mustish1
asked on
Union and intersection
I dont know how to calculate the dash also did i calculate right the A U B
needs help
A=1,2,3,4
B=3,4,5,6
U=1,2,3,4,5,6,7,8,9
(AUB)'
(AUB)=1,2,3,4
needs help
A=1,2,3,4
B=3,4,5,6
U=1,2,3,4,5,6,7,8,9
(AUB)'
(AUB)=1,2,3,4
Here is a good starting point to learn union (set theory) (Wikipedia).
ASKER
so (AUB)=1,2,3,4,5,6,7,8,9
but how to calculate the dash
(AUB)'
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.
"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
--------------------------
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.
ASKER
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)'
---------------- --------------
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.
Then, please explain what in the definition are you having problems with.
ASKER
it says in the book
calculate (A U B)'
(A' U B')
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.
>> 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.
ASKER
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
<==> 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?
--------- ----------
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.
>> 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.
SOLUTION
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>> 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.
In below figure, you see three sets: U, A, and B. A and B intersect, where A^B = {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
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 |
\-------------------/
ASKER
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
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
ASKER CERTIFIED SOLUTION
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Now, try checking your work by computing:
A' ^ B'
and see if the result is the same.
A' ^ B'
and see if the result is the same.