I'm proving that $P(X)$ (the set of the subsets of $X$) is a ring with the following operations: If $A, B \subset X$, then $A+B := (A \cup B) \backslash (A \cap B) $ and $A \cdot B = A \cap B $.
I have proven addition's associativity $ (A+B)+C=A+(B+C)$ by showing that
$$ ( (A \cup B) \backslash (A \cap B) ) \cup C \backslash ( (A \cup B) \backslash (A \cap B)) \cap C \\ = A \cup ((B \cup C ) \backslash (B \cap C)) \backslash A \cap ((B \cup C ) \backslash (B \cap C)) . $$
I wanted to know if there is an alternative way to prove this.