# Definition:Boolean Algebra/Definition 2

## Definition

### Boolean Algebra Axioms

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:

 $(\text {BA}_2 0)$ $:$ Closure: $\ds \forall a, b \in S:$ $\ds a \vee b \in S$ $\ds a \wedge b \in S$ $\ds \neg a \in S$ $(\text {BA}_2 1)$ $:$ Commutativity: $\ds \forall a, b \in S:$ $\ds a \vee b = b \vee a$ $\ds a \wedge b = b \wedge a$ $(\text {BA}_2 2)$ $:$ Associativity: $\ds \forall a, b, c \in S:$ $\ds a \vee \paren {b \vee c} = \paren {a \vee b} \vee c$ $\ds a \wedge \paren {b \wedge c} = \paren {a \wedge b} \wedge c$ $(\text {BA}_2 3)$ $:$ Absorption Laws: $\ds \forall a, b \in S:$ $\ds \paren {a \wedge b} \vee b = b$ $\ds \paren {a \vee b} \wedge b = b$ $(\text {BA}_2 4)$ $:$ Distributivity: $\ds \forall a, b, c \in S:$ $\ds a \wedge \paren {b \vee c} = \paren {a \wedge b} \vee \paren {a \wedge c}$ $\ds a \vee \paren {b \wedge c} = \paren {a \vee b} \wedge \paren {a \vee c}$ $(\text {BA}_2 5)$ $:$ Identity Elements: $\ds \forall a, b \in S:$ $\ds \paren {a \wedge \neg a} \vee b = b$ $\ds \paren {a \vee \neg a} \wedge b = b$

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

The operation $\neg$ is called complementation.

## Also defined as

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

It is a common approach to define (the) Boolean algebra to be an algebraic structure consisting of:

a boolean domain (that is, a set with two elements, typically $\set {0, 1}$)

together with:

the two operations addition $+$ and multiplication $\times$ defined as follows:
$\begin{array}{c|cc} + & 0 & 1 \\ \hline 0 & 0 & 1 \\ 1 & 1 & 0 \\ \end{array} \qquad \begin{array}{c|cc} \times & 0 & 1 \\ \hline 0 & 0 & 0 \\ 1 & 0 & 1 \\ \end{array}$

Hence expositions discussing such a structure are often considered to be included in a field of study referred to as Boolean algebra.

However, on $\mathsf{Pr} \infty \mathsf{fWiki}$ we do not take this approach.

Instead, we take the approach of investigating such results in the context of propositional logic.

## 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.