Thanks to all the proofs we've been having to do, I think I'm actually getting better at them! "Big O" and "Big Omega" are interesting concepts, and proofs involving them seems like a natural extension of what we were doing the weeks before. Most proofs involving Big O/Omega seem very formulaic, and not too difficult so long as you know what steps to take. In tutorial, it took me a long while to figure out the algorithm for building this new breed of proof, but once I figured it out, I realized how simple it was. The same goes for general proofs about functions. As interesting as I find the concept of Big O and its cousins to be, I am not really sure how it is related to the field of computer science, beyond being a convenient model for running time. I can see some connection to steps taken by a computer program, but the relationship is hazy in my mind and I don't quite understand.
The due date for the third assignment is coming up, and I still haven't figured out how to use Texmaker. I've installed LaTeX and everything, but the editor is still not working. Oh well, I guess it's back to MS Word for me.
I am not sure what I think of computablity and the halting problem yet. At this stage, the self-referential nature of a lot of halting problem still has my brain fried at times, but I think I'll get it eventually.
Wednesday, 27 March 2013
Monday, 11 March 2013
Assignment Two PROVED to be difficult!
Assignment 2 was, in my opinion at least, a lot harder than the first one. When it comes to proofs, I am often flabbergasted by the sheer number of options I have to approach it. I don't find it as intuitive as some people to know when to factor, or square both sides, or multiply by a constant. It feels to me like I'm grasping around in the dark for something that might have some potential to get me where I want.
That being said, I do think that the way proofs are approached in this course is a lot more my style than, say, high school math. Maybe it's because the course material was effective in introducing the foundations, but I generally feel more warmly towards proofs in this course than ones I previously encountered. Assignment 2 was still pretty hard, though.
In particular, I could not come up with a solution for question #4, so I simply wrote the structure, did the first step that I was sure was right, then wrote "I cannot solve this past this step." Hopefully the "substantial part marks" given to incomplete proofs really are substantial. Looking at the solutions, however, the answer seems almost embarrassingly obvious.
I had managed to get to the step where I squared both sides of the equation n = 11q + 5, and I knew that the only thing I could do from there was expand it into n^2 = 121q^2 + 110q + 25. After that, I couldn't figure out how to manipulate that polynomial into the form 11p + 3. The solution was to separate 25 into 22 + 3 then pull 11 out of each of the terms except 3. Unfortunately, I was just as dense solving this problem as I was during the tutorial quiz on February 26, where I ran into a similar situation and was unable to perform the simple arithmetic required to split an integer into two other integers. At least now I am aware that this is something I often overlook, so I (hopefully) won't make the same mistake in the future.
Also, interestingly, we were allowed to write all of the proofs for the assignment in plain English! I didn't realize that until I had already completed (er, almost completed) the assignment, but I think if I had written some of the proofs in English it would have helped me get a better grasp on what was required, and maybe I would have finished it sooner and with fewer mistakes.
Ah well. I will have to make sure I do the next assignment well in advance.
That being said, I do think that the way proofs are approached in this course is a lot more my style than, say, high school math. Maybe it's because the course material was effective in introducing the foundations, but I generally feel more warmly towards proofs in this course than ones I previously encountered. Assignment 2 was still pretty hard, though.
In particular, I could not come up with a solution for question #4, so I simply wrote the structure, did the first step that I was sure was right, then wrote "I cannot solve this past this step." Hopefully the "substantial part marks" given to incomplete proofs really are substantial. Looking at the solutions, however, the answer seems almost embarrassingly obvious.
I had managed to get to the step where I squared both sides of the equation n = 11q + 5, and I knew that the only thing I could do from there was expand it into n^2 = 121q^2 + 110q + 25. After that, I couldn't figure out how to manipulate that polynomial into the form 11p + 3. The solution was to separate 25 into 22 + 3 then pull 11 out of each of the terms except 3. Unfortunately, I was just as dense solving this problem as I was during the tutorial quiz on February 26, where I ran into a similar situation and was unable to perform the simple arithmetic required to split an integer into two other integers. At least now I am aware that this is something I often overlook, so I (hopefully) won't make the same mistake in the future.
Also, interestingly, we were allowed to write all of the proofs for the assignment in plain English! I didn't realize that until I had already completed (er, almost completed) the assignment, but I think if I had written some of the proofs in English it would have helped me get a better grasp on what was required, and maybe I would have finished it sooner and with fewer mistakes.
Ah well. I will have to make sure I do the next assignment well in advance.
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