# Definition:Boolean Algebra

## Definition

### Definition 1

A Boolean algebra is an algebraic system $\left({S, \vee, \wedge, \neg}\right)$, where $\vee$ and $\wedge$ are binary, and $\neg$ is a unary operation.

Furthermore, these operations are required to satisfy the following axioms:

 $(BA_1 \ 0)$ $:$ $S$ is closed under $\vee$, $\wedge$ and $\neg$ $(BA_1 \ 1)$ $:$ Both $\vee$ and $\wedge$ are commutative $(BA_1 \ 2)$ $:$ Both $\vee$ and $\wedge$ distribute over the other $(BA_1 \ 3)$ $:$ Both $\vee$ and $\wedge$ have identities $\bot$ and $\top$ respectively $(BA_1 \ 4)$ $:$ $\forall a \in S: a \vee \neg a = \top, a \wedge \neg a = \bot$

### Definition 2

A Boolean algebra is an algebraic system $\struct {S, \vee, \wedge, \neg}$, where $\vee$ and $\wedge$ are binary, and $\neg$ is a unary operation.

Furthermore, these operations are required to satisfy the following axioms:

 $(BA_2 \ 0)$ $:$ Closure: $\displaystyle \forall a, b \in S:$ $\displaystyle a \vee b \in S$ $\displaystyle a \wedge b \in S$ $\displaystyle \neg a \in S$ $(BA_2 \ 1)$ $:$ Commutativity: $\displaystyle \forall a, b \in S:$ $\displaystyle a \vee b = b \vee a$ $\displaystyle a \wedge b = b \wedge a$ $(BA_2 \ 2)$ $:$ Associativity: $\displaystyle \forall a, b, c \in S:$ $\displaystyle a \vee \paren {b \vee c} = \paren {a \vee b} \vee c$ $\displaystyle a \wedge \paren {b \wedge c} = \paren {a \wedge b} \wedge c$ $(BA_2 \ 3)$ $:$ Absorption Laws: $\displaystyle \forall a, b \in S:$ $\displaystyle \paren {a \wedge b} \vee b = b$ $\displaystyle \paren {a \vee b} \wedge b = b$ $(BA_2 \ 4)$ $:$ Distributivity: $\displaystyle \forall a, b, c \in S:$ $\displaystyle a \wedge \paren {b \vee c} = \paren {a \wedge b} \vee \paren {a \wedge c}$ $\displaystyle a \vee \paren {b \wedge c} = \paren {a \vee b} \wedge \paren {a \vee c}$ $(BA_2 \ 5)$ $:$ Identity Elements: $\displaystyle \forall a, b \in S:$ $\displaystyle \paren {a \wedge \neg a} \vee b = b$ $\displaystyle \paren {a \vee \neg a} \wedge b = b$

### Definition 3

A Boolean algebra is an algebraic structure $\left({S, \vee, \wedge}\right)$ such that:

 $(BA \ 0)$ $:$ $S$ is closed under both $\vee$ and $\wedge$ $(BA \ 1)$ $:$ Both $\vee$ and $\wedge$ are commutative $(BA \ 2)$ $:$ Both $\vee$ and $\wedge$ distribute over the other $(BA \ 3)$ $:$ Both $\vee$ and $\wedge$ have identities $\bot$ and $\top$ respectively $(BA \ 4)$ $:$ $\forall a \in S: \exists \neg a \in S: a \vee \neg a = \top, a \wedge \neg a = \bot$

The operations $\vee$ and $\wedge$ are called join and meet, respectively.

The identities $\bot$ and $\top$ are called bottom and top, respectively.

Also, $\neg a$ is called the complement of $a$.

The operation $\neg$ is called complementation.

## Equivalence of Definitions

These definitions are shown to be equivalent in Equivalence of Definitions of Boolean Algebra.

## Also defined as

Some sources define a Boolean algebra to be what on $\mathsf{Pr} \infty \mathsf{fWiki}$ is called a Boolean lattice.

Some sources define a Boolean algebra to be a set with two elements (typically $\set {0, 1}$ together with the two operations addition $+$ and multiplication $\times$ defined as follows:

$\begin{array}{c|cc} + & 0 & 1 \\ \hline 0 & 0 & 1 \\ 1 & 1 & 0 \\ \end{array} \qquad \begin{array}{c|cc} \times & 0 & 1 \\ \hline 0 & 0 & 0 \\ 1 & 0 & 1 \\ \end{array}$

## Also known as

Some sources refer to a Boolean algebra as:

a Boolean ring

or

a Huntington algebra

both of which terms already have a different definition on $\mathsf{Pr} \infty \mathsf{fWiki}$.

Other common notations for the elements of a Boolean algebra include:

$0$ and $1$ for $\bot$ and $\top$, respectively
$a'$ for $\neg a$.

When this convention is used, $0$ is called zero, and $1$ is called one or unit.

## Also see

• Results about Boolean algebras can be found here.

## Source of Name

This entry was named for George Boole.