For all sets A and B, A − (A − B) = A ∩ B
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Intuitively, A−B represents “the part of A that isn't in B ", and A∩B represents “the part of A that is in B .” When we combine “the part of A that is in B " and “the part of A that isn’t in B ,” we should just get A . What follows is a formal proof.
The definition of A−B is A∩BC , where BC denotes the complement of B , so we have:
(A−B)∪(A∩B)
=(A∩BC)∪(A∩B)
=A∩(BC∪B) This is the distributive property
=A∩U where U is the “universe” of sets you are working in
=A
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