The Principle of Inclusion and Exclusion

In the setting of Finite Combinatorics: Notation, let $X$ be a finite set, $m\in\mathbb{N}_{0}$ and $A_{1},\dots,A_{m}\subseteq X$, and let $\bigcup_{i=1}^{m}A_{i}$ be the union of the family $(A_{i})_{i\in[m]}$. Let $A_{\emptyset}=X$, and for nonempty $S\subseteq[m]$ let

$$A_{S}=\{x\in X:x\in A_{i}\text{ for every }i\in S\}.$$

(i) Elements in none of the sets.

$$\#\Big(X\setminus\bigcup_{i=1}^{m}A_{i}\Big)=\sum_{S\in\mathcal{P}([m])}(-1)^{\#S}\,\#A_{S}.$$

(ii) Elements in at least one of the sets.

$$\#\bigcup_{i=1}^{m}A_{i}=\sum_{S\in\mathcal{P}([m])\setminus\{\emptyset\}}(-1)^{\#S+1}\,\#A_{S}.$$