If $A-B = B-A$ what can we say about sets $A$ and $B$?

Question

I know this was a question that was previously asked, but what I'm asking isn't for the answer, I know what the answer is, and I know of another method of proving this. I'm wondering if my own rough work is correct because I showed them what amounted to my intuitive method of doing it to someone else and they said that the reasoning was somewhat "sketch", though admittedly it was not as structured. Below the rough work of my intended proof is outlined.

Answer 1

That's a really 'rough' proof indeed since it makes little sense to say something like:

$x \in A \cap B' = x \in A' \cap B$

because you are now putting an equality sign between two statements, rather than between two sets.

So, better would be:

$A-B=B-A$

$\Rightarrow A \cap B' = B \cap A'$

$\Rightarrow A - A \cap B = B - A \cap B$

$\Rightarrow A=B$

However, this 'proof' still suffers from a big problem. In the last step, you 'cross out' the '$-A \cap B$' on both sides, and the thinking seems to be:

If $A - X= B-X$, then we can cross out the '$-X$' part, and get to $A=B$

But note: that is not true in general!

For example, if $A=\{1\}$, $B=\{2\}$, and $X=\{1,2\}$, then $A-X=B-X=\emptyset$, but clearly $A\neq B$

So, you really can't do this kind of 'crossing out' when it comes to sets!

Or at the very least: you have to be very careful when 'crossing out', because you can only do this under certain conditions .... which, as it so happens, apply in this particular case ... but you would have needed to point that out to make the proof a proper proof.