Week 3

A new proof structure Well Ordering is introduced this week. Among three proof structures, I find well ordering is the hardest one to utilize. It is like prove something backwards when using well ordering, because we are going to use the "smallest elements", and in most of the time, derive a contradiction to what we assumed. Take A1 q3( the golden ratio) for example, if phi = n1/n2 = n2/ n1 - n2, so if we get the set cotaining all n1, n2, ..., nk, where n = n - n and we find this set is not going to cotaining any smallest element, thus by well ordering we find phi is not rational.

One thing worth to review is the relationship of three proof structures.

Week 2

We begin to learn an additional flavor of induction in this week. That is, complete induction. This idea is introduced alone with an example of proving every natrual number greater 1 has factorization. Simple induction can not be used for solving this problem because it's hard to judge which natrual number should be the suited base case and also, it is impossible to list all of them. Complete induction on the other hand, try to prove the last "domino" will fall by assuming all it's previous domino fall.

I think these idea are basically same, simple induction try to prove next statement by using it's neighbour while complete induction use all possible statements that proved before. So in this sense, the concepte of complete induction seems border than simple induction and I find it is easier to apprehand simple induction if we learn complete induction first.

Week 1

Most of week one's material is related with CSC165, Danny showed us some proof examples and I'm very glad we are not going to use the same proving model anymore.