 Solution

For finite sets $A_1,\dots ,A_r$, the inclusion-exclusion principle statesTo prove it, count the contribution of one element. If it belongs to exactly $q\ge 1$ of the sets, it is counted by $\left(\genfrac{}{}{0ex}{}{q}{k}\right)$ intersections of size $k$. Its total weight isusing the binomial theorem. An element belonging to none contributes zero. Summing these weights gives the cardinality of the union and proves the inclusion-exclusion principle. Equivalently, within a finite ambient set $U$, the number avoiding every $A_i$ is the sum over all $J$, with sign $(-1{)}^{|J|}$ and the empty intersection interpreted as $U$.