Definition:Logical Complement

Definition

The (logical) complement of a propositional formula $\mathbf A$ is the negation of $\mathbf A$, that is, $\neg \mathbf A$.

Conversely, the complement of $\neg \mathbf A$ is defined to be $\mathbf A$.

For any propositional formula $\mathbf A$, the set $\left\{{\mathbf A, \neg \mathbf A}\right\}$ is called a complementary pair of formulas.

The word complement comes from the idea of complete-ment, it being the thing needed to complete something else.

It is a common mistake to confuse the words complement and compliment. Usually the latter is mistakenly used when the former is meant.