Complement principle

Statement

Let $U$ be a finite set and let $A \subseteq U$. Let $A^c = U \setminus A$ be the complement of $A$ in $U$. Then $|A^c| = |U| - |A|$.

Remarks

The rule follows from the rule of sum, because $U$ is the disjoint union of $A$ and $A^c$. It is useful when the complement is easier to count. For example, the number of words of length $n$ over $\{0, 1\}$ that contain at least one $1$ is $2^n - 1$.

Explanations

L1 human

Sometimes it is easier to count what you do not want. To count the outcomes with at least one six in two dice rolls, count the outcomes with no six: $5 \cdot 5 = 25$. Then $36 - 25 = 11$ outcomes have at least one six.