Limits [to my patience]

This past week I've had real trouble with this course. Last lecture, we went more in-depth with proofs, and I won't lie, it is getting quite complicated.  Just when I felt confident about my 'proving' abilities, we were introduced to every CSC165 student's worst nightmare: proofs about limits.  

Up until yesterday, I was at loss.  Why is everything in Greek alphabets?  How does that compare to previously learnt material regarding calculus limits?  These and more questions overwhelmed my consciousness.
I finally gathered up the courage to go to the Help Centre yesterday.  I am definitely glad I did because assignment 2 has just been posted and most of the problems involve the concept.

For a problem of the sort,





I found that it is easiest to try and plot a graph.

(Courtesy of Professor Zhang)

Among the hardest things I found with limits is devising a plan to find the most appropriate δ that goes in terms of ε.  In other words, the goal is to have every possible δ ≤ ε.  As with everything else in this course, once the concept is clear, the actual proof becomes a little easier to fill out.

Achievement Unlocked: Test Results

I just got back my test results! I'm so proud: I did much better than expected.  I decided to document this since I am in such a great mood at the moment and it might help me in the future once things start to get discouraging.

The questions on the test were very similar to the ones that were covered in class and past tests.  At one point during my preparations, I almost completely ignored a note the professor had mentioned that I had documented in my notes because I found it too difficult to grasp.  I now appreciate my intuition for resolving the fear of failure and trying to understand the concept: that same concept, involving the use of functions in the form of any and all along with their negations in Python code, encompassed more than 60% of the test!

After resolving my issues with this concept I can in all honesty say that I found the tutorial questions much more challenging than those on the test.  I suppose hard work really does pay off!

Proof by Contraspositive

I remember the days when I used to boast and announce, "Proofs? Please, they come second nature to me!"  I can still recall that horrendously loud laugh with which I used to flood the rooms of my poor, confused friends.

It still haunts me... I mean them.

Only this week did I realize that in reality, I only knew how to write out the structure.  And not just that; I only knew how to write out direct proof structure.  My ego has been in hiding ever-since...

After a number of fruitless past trials that left me frustrated, I came to the realization that there could be times when a direct proof doesn't necessarily work.  In such cases, resolving to an alternative proof mechanism is required.  This is where proof by contraspositive comes into play.

To refresh your memory, a contrapositive of an implication is equivalent to the implication itself.

∀x ∈ L, P(x) => Q(x)   <====> ∀x ∈ L, ¬ Q(x) => ¬ P(x)

How does that help with proving something that doesn't 'respond' to a direct proof?  Proving the contrapositive instead of the implication will simultaneously prove the original implication!

The proof structure for the above contrapositive is:

Assume x ∈ L  # x is a generic L

     Assume ¬ Q (x)  # assuming antecedent

              Then ..... # finding a link between ¬ Q(x) and ¬ P(x) 
                              # using proven mathematical theories and logic

              Then ¬ P(x)   # then ¬ Q(x) => ¬ P(x)

      Then P(x) => Q(x) # since an implication is equivalent to its contrapositive

Then ∀x ∈ L, P(x) => Q(x) # Assumed x, got result.

In conclusion, if a direct proof doesn't work, try using the contrapositive (especially if it becomes easier to prove), before rejecting the claim.

Proofs

During our lecture on Tuesday, we were introduced to 'proofs': the mechanism used by mathematicians and computer scientists to accept or reject claims about the world.  It seemed a little arbitrary to myself as I had not been exposed to the concept before.  But at the same time, I was excited to challenge myself by learning something new.

Proving something true is not an easy task, especially when you are not aware of the reason of the validity  The first step towards proving  or disproving something should be to understand why it is true (or not); convincing yourself before anyone else is a vital step.  This is why I set out to make arguments from common beliefs that I could prove/disprove as this course progresses.

For now, here is my humble list:

1. If the users rated the book five stars, then they liked it
2. If the clouds are dark, it will rain.
3. Someone who doesn't eat meat can survive.