---
tag: SX87GXZBGG
kind: statement
role: proposition
name: Nonmonotone laws
sources:
  - wikipedia: en:54476844@r1368725040
    locator: Boolean algebra, section Laws > Nonmonotone laws
    note: adopted
    mention: wikipedia://en/54476844#SYNI
    sections: [Statement]
inventory: [V056JPD4BA]
---

## Statement

The complement operation is defined by the following two laws.

$$\begin{aligned} &\text{Complementation 1} & x \wedge \neg x & = 0 \\ &\text{Complementation 2} & x \vee \neg x & = 1 \end{aligned}$$

All properties of negation including the laws below follow from the above two laws alone.

In both ordinary and Boolean algebra, negation works by exchanging pairs of elements, hence in both algebras it satisfies the double negation law (also called involution law)

$$\begin{aligned} &\text{Double negation} & \neg{(\neg{x})} & = x \end{aligned}$$

But whereas *ordinary algebra* satisfies the two laws

$$\begin{aligned} (-x)(-y) & = xy \\ (-x) + (-y) & = -(x + y) \end{aligned}$$

Boolean algebra satisfies [[De Morgan's laws|09PKJKA6J0]]:

$$\begin{aligned} &\text{De Morgan 1} & \neg x \wedge \neg y & = \neg{(x \vee y)} \\ &\text{De Morgan 2} & \neg x \vee \neg y & = \neg{(x \wedge y)} \end{aligned}$$