Question

For all sets A, B and C 

Is (A – B) ∩ (C – B) = (A ∩ C) – B? Justify your answer.

Answer 1

Yes 

Let x ∈ (A – B) ∩ (C – B) 

⇒ x ∈ A – B and x ∈ C – B 

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

⇒ (x ∈ A and x ∈ C) and x ∉ B 

⇒ (x ∈ A ∩ C) and x ∉ B 

⇒ x ∈ (A ∩ C) – B 

So (A – B) ∩ (C – B) ⊂ (A ∩ C) – B ... (1)

Now, conversely

Let y ∈ (A ∩ C) – B 

⇒ y ∈ (A ∩ C) and y ∉ B 

⇒ (y ∈ A and y ∈ C) and (y ∉ B) 

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

⇒ y ∈ (A – B) and y ∈ (C – B) 

⇒ y ∈ (A – B) ∩ (C – B) 

So (A ∩ C) – B ⊂ (A – B) ∩ (C – B) ... (2) 

From (1) and (2), (A – B) ∩ (C – B) = (A ∩ C) – B