The week is finally over, which means it is time to write another slog! To be quite honest I almost forgot to write it due to all the assignments that I had going on. But now that I have finished all of them, I can focus on writing my slog for this week.

This week in CSC165, nothing was too difficult to understand which was nice, in fact most of the things we did this week was more or less review for my MAT137 course because most of the material we are learning is sort of the same as the material in CSC165. This week we mainly took a look at proofs and the structure of proofs. For me, I have been struggling with proofs in MAT137 so I'm not too confident about my proving skills yet. I hope that through more practice I can master proofs! 

In our lectures, we covered what a proof would look like, first we talked about what we need to know in order to start our proofs. The first thing was that we needed to understand why what we believe is true. This means we have to make our claim about what we know, introduce it, and then strengthen the weak parts of our belief. Next we needed to show what we believe is true. Basically we have to get into the meat of the proof and explain/ justify our claims to convince a skeptical peer.

After that, we looked at the outline of a proof, in other words, how it should look like on paper. The way that proofs are done in CSC165 are a little different than how we do them in MAT137, but it still helps to understand the structure of proofs. The following outline is what we did in lecture.

The proof starts off by assuming the generic part of the statement, in this case, we are assuming for all x's. Then we assume the antecedent. Notice that the structure has been indented once. This is basically to show that this is under the the previous assumption of all x's. Next we indent again, now we are in the meat of the proof. This is where we begin to explain/ justify our claim with results and examples. Each step is subsequent to the previous step, and this goes until we show the result that we need. Once the final result is achieved, we jump back in indentation to where it lines up with assumption of the antecedent. Here we say that the antecedent implies the consequent because we got the consequent result that we wanted. We then jump back in indentation again and conclude that for all x's, the antecedent does imply the consequent.

Now that the structure of the proof has been covered, we took a look at an example, 

In symbol form it would be: ∀ (x,y) ∈ R⁺, x > y ⇒ √ (xy) < (x+y)/2

I'm not entirely sure how to prove this statement, but the structure would look something like this.

First we assume the generic part of the statement, we assume that that x and y are real numbers. We then indent and assume the antecedent, which is that x is greater than y. Next we indent again and show the actual proof to show that the square root of x multiplied by y is less than x plus y all over 2. Once we get the right result we need, we jump back in indentation and say that since I assumed that x is greater than y and showed that the square root of x multiplied by y is less than x plus y all over 2, we introduce the implication that the antecedent implies the consequent. Lastly we jump back in indentation again and conclude that for every real number (x, y), this implication works. 

So this week's slog was definitely a bit more interesting now that we are talking about proofs. I even included some math symbols. I hope that perhaps by next week or the week after, I will be able to start proving questions and share them on my slog! That's all for this week so thanks for reading!