How can I write this in a different ways? $\vert(\bigcup_{i=0}^n Ai)\vert$

Question

I'm trying to write this equation in a more compressed way. For me, I think it would contribute a lot to my practice to see different ways of writing the same identity. How can I write this in a different ways?

$$\vert(\bigcup_{i=1}^n Ai)\vert = \sum_{i=1}^n \vert(Ai)\vert - \sum_{i=1\,\,i<j=2}^n \vert(Ai \cap Aj)\vert + \sum_{i=1\,i<j=2\,j<l=3}^n \vert(Ai \cap Aj \cap Al)\vert - ... + (-1)^{n-1} \vert(\bigcap_{i=0}^n Ai)\vert$$

Thanks!

Answer 1

Two different ways: $$ \left\lvert \bigcup_{i = 0}^n A_i \right\rvert = \sum_{\emptyset\ne S \subseteq\{0,\dots,n\}} (-1)^{|S|-1} \left\lvert \bigcap_{j \in S} A_j \right\rvert = \sum_{k = 1}^{n + 1} (-1)^{k-1} \sum_{0\le j_1 < \dots < j_k \le n} |A_{j_1} \cap \dots \cap A_{j_k}|. $$