how can I prove $A - B = A ⇔ B - A = B$? set theory I tried to do this operation, but I can't.
Here's my work so far:
First we start with what must be proven:
$$ A - B = A \iff B-A = B $$
Rewrite the LHS.
$$ A - B = A \implies x \in A - B \implies x \in A \land x \not\in B 
 \\ \implies x \in A \land x \in B^c \implies x \in A \cap B^c $$
Rewrite the RHS.
$$ B - A = B \implies x \in B - A \implies x \in B \land x \in A \implies x \in B \land x \in A^c \\ \implies x \in B \land x \in A^c \implies x \in B \cap A^c $$
 A: These are roughly good steps in roughly a good order, but I think you're falling into the classic trap: you don't need (nor should you strive for) an $\implies$ symbol between each step. Concentrate on what you're trying to prove: $A - B = A \iff B - A = B$. What do we need to do in order to establish this?
As you've noticed, it's an equivalence, so we need to prove two statements: "If $A - B = A$, then $B - A = B$", and "If $B - A = B$, then $A - B = A$". So far, so good. Let's concentrate on the first one, before we concentrate on the second (as you did in your attempt).
In your attempt, you wrote, as part of your first solution, $A - B = A \implies x \in A - B$. This implication doesn't follow! What is $x$? You haven't introduced it thus far. If you are assuming that $x \in A - B$, then you need to write that, using words. It certainly doesn't automatically follow from $A - B = A$.
Also, you need to consider if that is the assumption you want to make. What are you trying to prove here? You actually need to show that $B - A = B$! You are allowed to assume that $A - B = A$; you don't have to prove it. It therefore doesn't make much sense to assume that $x \in A - B$, because we're really only interested in $B - A$ and $B$. In particular, we wish to know that, given $x \in B - A$, we can conclude $x \in B$, and vice-versa.
So, I suggest beginning with: "Suppose $x \in B - A$...". How do we know that $x \in B$? From your attempt, I think you can conclude that $x \in B$. Note that we don't (yet) need to use the assumption $A - B = A$; we always have $B - A \subseteq B$.
Next, suppose that $x \in B$. Now, how do we know that $x \in B - A$? Since you already know $x \in B$, you only need to conclude that $x \notin A$. Here we will need to use the assumption. Assume for the sake of contradiction that $x \in A$. Since $A = A - B$, this means $x \in A - B$. What does this tell you? Why does this contradict our initial assumption that $x \in B$?
To write this, you are going to need words. You will need to write "assume" or "suppose", possibly followed by the phrase "... for the sake of contradiction". You will probably need some "hence" or "therefore". (Or, of course, their Portuguese equivalents!) Don't just write $\implies$ between each step. Only write $\implies$ when the previous statement (and only the previous statement) implies the next statement. If you need to bring in other established assumptions, then use words instead.
A: Think of $\mathbf A - \mathbf B = \mathbf A$ as a proposition $p$, and $\mathbf B - \mathbf A = \mathbf B$ as a proposition $q$. To prove $p \Leftrightarrow q$, we are performing what's called a proof of equivalence. To do this, you want to show that $p \rightarrow q$ and $q \rightarrow p$ are both true.
