Thread: need help!!!

If we have two sets, A and B, contained in the universal set
Can someone tell me why the smallest possible number of elements of
(A intersect B) occurs when A union B = universal set???

i know that
n(A intersect B) = n(A) + n(B) - n(A union B)
so to make it smaller, we need to make n(A union B) as big as possible.

But is there a way to explain this without using the formula above???

Originally Posted by acc100jt

If we have two sets, A and B, contained in the universal set
Can someone tell me why the smallest possible number of elements of
(A intersect B) occurs when A union B = universal set???

i know that
n(A intersect B) = n(A) + n(B) - n(A union B)
so to make it smaller, we need to make n(A union B) as big as possible.

But is there a way to explain this without using the formula above???

I'm not sure I understand the question. for instance let D={0,1,2,3,4} be
our universe of discourse (universal set), and let A={0,1}, B={2,3}, then
|A Intersect B|=0, but A Union B != D.

Similarly let A={0,1,2}, B={2,3,4}, then |A Intersect B|=1, and A Union B =D.

RonL

sorry, let me restate my questions,

If the sets A and B are not disjoint.
n(universal set)=60
n(A)=32
n(B)=34
then the least n(A intersect B) occurs when (A union B)=universal set.
WHY??

Originally Posted by acc100jt

sorry, let me restate my questions,

If the sets A and B are not disjoint.
n(universal set)=60
n(A)=32
n(B)=34
then the least n(A intersect B) occurs when (A union B)=universal set.
WHY??

Because n(A)+n(B)>n(U), there must be some elements shared between
A and B. The smallest number that could be shared is 6 (that is:
n(A)+n(B)-n(U), any smaller number of shared elements will leave n(A Union
B)>60, which would be a contradiction).

But if A and B share exactly 6 elements then n(A Union B)=60, and so
A Union B=U.

RonL

RonL