Question 343057: let A and B be subsets of U with n(A)=44, n(b)=32, n(A')=25 and n(A and B)=24
find n(AUB')
I am not sure how to find the compliment of just a number, ie how did they get n(A')
Answer by solver91311(24713)   (Show Source):
You can put this solution on YOUR website!
I don't know what the compliment of a number is either. I suppose it is something along the lines of "Gee, you are a nice round number" or "Oh, what handsome prime factors you have" or something like that. All that aside, you aren't asked to find the complement of "just a number." That is not what ) means.  means the set of everything in  except  , and means the number of elements that are in the set .
Step 1 in this process is to draw a rather simple Venn diagram. Draw a rectangle. Completely inside of the rectangle draw two circles that only partially overlap. Now you should have mapped out four regions. A region that is inside the rectangle but not inside of either circle. A region that is only inside of one of the circles. A region that is only inside of the other circle. And finally a region where the two circles overlap.
Label the inside of the rectangle (but outside of the circles) with , remembering that is the set of everything inside of the rectangle. Label one of the circles and the other 
Now, start in the middle. ) is the enumeration of the set represented by the piece of the two circles where they overlap. In otherwords, everything that is in both A and B simultaneously. We are given that this enumeration is 24, so write a 24 in the little overlap area.
Now we were also given that \ =\ 44) . Since the enumeration of A includes both elements that are only in A as well as elements that are also in B, we can determine that the part of A that is only in A must number 20 elements, because 44 minus 24 equals 20. Write a 20 into that part of the circle labeled A that doesn't overlap circle B.
Similarly, we can tell that the part of circle B that does not overlap circle A must have an 8 in it.
Next we want to determine the number that goes outside of both circles. The enumeration of the complement of A is given as 25. But the complement of A, in otherwords everything in the universe set that is NOT in A includes both this outer area plus the part of circle B containing the 8. Since we are given that \ =\ 25) , the part of the universe that is neither in A or B must have an enumeration of 25 minus 8 equals 17.
Now you are able to add up all of the numbers you see -- 20 plus 24 plus 8 plus 17 to get the total enumeration of this universe set. And then realize that the union of A and everything that is not B is just the universe set excluding that part of set B that is not shared with set A. Take the universe total and subtract 8 to get your answer.
John
My calculator said it, I believe it, that settles it
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