Let there be a collection of sets $\{A_{1},...,A_{m}\}$, where $A_{i}\subseteq\{1,2,3,...,n\}$ for every $i$.
Also let $\beta(J)=\begin{cases} 1 &\text{when } |\bigcup_{i\in J}A_{i}|\ge s\\0 &\text{otherwise } \end{cases}$
Where $s$ is some natural number.
Find a better formula for the following expression $$\sum_{i=1}^{m}(-1)^{i-1}\sum_{I\subseteq [m], |I|=i}\beta(I)$$
Thank you in advance