The principle of inclusion-exclusion provides a way of methodically counting the union of possibly non-disjoint sets.

Let $C=\{A_{1},A_{2},\dots A_{N}\}$ be a finite collection of finite sets. Let $I_{k}$ represent the set of $k$-fold intersections of members of $C$ (e.g., $I_{2}$ contains all possible intersections of two sets chosen from $C$).

Then

For example:

The principle of inclusion-exclusion, combined with de Morgan’s laws, can be used to count the intersection of sets as well. Let $A$ be some universal set such that $A_{k}\subseteq A$ for each $k$, and let $\overline{A_{k}}$ represent the complement of $A_{k}$ with respect to $A$. Then we have

thereby turning the problem of finding an intersection into the problem of finding a union.