First Post

Since this is the first post, I am going to talk about what I learned in the first few weeks. First lecture was about the mathematical induction, nothing too difficult. It is pretty much the same thing as what I learnt in CSC165. Then we learnt about the complete induction, which was a new thing to me. At first I don't really get it, how by proving just the base case, and assume everything up to n-1 is true, then prove n is true, the statement will be true. However, after Danny do a few examples, I kind of get the hang of it. It is not much of a difference than the mathematical induction. Since the base case is true, so if the next one can be broken into the composition of the base case, it must be true. Then just like dominoes, if n can be broken into the composition of smaller ones, it must be true also.

However, there is a little something confused me. It was the question about the postage of 3 and 5 cents that can make up every number after the the first few. What solution I would come up with would be proving 8 9 10 are true, and since adding 3s to each one, I can make up everything after. Another solution Danny suggested was to prove 8 is right, then assume everything up to n-1 is true, and assume every number (after the first few) have at least one 5 or three 3s, then prove n is true would complete the solution. This seems strange to me, since we assume that the numbers have at least one 5 or three 3s. Then what I have proved is that every number has one 5 or three 3s works. But what if the ones that do not have one 5 or three 3s, although there are none, but we didn't prove it.

Anyway, after that we have well ordering, then structural induction, which are the proofs I have learnt in math courses, so not so big of a problem to me. Lastly, the Fibonacci sequence, which is something I am not so familiar with, but still manageable, just need to be careful of the base cases, as it only works for n greater than two. And this ends my first post of CSC236 blog.