Question

Using the properties of sets and their complements prove that

A – (B ∪ C) = (A - B) ∩ (A - C)

Answer 1

Let,

x ∈ {A − (B ∪ C)} 

x ∈ A and x ∉ (B ∪ C). 

(x ∈ A and x ∉ B) and (x ∈ A and x ∉ C)

x ∈ (A − B) x ∈ (A − C) 

x ∈ {(A − B) ∩ (A − C)} 

A− (A − B) ⊆ (A − B) ∩ (A − C) …(i) 

Again, let 

y ∈ (A − B) ∩ (A − C) 

y ∈ (A − B) and y ∈ (A − C) 

(y ∈ A and y ∉ B) and (y ∈ A and y ∉ C 

y ∈ A and y ∉ B ∪ C) 

y ∈ {A − (B − C)} 

(A − B) ∩ (A − C) ⊆ A − (B ∪ C) …(ii) 

From eqs. (i) & (ii) 

A – (B ∪ C) = (A - B) ∩ (A - C)