Expert: Ahmed Salami - 4/17/2009

Question
Hi Ahmed, Can you help me figure this out?  I need to prove or disprove the following?
(A intersection B) - (A intersection C) is a subset of A intersection (B - C).  I come up with no it is not a subset.  Is this correct?  I go as far as using De Morgan's Law and then I have conjuntion and disjuntion and I am stuck there.  I want to believe that I am stuck because there is nothing else to try.
Thanks,
Evelyn

Answer
Hi Evelyn,
I'll rewrite both statements as
(A n B) - (A n C)
and
A n (B-C)
Now, to define set difference
X - Y = X n Y'            where Y' means the complement of Y
Going back to the first statement,
(A n B) - (A n C) = (A n B) n (A n C)'
By De Morgan's Law, (A n C)' = A' U C'
and so we have
(A n B) - (A n C) = (A n B) n (A' U C')
                 = [(A n B) n A'] U [(A n B) n C']
                 = [(A n A') n B] U [A n B n C']
                 = [0 n B] U [A n B n C']
Note: A n A' is a null set and i've represented that by 0
                 = 0 U [A n B n C']
                 = A n B n C'
Considering the second statement,
A n (B-C) = A n (B n C')
         = A n B n C'
This shows that both sets are in fact equal and subsequently subsets of each other.

I hope it helps.

Regards