Let A, B, and C be three events in a sample space . In each case below,

Let A, B, and C be three events in a sample space Ω. In each case below, find an event X such that the union operator on the right hand side is applied between events which are disjoint (for example, in (i) we seek X such that AB and X are disjoint sets):

(i) B = (AB) ∪ X;

(ii) A ∪ B = (A′B) ∪ X;

(iii) A ∪ B ∪ C = A ∪ (A′B) ∪ X.