CSC165 sLOG WEEK6

  Last week we continue to learn something about proof. I found out that proof is such a big topic which needs a lot of practices to manage it. This week I read the course notes and learn a lot from it.Here are some definations: a lemma is a small result needed to prove something we really care about. A theorem is the main result that we care about (at the moment). A corollary is an easy (or said to be easy) consequence of another result. A conjecture is something suspected to be true, but not yet proven. An axiom is something we assert to be true, without justi cation ( usually because it is "self-evident.") For direct proof of universally-quantified implication, When we assume that x is in D, we are in the "world" where x is a generic element of D. To prove the converse of a statement, you set up the proof of the contrapositive of the converse. I found it is a hard part for example of proving a statement about a sequence. I go through the progress several times and finally figured it out. For multiple quantifiers, implications, and conjunctions, I found that it is not difficult if you have a clear mind and do it step by step.

CSC165 SLOG week5

This week we have a term test for CSC165.Therefore, we only have two lectures. We continued to learn the structure of proof.Generally, there are two steps or phases to creating a proof:1. Understanding why something is true.2. Writing up your understanding. Sometimes these steps can be combined, and often these steps feedback on each other.For the proof outline, when we prove a statement leaded by the universally quantification, we assume the element is a generic one.And we conclude with the whole statement. In the middle, we assume antecedent and then try to prove the consequence of implication. When we prove a statement leaded by existential quantification, we just pick an element from the specific scale, then assume the antecedent and prove step by step until we get the consequence of implication. In the end, we conclude. I found out that writing comments is a good habit.Because if you could not understand one step, comments are very helpful.We also learnt how to prove by using contradiction. It seems that proof is not an easy chapter, we need to do a lot of practices to get to know how to do proof problems better.
  Btw, for the test I felt I did so so. Waiting for the result.

CSC165 SLOG week4

   This week I have learnt a bunch of new things of CSC165. I found that truth table is very useful for evaluating a complicated statement. And I found myself more skillful than before to change my expression for by-implication into the disjunction of two conjunctions.I have learnt the concept of transitivity. It should be well remembered that once the sequence of universal quantifier and existential quantifier has been changed, the meaning of the whole statement would be changed. To make us understand this clearly, the professor drew several graphs to explain this problem. By studying the graphs(such as the graph of approaching infinity) he drew, I gradually understand it. Since one factor is first chosen and another factor is chosen after we now the previous factor, it is not difficult to estimate the scale. Let us put it another way, if one is determined first, another one is going to win.By studying double quantifiers, I knew that there are at least 3 ways to claim that a certain subset of the cartesian product is non-empty. There are also 3 ways to claim that en entire catesian product has some property. In these cases, the order does not change the logic.That is the end of the first two chapters.
   Then we learnt some basic knowledge about proof.It is said that a proof communicates why and how you believe something to be true. Fist you need to understand why you believe the thing is true. Then you need to write it down. We learnt it by studying the example of finding proof of universally quantified implication. A proof outline contains some argument with 'assume' and 'Then'. Moreover, the comments which was leaded by # are very important in the format of proof.