Exponential Growing Pains

This week we were introduced to "big omega" and formal definitions for "big oh" and "big omega".  We looked at some examples about growth, and how functions need to be compared after the break-point to determine which grows faster.  Moreover, we learned what it means for a function to be "in" O(n^2) or Omega(n^2) by looking at some graphs and examples.  It was somewhat interesting and pretty straightforward, which was a nice change.  There was a slide that showed the order of speed of various functions, where f(x) = n^n is the fastest growing function.  I found it interesting to learn that slower growing functions are the most desirable algorithms for computer science, so that they are able to handle huge data sets efficiently.  It made sense after the instructor explained it, but I wouldn't have thought of that on my own.

We also took a look at some insertion sort functions, and how to produce an equation to determine the speed of the algorithm.  The equations came back to show how they were examples of "big oh" notation and growth speed, creating a well rounded lecture.  All in all it was fairly interesting, and I am starting to see the bigger picture about how "expensive" some programs can be and why.

M

O(n^2)wards

This week in 165 we wrapped up our introduction to proofs by looking at different introduction and elimination rules.  I found them simple enough because they are some concepts we've already seen before.  After that we moved on to Algorithm Analysis and Asymptotic Notation, which I enjoyed learning about.  We looked at different sorting methods and why one might take longer than another, and why certain algorithms are more favourable than others because of their speed.  It was nice to see something finally make sense right off the bat.

The tutorial this week was also very good, and I finally feel like I'm starting to get a grip of what we are learning.  I was able to solve the simple proofs in the tutorial, or at least know where to start.  The TA we have is also getting better at explaining certain concepts to us by getting a feeling for what we are confused by.  It also helped that my friend and I spent that afternoon working on assignment 2 together, and working with a partner helped me understand what I was missing.  So far assignment 2 is a good blend of challenging proofs and accessible proofs.  I was even able to get one on my own, which really surprised me.

I think overall this course has inspired me to give math another chance academically, and probably pursue computer science to a depth I hadn't considered before.  I feel like soon I'll be past an initial road block, and after that I'll grasp many new math concepts easier than before.  Who knows, maybe I'll even take the next proof course after this one.

M

Proof by Hoping

We received our midterms back this week, and unfortunately I didn't do as well as I hoped.  I sort of understand what I got wrong (at first glance) and I think I can do better next time.  I think it was an accurate assessment of what I know thus far, and at least it gives me an understanding of where to go from here.  This week in lecture we kept moving forward with proofs, and took a look at some more examples of strategies for proofs.  It seems pretty straight forward, but I still feel nervous about it all.  I just don't think I have the resources (or smarts, sometimes I worry...) to actually look at a problem and know what to do.  I am a bit eager for Assignment 2, because I think I'll feel more confident when I start to dive into problems.  The examples make sense in class, but looking at the statement before going through it I feel completely lost.  It might be time to attend some office hours too, but I should do more work on my own first to really find out exactly what I'm not understanding.  Overall the class is still really enjoyable, I just wish I was an expert!

M

Assume i = me

I got it.  I finally got it.  Last week's tutorial somehow made it all come together.  All of these laws I was fretting over suddenly made sense, and there was no memorization necessary.  It was like my brain was downloading information very slowly over the course of the first month, and then finally it hit 100%.  The TA was explaining the laws of distribution and how to negate an implication, and it finally just all came together for me.  Just in time too, because this week was our midterm.  I felt the midterm went very well actually, much to my relief.  After the test, we continued to dive into proofs even further.  Instead of the basic P => Q examples, we used more specific examples of equivalency.  The structure of the proof was a bit confusing because I didn't understand how the professor knew what to do for each step.  I decided not to worry too much about not understanding, because I'm sure there will be lots of time to learn what to do.  For now, I will look at how to structure proofs in general instead of worrying about individual examples just yet.

So far it's still very exciting and interesting.  I like the creativity required to move forward in a proof, and am looking forward learning more.  Speaking of which, I really ought to read the course notes, because I'm sure there is lots there that will be helpful.

M

Fire Proof

This week, we started off with more implications and laws.  First we looked at bi-implication, and a few different ways of writing these statements.  I think I am slowly catching on, and I feel somewhat confident about finding the negation of most statements.  Equivalency is still a challenge though, and I am not too sure how the laws of distribution work in some cases.  Next we covered transitivity which seemed straightforward.  Working through Assignment 1 gave me a bit of understanding with transitivity as well, so by the time the material came up it seemed fairly clear.  We moved on to mixed quantifiers next, demonstrating how switching around a statement might mean different things.  An example was shown about how this is done in math, and while the math equation was not too challenging, the way the instructor explained it was a bit convoluted.  After studying the example for a while I think I understand, but as he was explaining it I was completely lost.  After a break, we had our first introduction to proofs.  I had done a few proofs in calculus in high school, so this was not extremely new to me, but the types of transformations are.  I suppose these first few weeks have been about learning the building blocks that we will use to accomplish the proofs that are yet to come.  This is somewhat exciting actually, because it will help me learn the laws that we have been working on.  I think proofs are very beautiful, and I would love to be fluent in solving them. For now though, it does feel like my head is on fire, panicking as it can't escape the burning equations falling on top of it. 

M