22. Prove that for any two sets A and B,
(A – B) U (B - A) = (A UB) - (AUB).
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Intuitively, A−BA−B represents “the part of AA that isn't in BB", and A∩BA∩B represents “the part of AA that is in BB.” When we combine “the part of AA that is in BB" and “the part of AA that isn’t in BB,” we should just get AA. What follows is a formal proof.
The definition of A−BA−B is A∩BCA∩BC, where BCBCdenotes the complement of BB, so we have:
(A−B)∪(A∩B)(A−B)∪(A∩B)
=(A∩BC)∪(A∩B)=(A∩BC)∪(A∩B)
=A∩(BC∪B)=A∩(BC∪B) This is the distributive property
=A∩U=A∩U where UU is the “universe” of sets you are working in
= A
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