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Consider the following for any three non empty sets A, B and C.

1. A – (B ∪ C) = (A – B) ∪ (A – C) 

2. A – B = A – (A ∩ B) 

3. A = (A ∩ B) ∪ (A – B) 

which of the above is /are correct? 

(a) Only 1 

(b) 2 and 3 

(c) 1 and 2 

(d) 1 and 3

1 Answer

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Best answer

(b) 2 and 3

1. A – (B ∪ C) = (A – B) ∪ (A – C) 

(A – B) ∪ (A – C) 

⇒ For all (x ∈ A and x ∉ B) or (x ∈ A and x ∉ C) 

⇒ For all (x ∈ A and x ∉ B and x ∉ C) 

⇒ x ∈ A – (B ∩ C ) 

Hence not correct. 

2. x ∈ (A – (A ∩ B)) ⇒ x ∈ A and x ∉ A ∩ B 

⇒ x ∈ A and x ∉ B 

⇒ x ∈ (A – B) 

3. x ∈ [(A ∩ B) ∪ (A – B)] 

⇒ (x ∈ A and x ∉ B) or (x ∈ A and x ∈ B′) 

⇒ x ∈ A and x ∈ B and x ∈ B′ 

⇒ x ∈ A and x ∈ B ∩ B′ 

⇒ x ∈ A and x ∈ ϕ 

⇒ x ∈ A ∪ ϕ 

⇒ x ∈ A 

Hence (1) is incorrect.

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