Let A and B be sets Prove that if the intersection of A and

Let A and B be sets. Prove that if the intersection of A and B is not empty, then the intersection of P(A) and P(B) is not empty.

Write your proof clearly, using complete sentences! You can use anything proved in lecture or discussion as long as you write out the full statement first.

Solution

Given a set A, the power set of A, P(A), is the set of all subsets of A. Similarly, it is for B.

Now,

For all sets A and B, A B is a subset of A and subset of B.

The statement to be proved is universal:

sets A and B, AB is a subset of A and also a subset of B.

Suppose A and B are any (particular but arbitrarily chosen) sets.

A B is a subset of A, we must show

for all x, x belongs to A B --> x belongs to A and x belongs to B.

Suppose x is any (particular but arbitrarily chosen) element in AB.

By definition of A B, x belongs to A and x belongs to B.

Hence, P(A) and P(B) is not empty.