Wow! Final Assignment!

While preparing for the assignment 3,  I had a hard time in forming a chain of inequalities and simplifying the expression of  Ω(f). In case of big O, I could form a chain very smoothly but when I tried to form a chain in Ω(f), I was stuck on the final chaining field.

That is, when I tried to form a chain for 5n³ -3n² +2n +3 ∈ Ω(2³ - n² + n +1)

5n³ -3n² +2n +3≥ 5n³ - 3n²

                          ≥ 5n³ - 3n³

                          = 2n³

                          = c'n³

                          = c'(2n³-n³)

                          ≥ c'(2n³-n²) 

I was stuck here because if n + 1 is added, then c'(2n³-n²) ≤ c'(2³ - n² + n +1).

So, I had to do in a different way to connect the chain. I still don't know which chain I formed wrongly but I am going to figure out before the final exam.

I can't believe that I just have the final exam left. Doesn't time fly?  We should live life to the full.

Reading Week!

The professor dealt with a various methods of proving statements this week. Honestly, I could know the exact meaning of ⌊x⌋ through this course. Actually, I didn't know that "∀x ∈ R   y = ⌊x⌋"  has this complicated symbolic meaning  "y ∈ ℤ ∧ y ≤ x ∧ (∀z ∈ ℤ, z ≤ x ⇒ z ≤ y)" and conclusively, "∀x ∈ ℝ, ⌊x⌋ ﹥ x -1".

I realized that the symbolic expression which seems very simple could contain a complicated and sophisticated meaning in it.

Next week is reading week! I think I should read some reference books about proof and review mathematical terminologies for the next step, especially I am going to check the terms such as б, ԑ, ⌊x⌋, ⌈x⌉ and so on.

One more step into by contrapositive method

What I was really amused with this week was the proof by contrapositive method.

The fact that we can prove a statement ,which is very hard to prove in a direct way, by indirect way called contrapositive method made me so excited. Suddenly, the thought struck me that the problems of life we may face could be resolved by the same way. That is, when we face a serious problem of life, sometimes, we might resolve the problem by changing the way of looking at the matter. I might blow this method out of proportion but proof and life have something in common in terms of seeking truth.

Finally, the first term test is coming up. I am little nervous because I have been sick and couldn't have prepared well for the exam. Fortunately, the generous professor allowed us to bring an aid sheet to the test room so that we could take the exam being helped by it. However, in other words, even though we have help from sheet, it won't be easy, that is, it insinuate that the test will be very tough  I am really getting scared of taking the exam. Anyway, I will try my best trusting the maxim  "Man proposes, God disposes"

To the world of proof

This week's lesson was about the possibility of switching ∀ with ∃ without altering the truthfulness of the statement. It was very hard for me to understand and a challenging subject!

I still don't understand the graphical explanation and the difference of switching ∀ԑ with ∃б.

I think I have to see TA to figure out the difference when they are switched. I also made first step into the "Proof  World". I learned the format of proof and how to draw the consequence from the antecedent.

Looking carefully the format of proof, I was very impressed by the extent which  the human beings are trying to reach through their logical thinking. I guess that this lesson would be the most important base for the whole course and should study intensively to make strong basis.

The process of getting to know

This week, we learned about the difference between natural language and idiom and its logic interpretation into symbols.

It was very interesting that "unless" could be exchanged into "if not" but the position should be switched for the symbolic translation.

In professor's slides,

Don't knock it unless you've tried it

= Don't knock it if not you've tried it

= If not you've tried it, then don't knock it

¬ T ⇒ ¬K ≡ K ⇒ T

In idiom parts, it was amusing fact that just the subtle difference between all and some could change the choice of  '∧'  and '⇒' in symbolic interpretation such as

Every D that is  a P is also a Q

∀x ∈ D, P(x) ⇒ Q(x)

Some D that is a P is also a Q

∃x ∈ D, P(x) ∧ Q(x)

Also, we learned about how to negate and parse a logical expression.  I think the negation is very important skill to prove counterexamples and should practice as many as I could.

The first class and strain

As soon as the first class started,  I got so nervous because I had heard many times from my friends that CSC 165 was extremely difficult and one of them dropped this course, deciding to change his major.  

I thought they were smart guys so it was a big shock to me, besides, English is not even my mother tongue.

However,  now, a week later, the class is getting interesting and I find myself enjoying the class.  Yes, it is still hard to understand, sometimes so confused but definitely catching my interest.

Through the classes, I have realised that I have heard and understood other people's words and spoken to them without any logical thinking. Also, I learned how important it is to speak accurately and understand correctly in communication of professional fields.

During the first tutorial, our group members were talking about the question (d) of exercise 1,  the question was "There is one of the three python programs that fails all three test suites". We all were confused the meaning "There is one", we discussed whether "There is only one" or" There is at least one." I heard from TA later that "There is one" normally means "There is only one."

 It is very interesting fact that I never thought about the mathematical meaning of words carefully when I communicated with other people in the discussion of a meeting even where it was a very important matter to check the mathematical meaning of it .

I hope I could improve my logical thinking and speaking skills through this course, at least, to better understand others' idea and opinion as well as exactly convey my ideas to them.