Simplify each of the expressions below by the use of properties among event operators:

(i) A′B′AB;

(ii) (A′ ∪ B′) ∪ AB;

(iii) (A ∪ B)(A ∪ B′)(A′ ∪ B);

(iv) (A′B′)(A ∪ B);

(v) [(A ∪ B)(A ∪ B′)] ∪ [(A′ ∪ B)(A′ ∪ B′)];

(vi) (A ∪ B ∪ C)(A ∪ B ∪ C′)(A ∪ B′);

(vii) A ∪ (B − AB) ∪ (C − AC);

(viii) (A ∪ B ∪ C)(A ∪ BC′);
(ix) [ABC(A ∪ B ∪ C′)] ∪ (A ∪ B′).