Expert: Josh - 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
Hi Annie,

The cardinality of NOT(UNION(A,B)) is equal to #U-(#A+#B)+ #INTERSECTION(A,B).
So, #[(A u B)'] = u-(a+b)+x.
If the cardinality of set A (denoted "a") is smaller than the cardinality of set B, then the number of elements in their intersection is dictated by the size of A. It (the size of "A n B") cannot be larger than "a".
(In other words, "A and B" cannot have more things in common than what is already contained in set A, which must be smaller than B)

That is, [A n B] is a proper subset of A,
which means, #[(A n B)] < #A,
by definition, x < a.
So, y = u-(a+b)+x
     < u-(a+b)+a
     = u-b
Cheers.