Floor - Assignment 2

This is going to be my last post for this season.

I reviewed Assignment 2 which I got very low mark (eventually failed). I found I have a big problem with floor notation.

It seems obvious when the solution is in front, but I need to re-assure that I understand the two problems below.

This time, I use GoodNotes (iPhone) to publish the two problems.

Click image to enlarge.

 

Click image to enlarge.

Basically, I have to understand the definition of floor notation. I think now I'm good to go with final exam except those we haven't done any practice in either quiz/mid-terms or assignments, such as "onto".

Tomorrow, I will review past exams, and I should be okay for the final exam as I made enough mistakes to actually LEARN CSC165H1 (I think).

LaTex Copy & Paste / Assignment 3

I finally found the way to copy & paste LaTex codes.

Simply visit https://latex.codecogs.com/ this website, create a code, and copy + paste on the blogger's compose panel.


This is one of the assignment-3 questions which I have to review because I still misunderstood how the order of quantifiers matters.

I thought this is false, but it's actually true.

When I negated the claim as above, I could pick e, x and y. I picked

 and
 
It works, iff the claim is:

like this. I actually picked the variable "d" before "d" is introduced. The professor told us about the ordering of the quantifiers is significant when proving such claims in his review session. At that time, I didn't quite get it.

But now I get it!!

People learn from their mistakes. It's true. I've at least fixed this problem, and I hope this mistake helps my future courses such as CSC236.

There are several other things that I made mistakes on Assignment-2 need to be mentioned and noted in this blog to make sure I will not make the same mistakes during the final exam.

And again, check out latex.codecogs.com. This is a great and easy way to implement math equations on the web!!

Nice sLog - Blogger

Many students at CSC165 post their sLog in Blogger system. This is now owned by Google, and it's quite useful (as it seems Google always add Blogger updates within as early as 12 hours).

I actually went around other students' blogs and see how they build up their sLogs. I found one website that I want you to check out.

http://monicashver.blogspot.ca/

I found many students don't care about the style of their blogs. This is sometimes true for professor's websites too. (Or I would say it's true for many times). In business field, the desgin/style part of business projects is very important. I believe this is same for sLog. So I linked the one above, because that blog uses a cool style sheet (CSS), and attract vistors to her blog.

LaTex for Blogger

This is a post to test LaTex code within Blogger in order to add reviews for final exam.

Actually, I got 25% on the assignment 2. This is horrible. It seems I've misunderstood floor notation and the way to prove "there exists" and "for all". "For all" means really "for all", and the variable indicated for "for all" must satisfy "for all" really.

It sounds obvious, but it seems that's not compiled correctly in my brain. I will have to re-compile CSC165 concepts by reviewing all quizzes, tests and assingments where I made mistakes.

To do so, I should use LaTex coding in blog. But it has never worked.

Uuum... I don't have much time to dig around how to do LaTex... So I guess I will use images as usual.

There has to be something that's easy to implement just like copy/paste equations..

Anyways... my reviews are coming soon in the next couple days.

Week#11

The term is almost finished, but then I have to remember Calculus which I studied when I was in Glendon College almost a decade ago...

Again, since I'm pursuing Computer Science degree as my second degree, any course I've already taken that overlap U of T courses is in the list of exclusion. That means, I supposed to know the material.

Well... maybe I'm lucky or not, only Calculus I have to remember is L'Hopital's Rule. (And derivatives, of course).

Basically, when limit of n tends to sigma where 2^n / n^2 = sigma. It's obvious that 2^n is much larger than n^2. But this is still not quite intuitive. So we take derivatives of both numerator and denominator until we find a constant.

d(2^n) / d(n^2) = ln2*2^n / 2n
d(ln2*2^n) / d(2n) = ln2*ln2*2^n / 2

Voila! Now we can intuitively see that this function gets enoumously large when n tends to sigma (infinity).

Here is my course note for proving n^2 is not element of Big-Oh(2^n).

Click the image to enlarge.

Week#9/#10

University of Toronto has a holiday called Fall Break. So my sLog is also combined Week9 and Week10. I think this is fair?

Anyways, here are new topics in CSC165.

BIG DATA, O and Omega.

I think this is a very important field in Computer Science in the next few years because one chunk of data is getting so big these days as the availability of high speed internet at homes is also getting matured.

I found the following example is useful for my sLog note.

n^3 is not an element of O(3n^2). So there exists some postive number c, some natural number B, for all n that is natural number, such that when n is bigger or equal to B implies n^3 is less or equal to c3n^2.

Well... is it really false? Nowadays, you can plot graphs in Google search.
https://support.google.com/websearch/answer/3284611

Now, here is the plot.

https://www.google.ca/?gws_rd=ssl#q=x%5E3%2C+3x%5E3

It looks like x^3 is larger than 3x^2 when it's positive. So the statement supposed to be false, and it is false.

I found that plotting graphs help me understand clearer for O and Omega. It's worth it to try it in graphs!

Week#8

When a question asks us to prove "OR" or "AND" notation, we need to show both end of "OR" and "AND". Intuitively, it's simpler in "AND", and indeed it's simpler. Here are what we have discovered during tutorial. These questions are very useful to note in slog as these can be applied to many other problems.

For all real numbers r,s if r and s are both positive, then .
This can be re-writen as follows:

Now... this statement is false. So we have to prove its negation. Then "AND" notation comes into play. Here is how we have disproved this claim.

It so it's basically proving both side of "AND", because the both sides must be true for A and B to be true.

When it comes to "OR" notation, we need to prove both sides separately. Here is one tutorial example.

 It's in my hand writing, by the way.
The claim seems quite obvious, but it's harder to prove because it's intuitive. When I see this proof, I also convince myself root(r) + root(s) is not equal to root(r+s) indeed.