Q. For any two sets A and B, what does A – (A – B) equal?

Explanation:

Set difference A – C = {x | x ∈ A and x ∉ C}. Let C = A – B = {x | x ∈ A and x ∉ B}.
Consider x ∈ A – (A – B):
– Then x ∈ A and x ∉ (A – B).
– x ∉ (A – B) means not(x ∈ A and x ∉ B) ⇒ x ∉ A or x ∈ B.
– Together with x ∈ A, this forces x ∈ B.
– Hence x ∈ A and x ∈ B ⇒ x ∈ A ∩ B.
Conversely, if x ∈ A ∩ B, then x ∈ A and x ∈ B, so x ∉ (A – B), hence x ∈ A – (A – B).
Therefore, A – (A – B) = A ∩ B.