This post is regarding Question 3.7.3 of Velleman's How to Prove It. It is:
Suppose A and B are sets. What can you prove about P(A\B)
(P(A)\ P(B))? (No, it’s not equal to ∅. Try some ex am ples
and see what you get.)
The following is a proof that it is equal to { {} }.
What can be said about $P (A \setminus B) \setminus (P (A) \setminus P (B))$?
But, as the question suggested, I tried an example (see below) and it clearly isn't the same. Can someone point out where I went wrong.
Suppose A = {1,2} and B={2,3}.
A\B= {1,3} and P(A/B)= { {}, {1}, {3}, {1,3} }
Then P(A)={ {}, {1}, {2}, {1,2} } and P(B)={ {}, {2}, {3}, {2,3} }.
We have P(A)\P(B)= { {1}, {1,2} }
So, P(A\B)(P(A)\ P(B))= { {}, {3}, {1,3} }