Prove $B = A - (A - B)$

Question

To Prove: $B = A -(A-B)$

suppose $ A,B $ are sets. Let $x \in A - (A-B)$.

$$x \in A - (A \cap B^C)$$

$$x \in A \cap (A \cap B^C)^C$$

$$x \in A \cap (A^C \cup B) $$

$$ x \in (A \cap A^C) \cup (A \cap B)$$

$$ x \in \emptyset \cup (A \cap B)$$

$$ x \in (A \cap B)$$

In particular, $ x \in B$

Then, is it true, because $x$ is an element of $B$(in particular), or does it require $A = B$, since $x \in A \cap B $

Also, do I have to prove it the other way since there is an equal sign? Thanks!

Answer 1

The statement is false. Take $A=\{1\}$, $B=\{2\}$. Then $A-B=\{1\}$, and $A-(A-B)=\emptyset\neq B$.

Answer 2

The statement $$B = A -(A-B)$$ is not necessarily true.

Let$A=\text {{1,2,3}}$ and $B=\text {{4}}.$

Then $A-B= A.$

Thus $ A-(A-B)$ = $\phi \ne B $