Question

Using the properties of sets and their complements prove that

(A ∪ B) – C = (A - C) ∪ (B - C)

Answer 1

(A ∪ B) – C = (A - C) ∪ (B - C) 

Let,

x ∈ [(A ∪ B) - C] 

x ∈ (A ∪ B) and x ∉ C 

(x ∈ A or ∈ B) and x ∉ C) 

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

x ∈ {(A - C) or x ∈ (B - C)} 

x ∈ {(A - C) ∪ (B - C)} 

(A ∪ B) – C ⊆ (A - C) ∪ (B - C) …(i) 

Again, 

let y ∈ [(A - C) ∪ (B - C)] 

y ∈ (A - C) or y ∈ (B - C) 

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

(y ∈ A or y ∉ B and y ∉ C 

y ∈ {(A ∪ B) and y ∉ C 

y ∈ {(A ∪ B) - C} 

(A - C) ∪ (B - C) ⊆ (A ∪ B) – C …(ii) 

From equation (i) and (ii), 

(A ∪ B) - C = (A - C) ∪ (B - C)