Solution
= Solution
Take the finite ambient <set> $S=\bigcup_iA_i$ and the additive <set function> $f(A)=|A|$. Disjoint subsets have additive <cardinality>, so (a) applies and gives
$$
\boxed{\left|\bigcup_{i=1}^nA_i\right|=\sum_{\varnothing\ne J\subseteq\{1,\ldots,n\}}(-1)^{|J|+1}\left|\bigcap_{j\in J}A_j\right|}.
$$
This is the finite <inclusion-exclusion principle>.