Episode 1 (Pilot)

After trying to think of some witty name to put to my blog (and settling on some obscure 1998 game reference...fail), here goes the first post of the year...

Well, after 4 months of freedom I've come to a conclusion that there's no place like hell home... a 4 day headache isn't exactly the best way to start off the year == but I think I've settled ok. To be honest I think CSC236 is going to be one of my hardest courses this semester, along with STA247 (11% retake rate?!) and maybe CSC207 (I don't know a thing about Java... TT). I've never been good at proofs and proof-related theory (like CSC165 and MAT137) so its going to be a long long year ahead...

Anyway, the course has been okay this far, I'm still following along at a reasonable pace. As for course material, the most interesting thing so far has been the proof of:

P(n): 3^n has units digit in 1, 3, 7, 9

since originally I had thought you did need 4 separate base cases to prove it, but the fact that you only need 1 was a bit surprising. The other question that was quite intriguing was:

How many subsets of odd size does a set with n elements have?

I was surprised at the pattern that came out of investigating this question:

Case 1: {a}
Odd: {a}
Even: {Empty Set}

Case 2: {a,b}
Odd: {a}, {b}
Even: {Empty Set}, {a, b}

Case 3: {a,b,c}
Odd: {a}, {b}, {c}, {a, b, c}
Even: {Empty Set}, {a, b}, {a, c}, {b, c}

We can easily see that the odd subsets of a particular case contain the odd subsets of the previous case. However, a "hidden" pattern is adding the new element of the current set to each of the even subsets of the previous case, which makes them new odd subsets. For example, two of the odd subsets in Case 3, {a} and {b}, exist already from Case 2. To find the remaining odd subsets of Case 3, we must add the new element in Case 3 to the even subsets of Case 2, {Empty Set}, and {a, b}, creating {c}, and {a, b, c}. I was quite surprised that this method work, as shown in a proof during class.

On an unrelated note, I wish the lecture slides could be posted up perhaps a day or 2 before class, so I could have some time to look through them and prepare. It would be great to have the annotated version posted up soon after lectures as well.

Hope this is a decent enough first post ==...cheers to everyone else in 236,

fOrTT