A and B are subsets of the universal set U. Given n(A')=23, n(B')=16, and n((A ∩ B) U (AUB)'))= 24, find (A∩B)
I know the answer, I just don't know how to get it without guessing and checking.
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A and B are subsets of the universal set U. Given n(A')=23, n(B')=16, and n((A ∩ B) U (AUB)'))= 24, find (A∩B)
I know the answer, I just don't know how to get it without guessing and checking.
What does (A ∩ B) mean?
Sorry, n(A ∩ B)
Um... yea, but I was trying to know if you really understand what each of the terms in "n((A ∩ B) U (A U B)')) = 24" mean.
Can you draw this on a Venn diagram, and on separate diagrams, the other information provided?
http://fc06.deviantart.net/fs70/f/20...o8-d5e8s9v.png
1. n((A ∩ B) U (A U B)')) = 24
2. n(A') = 23
3. n(B') = 16
Okay, can you try to do a relation about those?
Initially I was going to say that n((A ∩ B) U (A U B)'))-n(B')-n(A')=(A ∩ B), but that doesn't work, since it's negative and I'm subtracting the area outside A and B twice. Since I was able to guess and check the answer, I know that the sum of n(A') and n(B') minus n((A ∩ B) U (A U B)')) is what is outside of A and B. However, I'm sure that's just luck.
Short answer: No, I can't...
Hmm, I think the easiest way for me would be to assign some variables to the different areas.
n((A U B)') = a
n(A ∩ B) = b
n(A ∩ B') = c
n(A' ∩ B) = d
The first picture says: a + b = 24
The second: a + d = 23
The third: a + c = 16
And you are asked to find b.
Well, I don't think that there is a single solution to this.
For instance, we can get:
b - d = 1
b - c = 8
d - c = 7
And get that:
8 < b <= 24
7 < d <= 23
0 <= a < 16
Taking b = 9:
a = 15
c = 1
d = 8
Taking b = 10;
a = 14
c = 2
d = 9
[... ]
Taking b = 24;
a = 0
c = 16
d = 23
Which are all valid possibilities.
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