If A and B are two events on a sample space such that 0 < P(A), P(B) < 1, verify that each of the following conditions is equivalent to the independence of A and B:

(i) P(A|B) + P(A′) = 1;

(ii) P(B|A) + P(B′) = 1;

(iii) P(B|A′) = P(B|A);

(iv) P(A|B) = P(A|B′);

(v) P(B′|A′) = P(B′|A);

(vi) P(A′|B′) = P(A′|B).