"Information on coming quiz on 19 Sep (Fri) is posted in the course document folder. Feel free to ask me or your tutors if you have any questions on Prob & Stats. Don't worry, the quiz is meant to pass all of you, and not to 'kill' you. :-) Know this course is compulsory for most of you, but hope you all find this course not too stressful. :-p"
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Not stressful ... riiiighhht. I think I'm gonna die soon...
Since Sunday or so, every day is just Maths, Maths and more Maths.
Every break in between lessons, every trip traveling and every day after school has been spent doing Maths and Maths...
My eye circles are getting worse...
So sad I couldn't accept some of the invitation... I really wanted to go... but I really couldn't...
In fact, I'm been putting aside almost every other subject (doing the bare minimum) in order to save my dying Mathematics...
...
And today is finally the last day of revision!
Wish me the best tomorrow...
Thursday, September 18, 2008
Monday, September 15, 2008
Do You Know What Is Infinity?
Infinity is very large.
Infinity is larger than the largest value you can think of.
It is larger than one zero zero zero zero zero zero zero zero zero...
It is larger than 1 million, larger than 1 billion...
...
It is bigger than nine nine nine nine nine nine nine nine nine nine nine nine nine nine...
Imagine I give you a piece of paper, and you write a lot of 9 on it.
You write many many many many 9 on it.
Infinity would still be much bigger than that.
You can write 9 on the paper for 24 hours...
Infinity would still be larger.
If someone comes and add one more 9 to the paper, that number would be bigger...
And another come and add one more 9...
and one more 9...
and one more...
Infinity is still bigger.
...
You can write 9 for as long as you can, and infinity is still bigger.
...
...
No matter how big is your number, infinity is much bigger.
...
...
So, do you know big infinity is?
...
...
Just a very big number.
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Sunday, September 14, 2008
A Discrete Maths Test
Discrete Mathematics is easily the most intriguing subject I've ever taken in my life. After studying this subject for 6 weeks, I think I just might have a little idea of what Discrete Maths is all about. Let me try to put it into perspective for you.
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The main idea is simple - To make a simple concept as complicated as possible, to the point where no ordinary people can understand it.
If it sounds too abstract, let me give you the simplest example.
Primary School: 1 + 1 = ? [1 Mark]
Secondary School: 1 + 1 = X. Find the value of X. [1 Mark]
That's what you've been getting all your life. Now, let's take a look at a sample Discrete Maths paper.
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Question 1: The set X is related to Y, denoted by XRY, where the universe of discourse of X is the set of all positive integers smaller than 2, and the domain of Y is the set of all positive odd integers smaller than 3, defined by the binary relation as follows: For All x in X and for all y in Y, XRY => x + y = 2. [50 Marks]
a) Write down the power set of X and Y. [2]
b) Is R reflexive? Is R transitive? Is R symmetric? Is R anti-symmetric? Justify your answer for each case. [8]
c) For each of the above case, use logical reasoning to prove whether XRY is true for all cases of x and all cases of y. If not, show a counterexample by proving there exists a x or y where R is not true. [5]
d) Least out the greatest, least, maximal and minimal elements with respect to R. [4]
e) Justify if R is a total order. [2]
f) Explain whether XRY is a Tautology and plot the truth table for XRY. [4]
g) Highlight all the critical rows of the truth table. [1]
h) Draw the Venn Diagram of the relation [2]
i) Draw the Arrow Diagram of the relation [2]
Now, expand X and Y to the set of all Z+ < 100, or the set of all positive integers less than 100.
j) Is the relation One to One? Is the relation Onto? Justify your answer and write down the inverse function if it exists. [6]
k) What is the last number of x required for all values of y? Prove and justify your answer [4]
l) Guess a general formula for the above relation, and prove it by mathematical induction. [10]
Now, supposed a random x is chosen out of X, where X is just a sample of a larger population, K, and K is the set of all positive integers smaller than 1000.
m) Find the probability, denoted by P(X = x), that x is equal to 1.
n) Find the mean and variance of the population of X.
o) Identify the probability distribution associated with X.
p) Use unbiased point estimate to infer the population mean.
q) Use interval estimate toe infer the population mean with a 95% confidence level.
r) If someone claims that the mean of x is 500, test at a significance level of 5% whether it is possible to reject his claim.
...
...
...
...
...
...
z) You have finally arrived at the last part of this one and only question in this test. Give yourself a round of applause and congratulations for sitting through this ****. [1]
------------------------------ END OF PAPER (PAGE 2/2)-----------------------------
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The main idea is simple - To make a simple concept as complicated as possible, to the point where no ordinary people can understand it.
If it sounds too abstract, let me give you the simplest example.
Primary School: 1 + 1 = ? [1 Mark]
Secondary School: 1 + 1 = X. Find the value of X. [1 Mark]
That's what you've been getting all your life. Now, let's take a look at a sample Discrete Maths paper.
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Discrete Mathematics Sample Examination Paper (2008/2009)
Section A
Question 1: The set X is related to Y, denoted by XRY, where the universe of discourse of X is the set of all positive integers smaller than 2, and the domain of Y is the set of all positive odd integers smaller than 3, defined by the binary relation as follows: For All x in X and for all y in Y, XRY => x + y = 2. [50 Marks]
a) Write down the power set of X and Y. [2]
b) Is R reflexive? Is R transitive? Is R symmetric? Is R anti-symmetric? Justify your answer for each case. [8]
c) For each of the above case, use logical reasoning to prove whether XRY is true for all cases of x and all cases of y. If not, show a counterexample by proving there exists a x or y where R is not true. [5]
d) Least out the greatest, least, maximal and minimal elements with respect to R. [4]
e) Justify if R is a total order. [2]
f) Explain whether XRY is a Tautology and plot the truth table for XRY. [4]
g) Highlight all the critical rows of the truth table. [1]
h) Draw the Venn Diagram of the relation [2]
i) Draw the Arrow Diagram of the relation [2]
Now, expand X and Y to the set of all Z+ < 100, or the set of all positive integers less than 100.
j) Is the relation One to One? Is the relation Onto? Justify your answer and write down the inverse function if it exists. [6]
k) What is the last number of x required for all values of y? Prove and justify your answer [4]
l) Guess a general formula for the above relation, and prove it by mathematical induction. [10]
Section B
Now, supposed a random x is chosen out of X, where X is just a sample of a larger population, K, and K is the set of all positive integers smaller than 1000.
m) Find the probability, denoted by P(X = x), that x is equal to 1.
n) Find the mean and variance of the population of X.
o) Identify the probability distribution associated with X.
p) Use unbiased point estimate to infer the population mean.
q) Use interval estimate toe infer the population mean with a 95% confidence level.
r) If someone claims that the mean of x is 500, test at a significance level of 5% whether it is possible to reject his claim.
...
...
...
...
...
...
z) You have finally arrived at the last part of this one and only question in this test. Give yourself a round of applause and congratulations for sitting through this ****. [1]
------------------------------ END OF PAPER (PAGE 2/2)-----------------------------
Saturday, September 13, 2008
NTU Y1S1, Week 6 In Review
Random Quotes Of The Week:
"And, as you go out into the world, I predict that you will gradually and imperceptibly, forget all you ever learned at this University."
- Dr Woo (Last Lecture)
"I think I have to set up an ERP gantry at my office."
- Dr Kwoh (Computer Systems Tutorial)
"Dong Fang Bu Bai."
- ??? (???)
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In The Class:
Yeah we're halfway there now. Had the last lectures by Dr Woo and Dr Chan as we head on to the more "deadly" topics... It almost feels like I'm suffocating...
I can't wait for Recess to come!!!
Computer Systems
What can I say?
...
I think somethings are better kept among ourselves...
My first "test" in 3 years... and it's something like this. -_-
Logic Design
This module is doing its best to catch up the Maths in terms of difficulty... If it continues to flip a few more flops, I think that goal will soon be within reach.
I think we are lucky to get 2 very "pro" lecturers for this module...
Haiz... why can't every lecturer be like them?
Data Strucutres Programming
All I can say is...
The best is yet to come.
Discrete Maths
Every single week this subject gets more and more intriguing.
First, we're talking about how the moon transform into a star.
Now, we're talking about birds flying into holes.
Are all these topic even remotely related?
It's like taking content from about 20 different areas of Mathematics and combining them into this.
...
...
Wait. Maybe they're related after-all.
...
Discretely related.
Foundation Maths
A picture speaks a thousand words.
This picture speaks a million.
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"And, as you go out into the world, I predict that you will gradually and imperceptibly, forget all you ever learned at this University."
- Dr Woo (Last Lecture)
"I think I have to set up an ERP gantry at my office."
- Dr Kwoh (Computer Systems Tutorial)
"Dong Fang Bu Bai."
- ??? (???)
-------------------------------------------------------------------------------------------------
In The Class:
Yeah we're halfway there now. Had the last lectures by Dr Woo and Dr Chan as we head on to the more "deadly" topics... It almost feels like I'm suffocating...
I can't wait for Recess to come!!!
Computer Systems
What can I say?
...
I think somethings are better kept among ourselves...
My first "test" in 3 years... and it's something like this. -_-
Logic Design
This module is doing its best to catch up the Maths in terms of difficulty... If it continues to flip a few more flops, I think that goal will soon be within reach.
I think we are lucky to get 2 very "pro" lecturers for this module...
Haiz... why can't every lecturer be like them?
Data Strucutres Programming
All I can say is...
The best is yet to come.
Discrete Maths
Every single week this subject gets more and more intriguing.
First, we're talking about how the moon transform into a star.
Now, we're talking about birds flying into holes.
Are all these topic even remotely related?
It's like taking content from about 20 different areas of Mathematics and combining them into this.
...
...
Wait. Maybe they're related after-all.
...
Discretely related.
Foundation Maths
A picture speaks a thousand words.
This picture speaks a million.
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The Last Temasek
Wah... really shagged from the outing last night. At first I wanted to leave earlier, but got someone just took my belongings on a tour to the other end of Singapore, without informing me first.
So bo bian... SK had to wait for me. So we decided to get something to do. At first just wanted to play while... but it's too addictive. Once you start you can't stop, just like Pringles.
Kept playing round after round after round... Then see the time, after 12 liao... makes no difference anymore. So play some more... LOL....
Loads of fun, great way to relive stress. ^_^
So bo bian... SK had to wait for me. So we decided to get something to do. At first just wanted to play while... but it's too addictive. Once you start you can't stop, just like Pringles.
Kept playing round after round after round... Then see the time, after 12 liao... makes no difference anymore. So play some more... LOL....
Loads of fun, great way to relive stress. ^_^
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