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.

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.

The number of elements in the intersection of A and B or Not in A nor B is 24.

Can you draw this on a Venn diagram, and on separate diagrams, the other information provided?

Can you draw this on a Venn diagram, and on separate diagrams, the other information provided?

The space (sometimes an oval) between A and B would be counted (or shaded), and the area outside of A and B would be shaded. I can't post pictures online because I don't have a camera or scanner connected to my desktop.

http://fc06.deviantart.net/fs70/f/2012/253/2/0/test_by_unknownoo8-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?

http://fc06.deviantart.net/fs70/f/2012/253/2/0/test_by_unknownoo8-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.