Union and intersection

Here is a good starting point to learn union (set theory) (Wikipedia).

so (AUB)=1,2,3,4,5,6,7,8,9

but how to calculate the dash
(AUB)'

from the wiki link:
"The union of two sets A and B is the collection of points which are in A or in B (or in both)"

>> so (AUB)=1,2,3,4,5,6,7,8,9
No.

>> A=1,2,3,4
>> B=3,4,5,6
>> U=1,2,3,4,5,6,7,8,9

A consists of 4 elements.
B consists of 4 elements.

I take it that by U, you mean the Universe of elements that you are considering, of which A and B are subsets of the Universe. Is that your understanding also?

If the elements in A were distinct from the elements in B, then at most AUB would have 8 distinct elements. Do you see that? Somehow, your union of AUB has 9 elements, so you should immediately raise a flag that something is wrong. Here is an example of X and Y having mutually distinct elements, and their resultant union.

   X             Y
---------    ----------
 1 2 3 4      5 6 7 8
---------    ----------

 XUY (has only 8 elements)
--------------------------
 1 2 3 4      5 6 7 8
--------------------------

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Now, try to draw and post the sets A and B, and then draw the union of A and B.

Also, provide a definition from your text of the dash. (You drew a tick; and there is a dash, '-', defined; but you are not showing it.) Provide a definition of tick also.

    A                             B
----------------         --------------
1 23 4                      3 4 5 6
-----------------        --------------

sorry A U B=1,2,3,4,5,6

NOW how to calculate (A U B)'

Provide a definition from your text of the dash. (You drew a tick; and there is a dash, '-', defined; but you are not showing it.) Provide a definition of tick also; and we'll take it from there.

Then, please explain what in the definition are you having problems with.

it says in the book
calculate (A U B)'
(A' U B')

>> calculate (A U B)'
>> calculate (A' U B')

Those are the problems. In order to solve set theory problems, you have to first start with definitions and work with them. So, look up the definitions and post them; and I'll try to help you understand them to work out the problem.

x belongs to (A U B)' <==> not(x belongs A U B)  Definition of '
                                 <==> not[(x belongs A) V (x belongs B)]     Definition of U
                                 <==> not(x belongs A) ^ not(x belongs B) De Morgan's law
                                 <==> (x belongs A') ^ (x belongs B') Definition of '
                                 <==> x belongs A' intersect B'   Definition of intersection

Therefore the sets (A U B)' and A' intersect B' are equal

  X             Y
---------    ----------
 1 2 3 4      5 6 7 8
---------    ----------
XUY (has only 8 elements)
--------------------------
 1 2 3 4      5 6 7 8
--------------------------

Now, suppose U = {0 1 2 3 4 5 6 7 8 9 10 11}

>> Definition of ':  x belongs to (X U Y)' <==> not(x belongs X U Y)

x belongs X U Y ==> x is an element in { 1 2 3 4 5 6 7 8 }

>> not(x belongs X U Y) means the elements, x, that do not belong to X U Y
So, in this example, x cannot be any of { 1 2 3 4 5 6 7 8 }.

So what is left in the universe, U = {0 1 2 3 4 5 6 7 8 9 10 11}, that is not in X U Y in this example?

I just saw that you wrote:
>> Definition of U:  not[(x belongs A) V (x belongs B)]

So, in your other notation, U = not( AUB ), which means that if x is an element of U, then x is not an element of A and x is not an element of B.

In your OP:
>> U=1,2,3,4,5,6,7,8,9
But here, U has elements that are in A as well as B

So, for clarication, could you double-check the definition of U, and, possibly, write down the text's written description of its meaning. I was originally taking U to be the Universe of all elements that you are concerned with.

>> intersection are the common elements of A and B in one set (3,4)
That's true. In your notation:
A^B = {3,4}

From your title, it might appear that you are interested in the intersection. However, your question is related to the tick operation:
    (A U B)'

Getting an answer in set theory is nice, but not as important as understanding the definitions and notation.

Take your conclusion, for example:
(A U B)'  <==> x belongs A' intersect B'   Definition of intersection

What is A' ? Here is a picture illustrating the complement of A (taken from http://en.wikipedia.org/wiki/Complement_(set_theory) ). The outer rectangle is the entire Universe, U; and the red color is the complement of A, A'.
 What is B' ?

If you know A' and B', then you easily figure out A' ^ B'
However, you should also compute (A U B)'  and verify that they are the same.
If you have problems getting the results identical, let us know where, and we will try to help.
=============
Rather than just deal with symbols, it is easier to visualize the sets in a diagram.

A=1,2,3,4
B=3,4,5,6
U=1,2,3,4,5,6,7,8,9

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In below figure, you see three sets: U, A, and B. A and B intersect, where A^B = {3,4}

U-------------------\
|                   |
|    A--------\     |
|    |   1 2  |     |
|    |        |     |
|  B-+--------+-\   |
|  | |   3 4  | |   |
|  | \--------/ |   |
|  |            |   |
|  |   5  6     |   |
|  |            |   |
|  \------------/   |
|     7   8   9     |
\-------------------/

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A' I think U -A
B' U - B

A U B=1,2,3,4,5,6

U=1,2,3,...9
(A U B)' = 1,2,3,4,5,6 - U

(A U B)' = 7,8,9
Is this is correct

Now, try checking your work by computing:
    A' ^ B'
and see if the result is the same.