Let \(A\) and \(B\) be events in a sample space \(S\). Let \(C\) be the set of outcomes that are in either \(A\) or \(B\), but not both.

1. Draw a well-labeled picture of \(S\), \(A\), \(B\), and \(C\).

2. Write down a formula for \(C\) in terms of \(A\) and \(B\) using any of the basic operations: union (\(\cup\)), intersection (\(\cap\)), complement (\(^c\)).

3. Use set theory and the probability axioms to show that

\[ P(C)=P(A)+P(B)-2P(A\cap B). \]

4. Explain this result conceptually (with words and pictures).