were presented to Equivalence, that in my opinion is just a special case of Implication, it simply is an implication that works both ways, so “P => Q” and “Q => P” merged into “P <=> Q” or “P iff Q”, in english you read “if and only if”. Back into the bird example, bird P and fly Q, “P <=> Q” if it is a bird, it can fly, and if can fly, it is a bird, so it can fly if and only if it is a bird. (I know, it is getting complex, I thought the same, but it needs some time and thinking to get into my mind.)

TUTORIAL

The tutorial was all about the Venn Diagram, and how to verify and falsify. Nothing I could not handle, but surprisingly during the quiz, that was composed by two questions I didn’t answer the first one correctly, and that is the warning, it can be very simple, but we are dealing with small letters and simple graphics that can represent a whole world, so it should be treated very carefully. I will show the question below and say exactly what happened, so everyone can be aware.

I will write just the key sentences of the question one:

• U is the set of all students currently enrolled at UofT;
• P is the set of all students enrolled in a programming course;
• L is the set of all students enrolled in a linguistic course;
• Claim: “all students enrolled in a programming course at UofT are also enrolled in a linguistics course at UofT”.

-> Prove the claim is correct.

Well, in a really fast thought I answered that “To prove it is correct, all students should be enrolled in both courses.”.
When I walking back home after the quiz I realized how wrong I was. Basically, the statement was a simple “P => L”, and not “P <=> L”, and just some time after the quiz I knew that my first question was wrong, mainly because of impatience, but most of all, due to the inability I had to transform a statement and detect what I was dealing with, if it was a “if P, then L” or “P if and only if L”. Sadly, it was too late.

#I hope you liked, and don’t forget to check the week 3 (soon), 
#I am actually late with my week post, so I hope I can finish week 3 by the end of monday, 29 Sept.
#Questions, comments, suggestion?! Write it below, I will answer as soon as possible.
#Thank you.

WEEK 1

Hello everyone, here I am to speak about the first week of class, it is going to be pretty short, since we had no tutorial, and just an introductory class was presented, but I have some points to discuss.

CLASS

My first thought about the class was, another late night class when I will be tired and just think about going home, then everything changed when I saw the internet meme, to be more precise, that one:

I don’t think I need to say that I realized that this class was not going to be like any other, and I highly hope that I am right.
But suddenly I was possessed by the same feeling of boredom that I felt at the beginning when the “Why?” slide was presented, because that is exactly what everyone else say when try to explain why math is important:

I know that I should have more faith in it, since I am at the university, but my feelings from high school are not easily replaced, and I will probably need some magical experience of enlightenment to break my fear of math, but do not worry, because one slide before we were presented to some information about specific fields and what kind of math each one is related. This information is something that I will keep for my life, because I was finally presented to something that tells me exactly where I am going to use math, thank you. It would be awesome if the professor could present some real examples about that utility and obligatory math for those fields of study, but for now, any concrete information that is more than the usual is enough to break the ice where my “faith in math” was kept. If you have any doubt check the slide below, it is beautiful.

Closing the introduction portion of the class, that is how we should feel about math at the end of this course (let’s hope it is true):

After that, the class it self started and those points were presented:

- Precision
- Problem solving
- Quantifiers

if you need to check the lecture to remember anything it is available here.

LAB

No lab this week, sorry guys.

#I hope you liked, and don’t forget to check the week 2.
#Questions, comments, suggestion?! Write it below, I will answer as soon as possible.
#Thank you.

WELCOME

Hello everyone, welcome to my Slog. I was thinking of what should be the best way to organize each post, and after reading the slog documentation I decided to adopt it partially. So, I will try to structure each post as much as possible, but I can’t promise, since when I am writing I use to just let my instincts take over, but as I was saying, the structure that I have in mind is the following:

• First Part - Subjective thoughts about classes and/or any material
• Second Part - Thoughts about exercises/labs and any interesting problem-solving situation.
• Last Part - Conclusion/non-related course writing*

*Since it is a blog, why don’t write something about ourself, or at least something not related to the course, for example, experiences/events/activities at the university?! Well, it is just a plus, because a student week is not just an infinite stack of classes.

I will make sure the non-related course material to be easily identified at the end, so if the reader is not interested on that part he/she can just skip it.

Once again, welcome to my slog.

#Don’t forget to check the week 1 post,
#Questions, comments, suggestion?! Write it below, I will answer as soon as possible.
#Thank you.