Need help on Set Theory

[Nipuna]

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

[Nipuna]

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

(4,1),(1,4)

[Neo]

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.

[Nipuna]

Yes, You Right. Main Reason to ask these Questions from you guys was to know how to prove those Stuff

[Nipuna]

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)

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 ?

[Herath]

Capture.PNG (5.66 KiB) Viewed 13277 times

[Nipuna]

Thanks Friend. But Still Didn't Get It

[Neo]

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

[Herath]

It is just simple expansion of (a+b)^2.

Capture.PNG (1.56 KiB) Viewed 13273 times

http://en.wikipedia.org/wiki/Binomial_theorem

[Herath]

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