SOLUTION: Let A and B be subsets of U with n(A)=12, n(B)=25, n(A')=67, and n(A intersect B)= 11 Find n(A U B'). I have created the Venn diagram and with two circles, where A is 12, B is

Question 450991: Let A and B be subsets of U with n(A)=12, n(B)=25, n(A')=67, and n(A intersect B)= 11 Find n(A U B').
I have created the Venn diagram and with two circles, where A is 12, B is 25 and the intersection of A and B is 11, but i'm not sure where to go from there. A' is given as 67, but I don't see how that provides me what I need to solve.
Thanks
Answer by jim_thompson5910(35256) (Show Source): You can put this solution on YOUR website!
Since

and

, this means that

which means that

So there are 79 elements in the universe U. This is a very important piece of information.

Now comes the task of creating the Venn diagram. This isn't required, but it really helps to see what's going on.

So start with drawing a rectangle. Now draw two overlapping circles within this rectangle. Label the circles as A and B and label the rectangle as U like so:

Now because

, this means that the number 11 goes in the region between the two circles (since this is where the two "intersect"). So fill in this region:

Now, we know that n(A) = 12 and we know that

. We want to know how many is in A alone and not in B. So just subtract off 11 from 12 to get 12-11 = 1. So there is 1 element in set A. So fill in the circle A with "1" like so:

Do the same for set B to get 25 - 11 = 14. So there are 14 elements that are only in B (and not in A). Fill this in to get

Now add up the numbers in the separate regions to get: 1+11+14 = 26

So there are 26 elements that belong in either set A, set B, or both. Since there are 79 elements total, this must mean that there are 79 - 26 = 53 elements that are in neither set. This number gets written outside the circles but inside the rectangle like so:

Now our Venn diagram is done. Our task is to now appropriately shade the region that pertains to

.

We do this by first shading all of set A (the first set listed) like so

then we shade

, which is everything but B, to get

Combine these shadings to get the final product of

What we get is that practically everything is shaded except the region that has a 14 in it. Now just add up the numbers that lie in the shaded regions to get: 1+11+53 = 65

So there are 65 things in the set

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