Are you Ready for the Test?!

Hello all. Tell me. Are YOU ready for the test on Monday?
I, personally, will be spending this weekend studying for it. It supposedly will contain everything we've learned up until now, but I think it will be dominantly about proofs. As in proving whether or not a statement is True in the taught format / style.

I've realised that as long as you do roughly the following, you will be fine.... structure wise.

Assume your variables.
Assume your antecedent.
Pick a variable if required.
Work out all that middle juicy stuff to your proof.
End up with your consequent.
Restate the original statement. # Assumed antecedent, got consequent.
Conclude the original statement this time containing any "for all"s and "there exists".
Note: Be sure to add comments for every line explaining yourself.
Done.

The same applies for proving the statement false, just be sure you're proving the original statement's negation instead. # The negation is True, therefore the original statement is False.

Careful with epsilon and delta proofs though. These are proven in a slightly different manner I've noticed which can be rather confusing. I think these one's require the most practice....

Anyways, hope this little blog was helpful in some way. Happy studying and good luck to all! See you Monday. ;)

Assignment 2

I have just recently started working on assignment 2, and I didn't realise how difficult proofs could be until now. There are so many unknowns and assumptions involved that one could get confused quite easily. However, I like the quote "If you don't understand why something is true, don't expect to be able to prove it!"

So basically, unless I fully understand these questions, I can't be expected to start solving them.
I have read "Chapter 4 - Proofs" though, which has proved quite helpful. There are so many ways to go about proofs that I wasn't aware of. I especially like how to disprove something. It's basically the same as proving the negation. I originally thought it would be harder than that.

Anyways, now that I have a better understanding, I'm going to go reattempt the problems. Wish me luck as I do the same for you. ;)

Proofs!

Joy to the world! The long awaited "Reading Week" has finally arrived!

So this week has dominantly been about proofs. The method / style being taught on how to set up these proofs is odd in my personal opinion, but easy to get used to. I like the structure of creating the assumptions and conclusions first, then working out the middle to connect the two ends. I find it trivial however to comment on every single line of the proof. I understand that it makes the code more understandable for others, as well as myself for when I come back to it later on, but still. Some steps are just obvious, but I guess it provides good practice for the more important steps.

Exercise 2: This exercise consisted of just proofs. I found the proofs in question 1 to be fun. It's easy to convert symbolic statements with the material given in chapter 3. You just need to mess with / convert the given statement until it looks like the given solution. Question 2 was also somewhat easy, however I'm not as fond of the steps to proving these types of statements. Having to state your assumptions, conclusions, and given information along with the middle portion is a little tedious to me, however important it is to mention.

Overall, proofs don't seem to be that hard to work with. I'm sure they'll get harder soon, but for now they're okay. That's all for now. 'night.

Tutorial 3

I found this tutorial lesson fun. The first two parts referred to rewriting equivalence statements, which wasn't difficult at all. The third had to do with writing proper proofs. Although the concept for these isn't hard, I noticed my format for writing them isn't completely correct yet. Just need practice I guess. My favourite part though was part four, where we had to solve a problem through logic. This reminded me of a game that I found in puzzle books about five years ago. I forgot how much fun they were. I hope we get to do more of them.

The Power of Negation

I find it interesting how negation can be used in several different ways. For example, I am referring to double, implication, equivalence, and quantifier negation. Double negation is pretty straight forward in the sense that two negatives make a positive. ("I'm not not going" means "I'm going.") Implication negation can change an implication statement, such as "P implies Q", to a "not P or Q" statement, where there is no implication. As for equivalence negation, a statement
such as "not(P if and only if Q)" can be rewritten as "not(P implies Q) or not(Q implies P)". Although resulting in a much longer statement, the two are still equivalent. As for quantifier negation, I found this one interesting as it lets you shift between "for all" and "there exists" statements. I had some trouble with understanding this one at first, but I soon got the hang of it.


Side joke about double negation: (I just wanted to add this in for fun....)

Teacher: "A negative and a positive make a negative. Two negatives or two positives will always make a positive."
Student: "Yeah right."

(For those of you who didn't catch that, "Yeah" and "Right" are both positive, but together imply something negative. Thus counterexample. Haha.)