Need help on Set Theory

Mathematics for Computing
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Nipuna
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Re: Need help on Set Theory

Post by Nipuna » Mon Jun 20, 2011 6:42 pm

Neo wrote:
I made a Simple way to Remember Those weird Set Algebra Names :)
Where is the last D for De Morgan's Law? Should be I A C D I I C D right?

Yes. That's Right. That's Why I added And before It since I only Made that method for I A C D I I C. I just memorized De Morgan Law without that. If some one says Me to write the Law Names then I will Write I A C D I I C and Then add De Morgan Separately.Because De Morgan one is in my mind even before I memorize the I A C D I I C :)
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Re: Need help on Set Theory

Post by Nipuna » Mon Jun 20, 2011 6:45 pm

Herath wrote:
Neo wrote:
i expected that nipuna would try to find the answer for the first one.that is why i didnt post a complete answer.
He hasn't done Maths for his A/Ls so he might not be knowing about solving quadratic equations, etc... I think it would be helpful for him to give a detailed description considering that. I know it is not easy to cover all these fundamentals. How do you think?
Sorry If I sounded rude. But I did not expect all answers from him. Just wanted him to post the elements of the set B and the final answer which is A and B. I just wanted to make sure that he understood the given notation for the set and few basic things. He might go through the given answer and think he is ok with it. But when in the exam he might get in trouble because he did not tested him self. :)

By the way neo, how do you type these mathematical symbols directly from keyboard. I thought that it was not possible to type directly. Copying and pasting from the character map is going to be a bit mess.

Thanks Friend

is these Right for the Q1 ? I feel Right because they are same as the ones in your answer. Only sides are changed :D
(4,1),(1,4)
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Re: Need help on Set Theory

Post by Neo » Mon Jun 20, 2011 6:52 pm

is these Right for the Q1 ? I feel Right because they are same as the ones in your answer. Only sides are changed
Correct. I just wanted to show you how to get these elements mathematically rather than guessing.

In mathematics, you need to prove that there can be only these two elements and nothing else. Simple guessing can't always do this.
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Re: Need help on Set Theory

Post by Nipuna » Mon Jun 20, 2011 7:27 pm

Yes, You Right. Main Reason to ask these Questions from you guys was to know how to prove those Stuff :)
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Re: Need help on Set Theory

Post by Nipuna » Wed Jun 22, 2011 10:52 am

Neo wrote:
First one
Here A is defined as all numbers x and y such that x^2+y^2=17. Also it is given that x and y are natural numbers, which means that x and y can only be numbers from zero to positive infinity(remember that infinity is not a number).
Notice that x and y are in ordered pairs (x,y). So (x,y) and (y,x) are two different elements of the set. That is (x,y)=(y,x) if and only if x=y.

For and example 1^2+4^2=17. Hence x=1 and y=4 . So you can get two elements easily which are (1,4) and (4,1).
Then we can write A={(1,4),(4,1)} intersection {rest of the elements of A}

Try to find the rest of the elements and the answer for the first question. i hope that thing are clear and any corrections are welcome. :)

I will try answering the rest soon. Now I am at the boarding. Although I have brought my notebook, internet access is a problem.
Herath wrote:First one

Try to find the rest of the elements and the answer for the first question.


Here is the Rest. I could only Find

(4,1),(1,4) :D
Just another addition to Herath's explanation on question 1...

There are two sets...
A = {(x,y)| x,y ?N and x2 + y2 = 17}
B = {(x,y)| x,y ?N and x + y = 5}

So the question is to find A ? B which is read as A intersection B,
Let's name a new set C where C = A ? B.

Based on the above definition for A and B, we can define C as below.

C = {(x,y)| x,y ?N and x2 + y2 = 17 AND x + y = 5}

Notice the 'AND' I put to denote the intersection. If it was ? (Union) I would put OR instead of AND.

Okay. Now, we can solve the equation.

x + y = 5
y = (5 - x) ------[EQ1]

Now we are going to substitute y to the other equation.

x2 + (5 - x)2 = 17
x2 + 25 - 10x + x2 = 17
2x2 - 10x + 25 = 17
2x2 - 10x + 8 = 0
x2 - 5x + 4 = 0 -------------[EQ2]

This is in the form of a quadratic equation.
ax2 + bx + c = 0

In our equation,
a = 1
b = -5
c = 4

We can take the value of x when our equation is in the form of quadratic equation using quadratic formula as given below.
quadeq.PNG
x = (-(-5) ± ?((-5)2 - 4x1x4)) / 2x1
x = (5 ± ?(25 - 16)) / 2
x = (5 ± ?(9)) / 2
x = (5 ± 3) / 2

x = 8/2 OR x = 2 / 2
x = 4 OR x = 1

From EQ1,
y = 1 if x = 4 OR y = 4 if x = 1

In other words, the two possible elements are (4, 1), (1, 4)
? C = A ? B = {(4, 1), (1, 4)}

In this way you can solve any type of similar question. Just simple Algebra.
Thanks Friend,

But In here

x2 + (5 - x)2 = 17
x2 + 25 - 10x + x2 = 17
2x2 - 10x + 25 = 17
2x2 - 10x + 8 = 0
x2 - 5x + 4 = 0 -------------[EQ2]

Where did that 10x come from ? :?
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Re: Need help on Set Theory

Post by Herath » Wed Jun 22, 2011 11:02 am

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Re: Need help on Set Theory

Post by Nipuna » Wed Jun 22, 2011 12:38 pm

Thanks Friend. But Still Didn't Get It :?
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Re: Need help on Set Theory

Post by Neo » Wed Jun 22, 2011 1:06 pm

See This Squaring Binomials Video.
[media]http://www.youtube.com/watch?v=w8smA_akWBY[/media]

BTW: If she was my teacher, her beauty would have been distracted me from learning this :lol:
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Re: Need help on Set Theory

Post by Herath » Wed Jun 22, 2011 1:10 pm

It is just simple expansion of (a+b)^2.
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http://en.wikipedia.org/wiki/Binomial_theorem
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Re: Need help on Set Theory

Post by Herath » Wed Jun 22, 2011 1:11 pm

Neo wrote:See This Squaring Binomials Video.
BTW: If she was my teacher, her beauty would have been distracted me from learning this :lol:
:D
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