I am asked to prove that
$$ (A\setminus B)\cup (B\setminus A)\subseteq (A\cup B)\setminus(A\cap B) $$ where $A$ and $B$ are sets. Could someone please check my solution and see if it is correct?
I suppose that sets $A$ and $B$ are subsets of a global set $X$. Fix $x\in (A\setminus B)\cup (B\setminus A)$. Then
$$ \begin{align} x\in (A\setminus B)\cup (B\setminus A)&\Rightarrow x\in A\setminus B\text{ or }x\in B\setminus A\\ &\Rightarrow (x\in A\text{ and }x\not\in B)\text{ or }(x\in B \text{ and } x\not\in A)\\ &\Rightarrow [(x\in A\text{ and }x\not\in B)\text{ or }x\in B]\text{ and } [(x\in A\text{ and }x\not\in B)\text{ or }x\not\in A]\\ &\Rightarrow [(x\in A\text{ or }x\in B) \text{ and } (x\in B\text{ or }x\not\in B)]\text{ and }\\ &\qquad [(x\in A\text{ or }x\not\in A)\text{ and } (x\not\in A\text{ or }x\not\in B)]\\ &\Rightarrow [(x\in A\cup B \text{ and } (x\in B\text{ or }x\in X\setminus B)]\text{ and }\\ &\qquad [(x\in A\text{ or }x\in X\setminus A)\text{ and } \neg(x\in A\text{ and }x\in B)]\\ &\Rightarrow [x\in A\cup B \text{ and } x\in B\cup(X\setminus B)] \text{ and }\\ &\qquad [x\in A\cup(X\setminus A)\text{ and }\neg(x\in A\cap B)]\\ &\Rightarrow [x\in A\cup B \text{ and }x\in X]\text{ and } [x\in X\text{ and }x\not\in A\cap B]\\ &\Rightarrow x\in A\cup B\text{ and } x\not\in A\cap B\\ &\Rightarrow x\in(A\cup B)\setminus(A\cap B) \end{align} $$