let A and B be subsets of U with n(A)=62, n(B)=37, n(A')=17 and n(AnB)=27. find n(AUB')

Question

finding the subset of A union B prime from the given data.

Answer 1

First of all, since n(A)=62 and n(A')=17, you know that n(U) = 62 + 17 = 79

 

Now, one way to answer questions like this is to draw a Venn diagram and identify the sets you're interested in.  If you do this, you can see that the only elements of U that are not in A∪B' are the ones that are in B but not in A -- in other words, they're ONLY in B.  That's like saying:  (A∪B')' = B∩A', which is a true statement about sets.

 

So there are 37 elements in B and 27 elements in A and B, so that leaves 10 elements that are only in B.  All the rest of the elements of U are in A or B'.  Since there are 79 elements in U, there must be 79 - 10 = 69 that are in A or B' -- so the answer is 69.

 

You could do this symbolically like this:

 

n(A∪B') = N(U) - n(A∪B')' = N(U) - n(A'∩B) = N(U) - [ N(B) - N(A∩B) ] = 79 - (37 - 27) = 79 - 10 = 69