# Definition:Logical Complement

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## 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 $\mathbf A$.

### Complementary Pair

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

## Linguistic Note

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.

## Sources

- 2012: M. Ben-Ari:
*Mathematical Logic for Computer Science*(3rd ed.) ... (previous) ... (next): $\S 2.6.1$: Definition $2.57$