Expert: Abe Mantell - 1/10/2005

Question
U is the universal set and A and B are two subsets of U.
#U=u, #A=a, #B=b, #(A n B)=x, #[(A u B)']=y.
Represent this info on a Venn Diagram and hence express y in terms of u, a, b and x.
Show that if a < b, then the maximum possible value of y is u - b.

I can express y in terms of u, a, b and x.
Here is what i think it is.
y = u-(a-x)-x-(b-x)
So y = u-a-b+x

But I cant see how y = u - b is  is the largest possible value for y.
Can you explain how this is the case?

thanks,
Annie

Answer
Hello Annie,

Well, it is tough for me to give a Venn diagram here!!!
But I'll try to explain it anyhow...
[I'll use 'U' for union, and 'n' for intersection)
x is how many items A and B have in common
(i.e. in the intersection)...
y is how many items are strictly outside of A and B.

So, #(AUB)=a+b-x
thus, u=a+b-x+y...now solve for y:
y=u-b-a+x -- which is what you obtained.
or y=u-b-(a-x)
but, x cannot be larger than a!!!
So, the smallest a-x could be is 0...making
the maximum y-value u-b-0=u-b.

OK?

TTYL, Abe