In this article, how do they get the complementary form of
$$\Big|S\setminus\bigcup_{i=1}^{n}A_i\Big|=\sum_{k=0}^{n}(-1)^k\binom{n}{k}\alpha_k$$
from $$\Big|\bigcup_{i=1}^{n}A_i\Big|=\sum_{k=1}^{n}(-1)^{k-1}\binom{n}{k}\alpha_k$$
The thing that confuses me is that we don't know what is $|S|$, do we even need to know it?
Could someone please clarify the steps or reasoning needed to get that complementary form?
It is not explicitly written in the article the steps that they took.
Many thanks for the help!
$$\left|S\setminus\bigcup_{i=1}^{n}A_i\right|=|S|-\left|\bigcup_{i=1}^{n}A_i\right|=\alpha_0-\sum_{k=1}^{n}(-1)^{k-1}\binom{n}{k}\alpha_k=\sum_{k=0}^{n}(-1)^k\binom{n}{k}\alpha_k\;.$$