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For all sets A, B and C

Is (A – B) ∩ (C – B) = (A ∩ C) – B?

Justify your answer.

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Best answer

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

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