The union, intersection and complement of events On the Probability chapter of a 1995 mathematical statistical book I am reviewing I have found the following exercise: 

Let A and B be arbitrary events. Let C be the event that either A occurs or B occurs, but not both. Express C in terms of A and B using any of the basic operations of union, intersection and complement.

Now, the book suggested answer is to describe the entire sample space as: 
$$\Omega=(A\cap B)^{C}\cap(A\cup B)$$
I think the correct answer is:
$$\Omega=(A\cap B)^{C}\cup(A\cap B)$$
and 
$$C=(A\cap B)^{C}$$
Where is the error that I have made?
 A: $C$ (the symmetric difference of $A$ and $B$) is obtained by overlaying (intersecting) $A\cup B$ and $(A\cap B)^c$, whence $C = (A\cup B) \cap (A\cap B)^c$:

Another expression frequently used is $C = (A\cap B^c) \cup (B\cap A^c)$.  The left-hand term is the pure red lune in the figure while the right-hand term is the pure blue lune; together, they form $C$.
A: Actually, you both got it wrong!
You're right in thinking that
$$\Omega=(A\cap B)^{C}\cup(A\cap B)$$
since it is true that, for any set $D$ in $\Omega$, $D^C \cup D=\Omega$.
However, $C$ is the part of $A\cup B$ such that only one of $A$ and $B$ occurs. In other words, you need both the event $(A\cup B)$ and the event $(A\cap B)^{C}$ to occur. Thus
$$C=(A\cap B)^{C}\cap(A\cup B)$$
which is what the book claimed was $\Omega$.
A: As MånsT has pointed out, what the book claims is $\Omega$ is actually $C$, the
event that exactly one of $A$ and $B$ occur.  An alternative expression for
$C$ is
$$C = (A\cap B^c) \cup (A^c \cap B)$$
which is the disjoint union of the events "$A$ occurs and $B$ does not"
and "$A$ does not occur and $B$ does".  Thus, we have that
$$P(C) = P(A\cap B^c) + P(A^c \cap B)$$ 
and since 
$$\begin{align*}
P(A \cup B) &= P(A\cap B^c) + (A^c \cap B) + P(A \cap B)\\
&= P(A) + P(B) - P(A\cap B)
\end{align*}$$
we can write
$$\begin{align*}P(C) &= P(A \cup B) - P(A \cap B)\\
&= P(A) + P(B) - 2P(A\cap B)
\end{align*}$$
