If A is a set, let A¯ denote the complement of A. Show the following (a) (A ∩ B¯) ⊂ A ∩ B (b) A − B = A if and only if B − A = B
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(a) The question is wrong. As we saw in the Venn diagram
Here we see that can never be a subset of , as they are disjoint sets.
(b) Given that
Now we know that if but
Since it is given that , it only happens when and disjoint sets.
Now if and are disjoint sets then also .
Similarly we can prove that if that also implies
So we can conclude that iff