\[\left|S\setminus(A_{1}\cup A_{2}\cup\cdots\cup A_{n})\right|=\left|S\right|+%
\sum_{t=1}^{n}(-1)^{t}\sum_{1\leq j_{1}<j_{2}<\cdots<j_{t}\leq n}\left|A_{j_{1%
}}\cap A_{j_{2}}\cap\cdots\cap A_{j_{t}}\right|.\]