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Question
For all sets A, B, and C
Is (A – B) ∩ (C – B) = (A ∩ C) – B? Justify your answer.
Solution
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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