Prove that for any sets $A$ and $B $ with $(A - B) ∪ (B - A) = A ∪ B$ , then $A ∩ B = ∅$

Question

So this is an assignment question. I'm not sure how to do it so I started off assuming that intersection of $A$ and $B$ is not empty so if $ p$ is an element in it, then $p$ is an element of $A \cup{ B } $ too.

Is this right and if so where do I go from there?

Thanks in advance!

Answer 1

Assume $\exists x\in A \cap B$. Then $x \notin A $ \ $B $ and $x \notin B$ \ $A$ (do you see why?)

Hence $x \notin (A $ \ $B)\cup (B$ \ $A)$. However $x\in A\cup B$ hence a contradiction.

Answer 2

Yes, it's a good starting point. Now, since $p\notin A\setminus B$ and $p\notin B\setminus A$, $p\notin(A\setminus B)\cup(B\setminus A)$. This proves that $A\cup B\neq(A\setminus B)\cup(B\setminus A)$.