SOLUTION: Let A and B be subsets of U={a,b,c,d,1,2,3,4} A={a,2} A ∩ B = {2} and A ∪ B = {a,c,d,2,4}. Find B

First draw a big rectangle for the universal set U:
 

 
Next draw a circle and label it A:
 

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

 
The overlapping part is the set A ∩ B which is given
to be {2}.
So we put "2" in the overlapping part, the sort 
of football-shaped area in the middle, shaped sort 
of like this: "()".


 
Now we are told that A={a,2}. A is the entire 
left circle.  Since we already have the "2" as
the only element of A ∩ B, then the "a" must go
in the leftmost part of circle A, which is sort of a 
moon-shaped region part of the circle    



We are told that A ∪ B = {a,c,d,2,4} which is the
whole figure 8 part, consisting of both circles
together, and "2" is the only element in the "()" part,
and a is the only element in the left "moon" part.
so all the rest of the elements of A ∪ B must go in the 
right part of B, the moon-shaped part of the 
circle on the far right, so we will put the other elements 
of A ∪ B, which are "c", "d", and "4", over there, like 
this:



We can stop there because we have placed all the
elements in both sets A and B.  B consists of
all the elements in the right circle, so the
answer is B = {c,d,2,4}

However let's finish the Venn diagram. All the rest
of the elements "b","1",and "3" of the universal 
set U go outside of both circles, so we will put
them outside the circles: 



Edwin