Hmmmm......

Well, after reviewing the blogs of my fellow classmates, I feel less guilty about not updating my blog more frequently. I have greater than one blog entry. That's a start! What? What's that you say? I shouldn't be comparing myself to others, and the assignment outline details that I should be updating at least once a week? Rubbish!

Just kidding. Please don't hurt me.

Anyways, the midterm is fast approaching, and proofs will be the death of me. Upon reviewing last year's assignment and midterm, I feel fairly confident. But proofs with multiple quantifiers still confound me. Unfortunately, the assignment this year is one that I find (personally) more difficult to parse than last year's version.  My way of approaching proofs with multiple quantifiers, with variables appearing in the statement out of the order in which they have been quantified, is to first pray that it's an epsilon delta proof, and then to assess which variable comes first (epsilon or delta). Whichever variable appears second "wins". That is my only approach. THE END.


... Okay clearly that will not help me. But one can dream.

Claim 1.2 appears to be very similar to a delta epsilon proof. However, the quantifiers are placed out of order, so I'm a little lost

I will update as soon as I figure out what I am doing with my life.

Proving a Proof

Tips and tricks!

Whenever you introduce a let statement in a proof (when you pick a variable), conclude the end of that margin with an existential.
Whenever you introduce a generic variable , conclude the end of that margin with a universal.

Brief review of floors:
- The floor of x is the largest integer that is less than or equal to x.
- The floor of x is not a variable. It is a function, and quantifiers only apply to variables, so you cannot use a quantifier directly on floor.

Definition of floor:
Assume y is an int. Then y is less than or equal to x. Among all ints that are less than or equal to x, y is the LARGEST.

Note that the symbolic definition of floor and the english definition of floor are equivalent; one is just simpler to parse as a native english speaker.

Encountering failure.

" So why do I talk about the benefits of failure? Simply because failure meant a stripping away of the inessential. I stopped pretending to myself that I was anything other than what I was, and began to direct all my energy into finishing the only work that mattered to me. Had I really succeeded at anything else, I might never have found the determination to succeed in the one arena I believed I truly belonged. I was set free, because my greatest fear had been realised, and I was still alive, and I still had a daughter whom I adored, and I had an old typewriter and a big idea. And so rock bottom became the solid foundation on which I rebuilt my life.

You might never fail on the scale I did, but some failure in life is inevitable. It is impossible to live without failing at something, unless you live so cautiously that you might as well not have lived at all – in which case, you fail by default."

- J.K. Rowling

It's odd to discuss failure in a CSC165 post given that my grade in the course is currently above 99%. (This is not gloating-- as I am wholly aware that this mark will plummet after I encounter what is next to come. Guaranteed. This becomes more obvious as this post goes on.) However, I live in perpetual fear that I will be discovered to be a "fraud". I am impossibly bad at math, and it is a constant, pulsatile reminder; that I am attending a university revered for it's difficulty, in a faculty renowned for its focus on computation, mathematics, and logic. Today, a post on the UofT Computer Science FB page further perpetuated this fear. To hear that the difficulty of Computer Science rises disproportionately each year was a shock in and of itself. A friend at Waterloo's program had said that he had found his first few years in Computer Science to be the most difficult. Whether or not the difficulty of the program increases as the years progress, regardless of my weaknesses in math, I will continue to press on -- to learn more about the things that I am passionate about. Life is a continuous struggle to reach the goals we set for ourselves. I am finally learning things that I am passionate about. Although I am at a disadvantage in terms of my learning background, and the years that I have had away from my math, I am confident that I am well versed in the art of Try-Harding. I spent so much time being disinterested in the things that I-did-not-want-to-do, that my second time around in a post-secondary institution has been all about spending all of my time studying-the-things-that-i-want-to-do. Even so, I ultimately decided to drop MAT223 this semester. This is the first time in my academic career that I have ever dropped a course. My first true encounter with failure. However, it made me realize that my desire to do well, and my efforts to do well, were not entirely realistic. Majoring in a field completely void of the mathematics, and working in a hospital for a year and a half is not conducive to writing algebraic proofs. We're not in Kansas anymore. It's time to start from square one in terms of math. To learn from the beginning, and to learn it well. At first, I was ashamed. Now? I am determined.

It is currently 3 AM. Further updates will be made this week as I pursue the dread mathematical proofs and the big-Oh(my god).

Slog: Week 5: Happy Ending... for now.

 Greetings, audience.

This blog feels more and more like an intimate space for my inner thoughts about CSC165. Mostly because it often feels like I have 0 traffic and 0 post views (literally, I actually do). That's okay though. The company of me, myself, and I is more than enough. I mean, three is a crowd, right? There's some kind of english proverb about three being a crowd, no? *crickets chirping*

Moving on, then.

Today was the day of reckoning; a true test of skill. Yes, today was CSC165's Test 1 day. Armed with a double-sided, size 6 font cheat-sheet, I went into the test feeling overwhelmed, and frightened. I left feeling... ecstatic? Today was like the culmination of all of the concepts that I had been struggling with. However, the difference was that the test was formatted in such a way that it was very reflective of the practice test that Professor Heap had provided us with. It was more than fair, and still very challenging -- especially to people who were not prepared. However, it was very do-able to the people who had prepared by reviewing lectures, assignment one, past tests, and the course textbook. Kind and benevolent Heap. Compassionate Heap. Danny? Can I call you Danny?You took pity on these pathetic, miserable First Years, and you showed them the light. Hurrah.

For my cheat sheet, I prepared by consolidating information from all of the above mentioned resources, and doing my best to categorize this information by subject and concept. I included explanations or simple english translations of each law, including Transitivity, which ended up not being on the test. I feel that, despite not being on the test, that I have a stronger grasp of Transitivity. I do not regret spending time on this concept. YOLO.

I found that one of the questions on the test was very reflective of question 4 of Assignment 1. I had already had a strong grasp on the concepts of proving or disproving existential and universal statements, so this question ended up being very straightforward. In summary:

To prove a universal statement: you must prove that every element is an example -- that there is no counter example.
To disprove a universal statement: you must provide at least a single counter-example
To prove an existential statement: you must provide a single example
To disprove an existential statement: you must prove that every element is a counter-example -- that there is no example.

I believe that it's very important to be able to visualize statements in terms of sets -- to be able to determine whether proving a statement implies that a set must be occupied, or that a set must not be occupied. Not because every statement should be interpreted as a set -- but because it is important to be able to determine how to prove a statement true or false, and this is glaringly obvious with the aid of sets, especially at an introductory level. It is another aid to ensure that you are interpreting and understanding logical statements correctly. This concept becomes increasingly more difficult with the introduction of multiple quantifiers and multiple predicates.

Another concept on Test 1were the concepts of logic in Python (programming language). In summary:

(True not in ____) is logically equivalent to (not any ____)
- True not in = there is no true example, all are false
- This is similar to saying "all are not", "none are true", "not any"
- For all, not P(x)

(False not in _____)is logically equivalent to (all _____)
- False not in = there is no counter-example, all are True
- This is similar to saying "all are true", "none are false"
- For all, P(x)

(False in ____) is logically equivalent to (not all ______)
- This is similar to saying "there are some that are not", "there is a False", "not all are"
- There exists x such that P(x) is False

(True in ______) is logically equivalent to (any _____)
- This is similar to saying "There exists at least one example", "Any"
- There exists x such that P(x) is True

Therefore,
- not in = universal quantifier.
- in = existential quantifier
- any/not all = existential quantifier
- all/not any = universal quantifier

Another way to think about this: In English, when you ask someone about the existence of midnight snacks, you can phrase the question in a variety of ways:

"Are there any chips?" ---> any: is there at least one bag of chips? at least one example
"Aren't there any chips?" ---> not any: there are no chips? no example
"You ate all of the chips" ---> all:  all of the bag were eaten, without any exception. the entirety. no counter-example.
"So, what you're saying is that not all of the chips were eaten" ---> not all: there is at least one counter-example. Yay! a bag of chips.

I think the moral of the story is that there better be some leftover chips. Or something like that. Overall, today was a lovely day. I will celebrate by eating copious amounts of junk food, and studying for my linear algebra quiz tomorrow. A quiz that I will most likely fail, because I have neglected it in lieu of studying for 165. Curses.

Emergency Update

Captain, the sink is shipping.

I have yet to to be able to consistently analyze and interpret statements with multiple quantifiers. This is particularly the case when the operands are not presented in the order that they were introduced, or when the operand appears in multiple instances.

I attended a help session at the Undergraduate Computer Science Help Centre, and with the kind assistance of TA, I was able to understand one of the examples for a whole... 2 hours. Any attempts to recollect this session have been unsuccessful. All that remains of my momentary success are unintelligible scribbled notes on 2 sheets of paper. Disaster.

Abort! Abort! The midterm is on Wednesday!!

SLOG 3: Week 4

Today, the first assignment for CSC165 was due. For the assignment, I used LaTex to produce mathematical statements for the first time. Overall, it was an interesting experience, and I hope to continue to learn more about LaTeX and become more versed in this program.

A realization that I came to this week was that, although I'm struggling in this course, I'm not alone in this struggle. While discussing the assignment with students, and attending today's lecture, I became significantly more confident in my capabilities. Grasping some of the concepts in CSC165 has been a uphill climb, and there are often times that I had wondered if I was hopelessly alone in my incompetency. However, hearing that many other students were stuck on some of the same things that I had been stuck on a week ago was heartening, and has given me newfound confidence. 

The midterm next week will allow us the aid of an 8.5x11" double sided cheat-sheet. I am both excited and nervous that the midterm will be testing us on our capacity to produce and comprehend logical statements, rather than brute memorization. 

I will most likely update this blog with another post in the upcoming days as the day of the midterm approaches. Winter is coming.