Principle of Inclusion-Exclusion

$$\begin{array}{rl}|A_1\cup A_2\cup \cdots \cup A_n|& =\\ \sum _{i=1}^n|A_i|& -\\ \sum _{1\le i<j\le n}|A_i\cap A_j|& +\\ \sum _{1\le i<j<k\le n}|A_i\cap A_j\cap A_k|& -\\ \dots & \\ & \\ +(-1{)}^{n+1}|A_1\cap A_2\cap \cdots \cap A_n|& \end{array}$$

For 2 sets, for example $|A\cup B|=|A|+|B|-|A\cap B|$

For 3 sets, for example $\mid A\cup B\cup C\mid =\mid A\mid +\mid B\mid +\mid C\mid -\mid A\cap B\mid -\mid B\cap C\mid -\mid C\cap A\mid +\mid A\cap B\cap C\mid$ Note that we add $|A\cap B\cap C|$