Katie W.
asked • 10/13/15Let A, B, and C be sets in a universal set U.
We are given n(U)=73, n(A)=35, n(B)=43, n(C)=42, n(A intersect B)=18, n(A intersect C)=27, n(B intersect C)=27, n(A intersect B intersect C')=3.
Find (a) n((A u B u C)') and (b) n(A' intersect B' intersect C).
I think I found the answer to the first part of the question (part a). I got 10 which I believe is right.
Then when I went to do part b I thought I did it right (I got 63) but it was wrong. Any help is appreciated thanks.
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1 Expert Answer
Arnold F. answered • 10/13/15
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College Professor & Expert Tutor In Statistics and Calculus
You are looking for n(A'∩B'∩C') and you already found n[(AυBυC)'].
Did you learn DeMorgan's Laws: one of states (A'∩B'∩C') = (AυBυC)' that is these are the same sets so the answers to both parts are the same.
(By the way part (a) says find how many elements are in A, B and C combined and take it's complement and part b says find how many elements are in A', B' and C' at the same time; same thing as (a).)
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VENN Diagram: I've been trying to send you a link to my Venn diagram:
https://www.wyzant.com/resources/files/391685/venn_diagram
https://www.wyzant.com/resources/files/391685/venn_diagram
Katie W.
Part b is not A', B', C' it is A', B', and C
I did learn DeMorgan's Law and that's how I started to solve it.
I got (A' ∩ B' ∩ C) = (A ∪ B ∪ C')'
n(A' ∩ B' ∩ C) = n(A ∪ B ∪ C')'
= n(U) - n(A ∪ B ∪ C')
and that's where I am stuck. I think you made a mistake when reading my question becaues your symbols are wrong in the first line
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10/13/15
Arnold F.
Sorry Katie,
Of course you're right.
Then if your teacher allows it I could use a Venn diagram, unfortunately I can't seam to paste an image in this box so I'll describe what I've done.
The A,B,C divide U into 8 mutually exclusive pieces. As you go through the given you can fill in the 'n' of each piece.
1 P(ABC) =15 the innermost piece
2 P(A'BC) = 12
3. P(AB'C) = 12
4. P(ABC') = 3 (one of the given values)
5. P(A'B'C) = 4
6. P(A'BC') = 12
7. P(AB'C') = 5
8. P(A'B'C') = 10
From all of these you can figure out n of any piece(s).
Please comment back if you want to know how I got any of these.
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10/13/15
Katie W.
Okay I believe I drew a correct Venn diagram based on your description. I would post it but I can't paste an image as you mentioned.
So I'm assuming that what I am looking for is the section of the Venn diagram that includes A, B , and C and then the answer is everything else because it is a complement?
I am getting 70. Do you think that is right?
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10/13/15
Arnold F.
(b) says to look at the elements outside of A and at the same time outside of B but at the same time inside C.
A'B'C is line 5 above so I get n=4.
If you don't get that see: https://www.wyzant.com/resources/files/391685/capture
Let me know if your diagram agrees with mine.
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10/13/15
Arnold F.
We need A'B'C. Outside of A and outside of B and inside of C which is line 5 above so n=4.
Check out my Venn diagram here:
https://www.wyzant.com/resources/files/391685/capture
Let me know if yours agrees.
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10/13/15
Arnold F.
Check out this Venn diagram:
www.wyzant.com/resources/files/391685/capture
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10/13/15
Arnold F.
I've been trying to send you a link to my Venn diagram:
in your browser type in www.wysant.com/ and add the next line after the slash
resources/files/391685/capture
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10/13/15
Arnold F.
I added a link to my Venn diagram in the main answer section
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10/13/15
Katie W.
What is the file called? Your link takes me to the general resources page
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10/13/15
Katie W.
I searched Venn diagram and nothing came up under your name
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10/13/15
Arnold F.
Go to my original answer at the top of this screen.
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10/13/15
Katie W.
Ya it takes me to a generic file page with multiple files from different people. I don't see your name or a file that has to do with venn diagrams
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10/13/15
Katie W.
I got it!
That is the venn diagram I drew when you described it earlier. Glad to see I understood that.
So for the answer to part B is it 70? Everything in the Venn diagram added together except C?
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10/13/15
Arnold F.
You want everything outside of A and outside of B and inside C. That should be A'B'C which is line 5 back up in one of my early comments so it = 4.
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10/13/15
Katie W.
For n(A' ∩ B' ∩ C) ?
Oh okay I see what you are saying.
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10/13/15
Katie W.
Those are intersection symbols
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10/13/15
Arnold F.
yes A'B'C is the same as A'∩B'∩C
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10/13/15
Katie W.
okay!!
Thank you so much for your help. I really appreciate it. Thanks for making everything clear for me!
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10/13/15
Arnold F.
Sorry it took so long!
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10/13/15
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Arnold F.
10/13/15