TITLE: What is the inclusion-exclusion principle for 4 sets?
QUESTION [9 upvotes]: Proofs class homework question - It doesn't ask for us to prove, derive, or even illustrate the inclusion/exclusion principle - Just to jot it down.
We're learning about sets and inclusivity/exclusivity (evidently)
I've got the inclusion/exclusion principle for three sets down to 2 sets. I'm sort a bit confused as to how I'd go about getting 4. Is there a systematic/elementary manner to prove it or is this more to do with general knowledge and research?

REPLY [24 votes]: $$
\begin{align}
&|A\cup B\cup C\cup D|\\[3pt]
&=|A|+|B|+|C|+|D|\Big\}\text{ all singletons}\\
&-(|A\cap B|+|A\cap C|+|A\cap D|+|B\cap C|+|B\cap D|+|C\cap D|)\Big\}\text{ all pairs}\\
&+(|A\cap B\cap C|+|A\cap B\cap D|+|A\cap C\cap D|+|B\cap C\cap D|)\Big\}\text{ all triples}\\
&-|A\cap B\cap C\cap D|\Big\}\text{ all quadruples}\\
\end{align}
$$
This is an instance of a special case of the Generalized Inclusion-Exclusion Principle.