SOLUTION: Illustrate a Venn Diagram using this information to fill in the number of elements for each region. n(A union B) = 17, n(A intersect B) = 3, n(A) = 8, (A' union B') = 21.

First draw a big rectangle for the universal set:
 

 
Next draw a circle to and label it E:
 

 
Next draw a circle overlapping the first circle and
label it B. 
 

 
The overlapping part is the set "A ᑎ B".
We are given n(A ᑎ B) = 3 , so we write "3" in the 
region that's shaped like this "()", the overlapping 
part opf the two circles: 
 

 
Now since n(A) = 8, and the number of elements in
the overlapping part, shaped like thisa "()"
is 3, the probability of being in the rest of circle
A is 8 - 3 or 5, so we write 5 in the left part
of circle A, so that the total number of elements of 
in the two parts of circle A is 8.
 
 
 
We are given that

n(A U B) = 17

The union consists of all elements in the circles.
We already have the number of elements in 2 of the
three regions so there must be 9 in the right part
of circle B so that there will be 17 inside the
circles.


 
Any remaining elements are located outside the two
circles, yet inside the rectangle.

We are given that n(A' U B') = 21
 
By DeMorgan's law, A' U B' = (A ᑎ B)'



This means that all the elements in all the regions other
that the 3 in the intersection must total 21. 

Other than the 3, we have so far 5 and 9 or 14 elements besides
the 3 in the intersection so there must be 21-14 or 7 elements 
outside the two circles.  So the final Venn diagram is:




 
Edwin