week # 7

It's been seven weeks since this semester started, which means we are already in half way through the whole term. Every time I look at the stuff learned before, I feel like that the time in university goes so fast. This week, we are still continuing on proofs but more complicated than before, which is to prove by cases. This kind of proof is not hard but we have to separate it into several cases and prove every case in order to make a good proof. What's more, I think the most useful part during this week is to introduce some rules, which can also be seen as the conclusion of some basic and necessary rules of proof.
Elimination:
conjunction elimination: If you know A ^ B, you can conclude A separately (or B separately).
existential instantiation: If you know that there exists k in X, P(k), then you can certainly pick an element with that property, let k' in X, P(k').
disjunction elimination: If you know A or B, the additional information :A allows you to conclude B.
implication elimination: If you know A implies B, the additional information A allows you to conclude B. On the other hand, the additional information :B allows you to conclude :A.
universal elimination: If you know for all x in X, P(x ), the additional information a in X allows you to conclude P(a).
Introduction:
implication introduction:If you assume A and, under that assumption, B follows, than you can conclude A implies B.
universal introduction: If you assume that a is a generic element of D and, under that assumption, derive P(a), then you can conclude for all a in D, P(a).
existential introduction: If you show x in X and you show P(x ), then you can conclude not x in X, P(x ).
conjunction introduction: If you know A and you know B, then you can conclude A ^ B.
disjunction introduction: If you know A you can conclude A or B.
The most difficult part this week, which I think is the worst case by introducing two functions which are the upper bound O(U) and the lower bound. The formal definition was pretty complicated and confused when I first saw it. However, after understanding the actual meaning of them, it is much clearer to me. The issue is that I can understand it when I'm looking at the definition but can hardly write it down by myself. Therefore, it is probably a good idea by practicing related problems in order to get familiar with it.

week # 6

We have a long weekend  as Monday is a thanksgiving day, so we only have two lectures  and no tutorials this week. Personally I think the work in this week is much easier than that in previous weeks. And we are continuing on proof of different types of problems including the proof of non-boolean functions and limits as well as the proof of something false. Since we've already learned how to write the outline of a good proof on last week, it's not  as confused as I thought at the time in which I learned proving in MAT137. It's really helpful for me when the professor taught us the proof about limits with an example of asking us to proof the definition of the functions which is exactly the same as what I learned during MAT137 lectures. In fact, I've been frustrated and confused about those kind of questions for a long time and eventually, I chose to totally memorize them instead of understanding them. However, I got really excited when I saw this proof in165 lecture on Friday using the different way of thinking but the same solution. As a result,  I'm not as confused as before, and I even wanna go back to do all the questions that I didn't get one more time using the method I was learned in 165.

The following graph is a typical graph to illustrate the definition of limit:

week #5

Good news in this week was that I didn't lose mark on my quiz, but bad news was that we just had a test which is my first term test in university and more importantly I didn't even finish it. I focused on the first two questions which spent me lots of time and I didn't realize that the time passed so quickly. As a result, I had little time to think about the last question, which again, made me frustrated after the test. I think the most possible reason of this situation is that I'm not so familiar with those knowledge so that I'm afraid to make mistakes on them which at last waste me a lot of time. I'm on the way back and I have to catch up with other classmates and then keep pace with them. That's my foremost goal of 165 now. Thankfully, this week's work is to proof some fundamental questions, which means that I'm able to spend more time focusing on reviewing previous knowledge through my notes and lecture slides on course webpage.

Problem solving:

week #4

It has been four weeks since this course began. And what I found interesting is that this course is kind of similar to the course MAT137 which I'm also taking in this semester, such as the proof of limit. Although the ways of teaching might be slightly different, I'm still quite happy about that because that helps me to improve understanding of this field of knowledge.
Basically, what I have learned this week is about proof. Not the proof of a real problem like what we learned in MAT137 but the outline of proof, which I think is helpful on both course in the future because what we actually need is not to solve a real question correctly on the test but to understand the problem and design a plan to solve it. This week's stuff is pretty clear to me and hopefully I can still be like this next week.
However, the issue of this week is not about the lectures during the class but about the first assignment. This assignment only includes 5 questions with several subquestions each, but the fact is that I have been doing it with a group of 3 for two days. It looks like pretty easy but once we do that, lots of problems come out as each of us has different understanding of those questions. In particular the last two questions, none of us is pretty sure about them. As a result, we studied the notes from lectures over and over again in order to find the solutions, which spent us plenty of time.
         4. For each pair of statements below, given an example of sets D; P, and Q that make one statement true
and the other false. Explain the difference in words, and show it with a Venn diagram.
(a) The pair 8d 2 D; P(d) ) Q(d) and 8d 2 D; P(d) ^ Q(d).
(b) The pair 9d 2 D; P(d) ^ Q(d) and 9d 2 D; P(d) ) Q(d).
We still don't get the satisfying solution of question 4 above until the answer key has been out, which is quiete simple using the basic knowledge we learned. I looked through the solution carefully in order not to get confused next time after I meet a question similarly.