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?
Best Answer
$C$ (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 $C$.