the inclusion-exclusion principle is a way to count things in the world. it doesn't work on memories. trust me i've tried
$
\left | \bigcup^n_{(i=1)} A_i \right | =\sum {i=1}^{n}|A{i}|-\sum {1\leqslant i<j\leqslant n}|A{i}\cap A_{j}|+\sum {1\leqslant i<j<k\leqslant n}|A{i}\cap A_{j}\cap A_{k}|-\cdots +(-1)^{n+1}\left|A_{1}\cap \cdots \cap A_{n}\right|.
$