How to show that symmetric difference operation is binary operation of a power set.

Question

Let $A,B \subseteq S$. That means $x\in A\Rightarrow x\in S$ and $x\in B\Rightarrow x\in S$. We have to show that $A\Delta B\subseteq S$.

Suppose $x\in A\Delta B$. That implies $x\in A \wedge x\notin B$ or $x\in B\wedge x\notin A$.

I have my confusion here, can I imply from $x\in A\Rightarrow x\in S$ that $x\in A \wedge x\notin B\Rightarrow x\in S$ and similar for the next statement to eventually conclude $x\in A\Delta B\Rightarrow x \in S$

Answer 1

Yes.

if $x \in A \Delta B$, then we have $x \in A \land x \notin B$ or $x \in B \land x \notin B$.

Suppose it is the first case, that is $x \in A \land x \notin B$, then we can conclude that $x \in A$ and hence $x \in S$. Similary for the second case.