Slog for Week 12

Since we did not have lecture last week because of the reading week, this week's lecture and tutorial was the last one in this semester. For the last lecture, I learned about countability and Induction. But the professor just skimmed through induction part, which means he did not explain much details for induction, therefore I do not have much ideas about induction I learned.

For countability, when the professor Larry first asked us about what do we think the size of all even numbers and all natural numbers, of course, I thought the size of all natural numbers would be greater than the size of all even numbers, but actually the result was the same. Of course, this made me surprised, since it is obvious that the size of even number is only half of size of all natural numbers. However, soon I understood why those two are same: because of satisfaction of three characteristics. Since f is a well-defined function, '1-1', and 'onto', the two function is said to be same, therefore the size of natural numbers and even numbers can be said as the same.

But the part of combination of computability and countability, I hardly get the ideas what is going on there, so I think I need to read the lecture slide and course notes over and over until I understand it. Also the induction, even though I totally understood the proof of domino falls, I do not know how it can be used in the proof of some questions like a question in our assignment. But I hope this kind of problems I may see on the next semester in CSC240.

This was what I learned in this week, and all I need to do now is reviewing the whole chapters, especially what I learned in the beginning of this course(since I forgot many of them), and then practicing them by doing past exams in the University of Toronto website.

Slog for Week 10

For this week, we learned about the big omega proof and the big oh proofs for general functions. Since big-Omega proofs were mostly similar to big-oh proofs, therefore proving big-omega was not hard as we first learned about proofs of big-oh. Professor's summary about under-estimation and over-estimation really help me to follow the step: for under-estimation, removing positive terms or multiplying negative terms while removing negative terms or multiplying positive terms for over-estimation. But we never changed the highest degree.

During the lecture, I felt uncomfortable about finding the values of C and B, but I have become confident to choosing the values of C and B after the tutorial I had on Thursday because TA explains detailed steps to find the values and proof structures too. Moreover his teaching of various ways to approaching method helped me to think and prove in right way which is an appropriate method to the problem.

For the general statements about big-Oh, similar to proofs of big-Oh and big-Omega, I still confused to choose the values of B and C because now I have big-Oh and big-Omega in just one statement. I do not know when I need to use max(B, B') and choose B' = B something like that.

I believe I feel more comfortable as I solve many problem sets, and read course notes. Also next Tutorial would help me to figure out how to choose the appropriate values of B and C.

Slog for Week9

For this week, we had term test 2, which was about 3 proofs question. Since all three questions were pretty similar to the assignments, and the proofs the professor Larry Zhang explained during the lecture, I think this was not that hard. But I still feel uncomfortable with the question 2 in the term test2, which was about the mixture of definition of limit and definition of floor function. I think it would be better to go to the office hour after I received the test back to make sure the question 2.

For the lecture, the professor went over the proofs of big oh: the proof of big oh without constant value, the proof of big oh with constant value, and the proof using limit definition. He explained how to pick the value of c and b, which I mostly feel difficulties although I still don't feel comfortable about it. Especially on the part of disproving, I do not understand why he picked the value of n as maximum of the ceiling function of '3c' and B. Since I find the same difficulty again and again, I feel I must go to the office hour to perfectly solve the difficulty I have: picking the value(I don't know why sometimes we need to pick the maximum or minimum of two given values, while sometimes we just picked one specific value.)

Moreover, we learned L'Hopital's rule for non-polynomials big-oh proof. Because I have already known this rule from the calculus, this part was quite understandable. However I feel I need lots of practices to become familiar to big-oh proofs, and should go for office hour to get the help.