Let \(S\) be a sample space, and consider events \(A,\, B,\, C\subseteq S\). Recall that the law of inclusion/exclusion says that
\[ P(A\cup B)=P(A)+P(B)-P(A\cap B). \]
How should this be extended to unions of three events? \[ P(A\cup B\cup C)=P(A)+P(B)+P(C) +\,...{???} \] Explain your conjecture with words and pictures.
Use set theory and the probabiity axioms to prove your conjecture in the previous part.