# Definition:Boolean Algebra/Definition 1

## 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:

\((\text {BA}_1 0)\) | $:$ | $S$ is closed under $\vee$, $\wedge$ and $\neg$ | ||||||

\((\text {BA}_1 1)\) | $:$ | Both $\vee$ and $\wedge$ are commutative | ||||||

\((\text {BA}_1 2)\) | $:$ | Both $\vee$ and $\wedge$ distribute over the other | ||||||

\((\text {BA}_1 3)\) | $:$ | Both $\vee$ and $\wedge$ have identities $\bot$ and $\top$ respectively | ||||||

\((\text {BA}_1 4)\) | $:$ | $\forall a \in S: a \vee \neg a = \top, a \wedge \neg a = \bot$ |

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

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

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:

or

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.

## Sources

- 1964: W.E. Deskins:
*Abstract Algebra*... (previous) ... (next): $\S 1.5$ - 2008: Paul Halmos and Steven Givant:
*Introduction to Boolean Algebras*... (previous) ... (next): $\S 2$