Question

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

Answer 1

(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 $(B-A)=B.$