Question 459087
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

<pre><font face = "consolas" size = 2 color="indigo"><b>
First draw a big rectangle for the universal set U:
 
{{{drawing(300,300,-4,4,-3,6, locate(-4,4.5,U),

rectangle(-4,-1.2,4,4) )}}}
 
Next draw a circle and label it A:
 
{{{drawing(300,300,-4,4,-3,6,
rectangle(-4,-1.2,4,4), locate(-4,4.5,U),
 locate(-3.5,2.5,A),
circle(-sqrt(2),sqrt(2),2) )}}}
 
Next draw a circle overlapping the first circle and
label it B. 
 
{{{drawing(300,300,-4,4,-3,6,
rectangle(-4,-1.2,4,4), locate(-4,4.5,U),

 locate(-3.5,2.5,A),
circle(-sqrt(2),sqrt(2),2),locate(3.5,2.5,B),
circle(sqrt(2),sqrt(2),2)
 )}}}
 
The overlapping part is the set A &#8745; 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: "()".

{{{drawing(300,300,-4,4,-3,6,
rectangle(-4,-1.2,4,4), locate(-4,4.5,U),

 locate(-3.5,2.5,A), locate(-.1,1.8,2),
circle(-sqrt(2),sqrt(2),2),locate(3.5,2.5,B),
circle(sqrt(2),sqrt(2),2)
 )}}}
 
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 &#8745; 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    

{{{drawing(300,300,-4,4,-3,6,locate(-4,4.5,U),

rectangle(-4,-1.2,4,4), locate(-1.7,1.8,a),

 locate(-3.5,2.5,A), locate(-.1,1.8,2),
circle(-sqrt(2),sqrt(2),2),locate(3.5,2.5,B),
circle(sqrt(2),sqrt(2),2)
 )}}}

We are told that A &#8746; 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 &#8746; 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 &#8746; B, which are "c", "d", and "4", over there, like 
this:

{{{drawing(300,300,-4,4,-3,6, locate(-4,4.5,U),

rectangle(-4,-1.2,4,4), locate(-1.7,1.8,a),
locate(1,1.7,c), locate(1.5,1.7,d),locate(2,1.7,4),
 locate(-3.5,2.5,A), locate(-.1,1.8,2),
circle(-sqrt(2),sqrt(2),2),locate(3.5,2.5,B),
circle(sqrt(2),sqrt(2),2)
 )}}}

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: 

{{{drawing(300,300,-4,4,-3,6, locate(-4,4.5,U),
locate(-3.7,-.5,b), locate(-3.2,-.5,1),
locate(-2.7,-.5,3),
rectangle(-4,-1.2,4,4), locate(-1.7,1.8,a),
locate(1,1.7,c), locate(1.5,1.7,d),locate(2,1.7,4),
 locate(-3.5,2.5,A), locate(-.1,1.8,2),
circle(-sqrt(2),sqrt(2),2),locate(3.5,2.5,B),
circle(sqrt(2),sqrt(2),2)
 )}}}

Edwin</pre>