Answer to Question #162410 in Discrete Mathematics for Nomhle Lukashe

Question #162410

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


1
Expert's answer
2021-02-10T14:48:02-0500

(a) The question is wrong. As we saw in the Venn diagram


Here we see that "(A\\cap\\bar B)" can never be a subset of "(A\\cap B)" , as they are disjoint sets.

(b) Given that "A-B=A"

Now we know that if "x \\in (A-B)\\implies x\\in A" but "x\\notin B"

Since it is given that "A-B=A" , it only happens when "A" and "B" disjoint sets.

Now if "A" and "B" are disjoint sets then also "(B-A)=B" .

Similarly we can prove that if "(B-A)=B" that also implies "(A-B)=A"

So we can conclude that "(A-B)=A" iff "(B-A)=B."


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