Proof and problem solving - set theory

Question

Prove that $(A\Delta C)\backslash B = (A\backslash (B \cup C))\cup (C\backslash (A\cup B))$.

I tried with an $x$ that can be in $(A\Delta C)\backslash B$, so $x$ is in $A \Delta C$ but not in $B$.
If $x$ is in $A \Delta C$, it can be in $A$ or $C$. If it's in $A$, it's not in $B \cup C$, that means $x$ is in $A\backslash (B\cup C)$. I use the same idea and I can prove it's in $C\backslash A\cup B$, but I just don't know how to get the union of those two solutions.
The opposite implication also needs to be proved.

Thank you!

Answer 1

Remark that for any two sets $X,Y$, if $x \in X$, then $x \in X \cup Y$. Using this, you can conclude that in both of your cases, the element is in $(A\backslash (B \cup C))\cup (C\backslash (A\cup B))$.

Answer 2

$$(A\Delta C)\backslash B = $$

$$((A \cap C^C) \cup (A^C \cap C)) \setminus B =$$

$$((A \cap C^C) \cup (A^C \cap C)) \cap B^C =$$

$$(A \cap C^C \cap B^C) \cup (A^C \cap C \cap B^C) =$$

$$(A \cap B^C \cap C^C) \cup (C \cap A^C \cap B^C) =$$

$$(A \cap (B \cup C)^C) \cup (C \cap (A \cap B)^C) =$$

$$(A\backslash (B \cup C))\cup (C\backslash (A\cup B))$$