WEEK 7

6 weeks have been past, which means we have spent half of semester already. How time flies! And we have already learned so many things; I think it is time for me to have a mid-term review!

What I Learnt

        This week we focused on more complicated proofs: proof by cases. It was not hard to me. I needed to take several situations into account. Just like cutting one problem into pieces and solving them one by one as what I did before. Thus, this was just the further application of the old knowledge, nothing new actually.

Here, I learnt these two parts together with comparison and contrast:

Part A:

To prove, we let k = 7i^2 + 2i in order to connect antecedent and consequence together.

Part B:

To disprove, we use the theorem that the quotient and remainder are unique.

Just find an n^2 equals to 7k+1, and n not equals to 7i+1

          1^2 equals to 7 * 0 +1, and 1 equals to 7 * 0 +1

          6^2 equals to 7 * 5 +1, and 6 not equals to 7i+1

Then we use n = 6

What I Need To Do

    Like what Abhinav Chawla mentioned: ” It made me realize that it's only practice that would help me out to harden my concepts of writing a thorough proof.”(http://abhinavchawlacsc165.blogspot.ca/2014/11/week-7.html) Till now, we have a number of manipulation laws and many different-typed problem needed to solve. I need more practice to familiar them and improve my speed of solving a problem.  

WEEK 6

What I Learnt

This week we continued on proofs, including contrapositive, contradiction, proof about non-boolean functions, proof about limits, and disprove.

One of my classmates, Eujin Choi, felt confused about chapter 4, because she was only comfortable of proofing in direct ways, rather than indirect or other ways (http://comscislog.blogspot.ca/2014/10/slog-6.html). At first, when I read the course notes before the lecture, I shared the same problem with her. It was hard for me to tell when I should use the direct way and when I should use other approach until professor used the example to explain them.

Contrapositive   

When the reverse direction in the problem is easier for us to proof, we proof by using contrapositive. That is because contrapositive is equivalent to the original statement, and inverse the antecedent and consequence.

        Contradiction

        As for the contradiction, I think we should use it when the antecedent is implicit and so general. For example, when Q is “There are infinitely many even natural numbers”, we cannot find a certain P to imply Q.

Floor x

What is more, another confusion for me at first was the non-boolean functions. Our professor used floor x as the example. The definition was complicate and made no sense to me. Therefore, the proof based on this definition I did not understand either. 

However, when I combined the graph of the function floor x together with the definition and the proof, I suddenly understood the whole thing!

Inspiration

    Through this week’s lecture, I realized that we should apply flexibly what we learnt before and try varies kinds of method to solve a problem.