Solution to If A is a set, let A¯ denote the complement of A. Show the following … - Sikademy
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Archangel Macsika

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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Here's the Solution to this Question

(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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Question ID: mtid-5-stid-8-sqid-3374-qpid-2073