Exam

Ah, this exam felt so good!
After the STA257 exam, and an unexpectedly brutal CSC207 exam, this exam was like a blessing.

I really like the way this course is set-up, with all the required problem sets and other things, just because it really forces you into the material.

I'm glad I had this exam as my last.

Polya Goes to Work

Not quite sure where we were supposed to be pulling the problem from and no mathematical problem immediately jumps to mind, so I looked back at old exams for one.

Question 1 of the December 2006 exam seems intriguing.

Q: Prove that 4^(n+1) + 5^(2n-1) is a multiple of 21 for all natural numbers n >= 1.

Understanding the problem:
An incoherent and unintuitive mixture of 4 and 5 to some weird combination of exponents gives a multiple of 21.

Plan:
Try to see a pattern. Or just mess around with some induction and hope something pops out. I do like it when a seemingly random mess generates order. Chaos into something intelligible. Scrambled... Ah, off topic.

Carrying it out:
Well, we must begin where the darkness meets the light, where total balance exists, because nothing exists at all - zero.
So, popping in n = 0:
4^(0+1) + 5^(2(0)-1) = 4^1 + 5^(-1) = 4/5.
Eh? Oh wait, yes, that is right, the question says n >= 1... Phew.
n = 1:
4^(1+1) + 5^(2(1)-1) = 4^2 + 5^1 = 21.
Yay, we have our foundation. 21 is a multiple of 21. At least in this universe. And I do exist here. I assumed that when I created this blog.

Now, for some induction.
We assume that for n it's, true, and prove it is true for n+1.

4^(n+1+1) + 5^(2(n+1)-1) = 4^(n+2) + 5^(2n+1)

Now how can I make this so that what we assumed appears within...
So I guess the obvious step is to pull apart the exponents.
Hm...
= 4*4^(n+1) + 5^2*5^(2n-1) Doesn't look good.
We can see what we assumed inside this, but I don't see a way to get at it.
= 4^2*4^n + 5*5^2n Neither does that.
We have 21 from 4^2 + 5 but I don't see a way to work with that either.

It's a exam Q, so it shouldn't be that hard.
Speaking of exams, I need to study for them... Hopefully will finish this later.

Test - Eep!

Hm, I just finished writing the test and I think I did very badly.
I hadn't studied much for CSC236 because of just having a MAT237 test last night and I hadn't focused on anything but that...

And I ended up just studying some of the program correctness stuff.
When I saw the first Q on the test I had no idea what to do.
Then I started writing something that seemed to make sense to me, and then I cut that short... when I re-read over the functions they ended up seeming exactly the same.
So that question left me very confused. And now I'm stressed over it, since I seemed to have missed something obvious.
Since no one else seemed to have a problem... =_=
Ah!

Then I got stuck on the last Q as well, with the random m=0 hanging about. I was trying to figure out if there was some magical purpose to it. Until Danny Heap wrote on the board about it.
Then it was, Okay, I'm not totally losing it at least.

Oh right, there is this blog.
I kind of wish there were problem sets for this course every week.

Finding the closed form sort of confused me, but when I went to the TA hours yesterday and asked about it, I got some more understanding.
And what helped the most was figuring out you should unwind all the way down to your base case (skipping most of the steps inbetween of course) - that is when you can really see what is going on.
And after talking to some friends, it seems that hadn't seen this either.

Hoping to make like an alternating series that converges...

CSC165 for me, seemed to contain a lot of flailing about and stabbing in the dark.

I feel less like I am randomly messing around with numbers and logic symbols, and more like I have an organized attack, and I find I am actually enjoying the material now.
Which honestly, is something I never thought I would think.
I was... prepared to hide in a fortress and battle the course down when I began this semester.

But, I started on a totally different mindset this time around (I think I gave up before I even tried in CSC165), and I enjoyed doing the assignment 1, well mostly, I still have part 3b left.

I wrote out proofs for questions 1 and 2 from A1 on Wednesday as the actual intuitive reasoning behind them wasn't that hard to see, and went to the TA hours on Thursday morning, to make sure someone besides me could understand it, and that I was indeed actually proving something.
(And with some advice to clear some fog in my writing, it seemed that I was.)

I also didn't understand what I actually had to prove for 3a, so I went to ask about that as well, but another student asked about it before me, so it's nice to know I was not the only one who couldn't figure out the question.
I wrote the proof up yesterday morning, and I have some confidence in it...

Yesterday, after seeing the second invalid proof in the lecture, I started to get really scared that I would write proofs like that. I think I may be forever haunted by the hexagon proof. The seemingly glaringly obvious truth that hides lies?
AH!!

--------------------

And something that has been bugging me is using n = 0 as a base case.
Because the conventions used like n^0 = 1 and 0! = 1 can sometimes make a base case valid when it shouldn't be, or invalid when the statement works for other numbers. And sometimes it seems that n^0 = 0 makes more sense.

And also, the fact that the set of all real numbers can be defined as open or closed, with the empty set then defined as the opposite, makes me confused as well.
Why this flexibility here?
In my MAT237 lecture, we were given a counter example to a statement that included the fact that the reals was a open set. But if the reals can be defined as open or closed, is that really a good enough counter example?
My first reaction was that it was very weak and I was rather disappointed with the example since it didn't lead to a very concrete reason why the statement failed, but instead left me more confused.

3^n

I went in for help this morning because I didn't understand the 3^n proof, but now I do, clearly. I didn't see how we were assuming the pattern 1, 3, 7, 9 and I wanted to know why and how we were.
But now I see that you write the case for one, to imply three, and the case for three to imply seven... etc.
I didn't realize that was what we were doing.
Seeing more formal math of the steps helped cleared my confusion.
Poof.

And I also wanted to check about how I did for question two on the problem set, so I got the math down, that was the easy part. Explaining how I got there was a problem.
I wrote out the proof in words and got the TA to look at it.
He didn't get it. It made sense to me, no one else.
Definitely not good when the whole point of the course is quite the opposite.
Want to make the point clear and not obscure it.
But he helped me make it more concise and clear, though, I still have a bit of trouble deciding just what needs to be said in the proof, and what doesn't.
I don't think I'd want to read my own proof.
That I'm sure, signifies something bad...

So I feel a bit less like I don't know what I am doing now.
I think I will be in for help every week that I can though...
(Did I learn anything in CSC165? It's really feeling like I didn't.. -_-)

Convention Confusion and Problem Set One

So I was trying to read the textbook.
Chapter 0. Done.
Then Chapter 1 came along, seemingly, innocently, skipping along.
And it brought back some of my confusion (like a mudslide over my brain) about proving such simple things.

Did we not create the number system?
Is it not the way natural numbers work just the way the system was made?
Why do we have to prove it?
In what way could it possibly happen that adding one suddenly doesn't not make the next number exactly one larger?
No matter how large the number, is it not just the way of numbers to be that way?
Why, why, WHY, must it be proved?
And yet I know, without proving those simple things, we could say that everything we know in math could be wrong, because we haven't justified the most basic parts of the system.
And yet...
...

And I started working on problem set one.
I didn't understand quite why the proof worked the way it did for 3^n in class, it seems so simple and yet I don't see why it can imply that for all the numbers, why it can imply there is a pattern.
But the second question I think I actually managed to somehow workout.
Of course I have no idea what I am doing, like blindly waving about my hands, in some feeble way hoping to reach for the sun like a wilting tree.
-_-
So I am planning to go in for TA hours on Thursday to resolve my confusion and sort out the fuzziness of my head.
(And ask a lot of questions I probably should have asked in CSC165.)