Is this an Integral Domain? 
Let X be a non-empty set, and set $P(X)$ the set of all subsets of X
  with addition and multiplication:
$A+B = (A \cup B) \setminus (A \cap B) $ and $A \cdot B = A \cap B $

I am just wondering if I have thought about this correctly - I have that this a commutative, unital ring - but NOT an integral domain.
Checking the ring axioms, I have that
(i) Commutativity of addition is straightforward, clearly addition is closed.
(ii) Checking $(A+B)+C = A+ (B+C)$ I have that:
$$ (A+B)+C = [[(A \cup B) \setminus (A \cap B)] \cup C] \setminus [[(A \cup B) \setminus (A \cap B)] \cap C]$$
Thus if $x \in (A+B)+C$ then there are 3 cases: either (1) $x \in A \setminus (B \cup C)$, (2) $x \in B \setminus (A \cup C)$, (3) $x \in C \setminus (A \cup B)$.
Then,
$$ A+(B+C) = [A \cup [[(B \cup C) \setminus (B \cap C)]] \setminus [[A \cap [(B \cup C) \setminus (B \cap C)]]$$
and checking if $x \in A+(B+C)$ yields the same 3 cases as above. Would this be correct?
(iii)I take the empty set as the zero element: $A + \phi = (A \cup \phi ) \setminus (A \cap \phi ) = A$
(iv) I take each set A as it's own inverse $A + A = (A \cup A ) \setminus (A \cap A ) = \phi$
(v) Multiplication is closed
(vi) Multiplication is associative
(vii) Checking $A \cdot (B+C) = A \cdot B + A \cdot C$ I have: 
$$A \cdot (B+C) = A \cap [(B \cup C) \setminus (B \cap C)] $$ So there are two cases: (i) $ x \in A \cap B$ and $x \notin C$, or (ii) $ x \in A \cap C$ and $x \notin B$
On the otherhand: 
$$ A \cdot B + A\cdot C = [(A \cap B) \cup (A \cap C)] \setminus [(A \cap B) \cap (A \cap C)]$$ yielding the same two cases as above. So it is indeed a ring. 
The ring is clearly commutative under addition, and I take the identity as the set $X$ itself, since $A \cap X = A$. However, $A \cap B = \phi$ whenever $A$ and $B$ do not intersect. So it is not an integral domain. 
Have I thought about this correctly? Thanks in advance!

you may think of the power set of $X$ as a space $2^X$ of functions  from $X$ to the two-element field $\mathbb{F}_2$. since the codomain is a field there is a natural way of defining multiplication and addition as pointwise operations in $\mathbb{F}_2$. each set may be identified with its characteristic function. the associative law for addition, for example, or the distributive law, are a consequence of the same laws in $\mathbb{F}_2$
