I have a finite sequence of (not necessarily disjoint) sets $A_1, A_2, \cdots , A_n$. I have another finite sequence of sets $E_1, E_2, \cdots , E_n$ such that $E1, E2, \cdots, En$ are disjoint, and, for all $k : 1 ≤ k ≤ n$.
So essentially: $A_1 \cup A_2 \cup\cdots \cup A_k = E_1 \cup E_2 \cup\cdots\cup E_k$
So I can say:
$$\sum_i |A_i|- \sum_{i < j} |A_i \cap A_j|+ \sum_{i < j < k} |A_i \cap A_j \cap A_k|+ (-1)^{n-1} |A_1 \cap A_2 \cap\cdots \cap An| \\= |E_1| + |E_2| + |E_3| +\cdots+ |E_n|$$
Now my question is how can I represent $E_i$ in terms of $A_1 A_2 \cdots$? Am I headed in the right direction?