Let A and B be subsets of a given set X Show that A B PA PB

Let A and B be subsets of a given set X. Show that A B = P(A) P(B) = {}. Hint: Prove (=) by contradiction and (=) by contraposition.

Solution

Let us assume that  A B =

This implies prob of  A B = 0

since for null set prob =0

This implies there is no element common to A and B.

Hence P(A) and P(B) also have nothing in common