SOLUTION: Let A, B, and C be sets. Show that (A − B) − C = (A − C) − (B − C).

Question 1026273: Let A, B, and C be sets. Show that (A − B) − C =
(A − C) − (B − C).
Answer by robertb(5830) (Show Source): You can put this solution on YOUR website!
We use the fact that X - Y = X ∩ Y', where Y' = set complement of Y.
==> (A-C)-(B-C) = (A∩C')-(B∩C') = (A∩C')∩(B∩C')'
=(A∩C')∩(B'∪C'') =(A∩C')∩(B'∪C) , by using de Morgan's Law and the fact that C'' = C
=(A∩C'∩B')∪(A∩C'∩C) = (A∩C'∩B')∪� = A∩C'∩B', by the distributive property, and the fact that C'∩C = �.
= A∩B'∩C', by the commutative property.
= (A-B)∩C', by definition of A-B.
= (A-B)-C, by definition of (A-B)-C.
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SOLUTION: Let A, B, and C be sets. Show that (A − B) − C = (A − C) − (B − C).