Let A and B be two events on a sample space. Then, show the following:

(i) If P(A) = 0, this implies that P(AB) = 0.

(ii) If P(A) = 1, then we have P(A ∪ B) = 1; use this to establish that P(AB) = P(B).

Use the above results to prove that if P(A) = 0 or P(A) = 1, then the event A is independent of any other event B on the same sample space.