# Definition:Boolean Algebra

## Definition

### Definition 1

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 | ||||||

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

### 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:

\((\text {BA}_2 0)\) | $:$ | Closure: | \(\displaystyle \forall a, b \in S:\) | \(\displaystyle a \vee b \in S \) | ||||

\(\displaystyle a \wedge b \in S \) | ||||||||

\(\displaystyle \neg a \in S \) | ||||||||

\((\text {BA}_2 1)\) | $:$ | Commutativity: | \(\displaystyle \forall a, b \in S:\) | \(\displaystyle a \vee b = b \vee a \) | ||||

\(\displaystyle a \wedge b = b \wedge a \) | ||||||||

\((\text {BA}_2 2)\) | $:$ | Associativity: | \(\displaystyle \forall a, b, c \in S:\) | \(\displaystyle a \vee \paren {b \vee c} = \paren {a \vee b} \vee c \) | ||||

\(\displaystyle a \wedge \paren {b \wedge c} = \paren {a \wedge b} \wedge c \) | ||||||||

\((\text {BA}_2 3)\) | $:$ | Absorption Laws: | \(\displaystyle \forall a, b \in S:\) | \(\displaystyle \paren {a \wedge b} \vee b = b \) | ||||

\(\displaystyle \paren {a \vee b} \wedge b = b \) | ||||||||

\((\text {BA}_2 4)\) | $:$ | Distributivity: | \(\displaystyle \forall a, b, c \in S:\) | \(\displaystyle a \wedge \paren {b \vee c} = \paren {a \wedge b} \vee \paren {a \wedge c} \) | ||||

\(\displaystyle a \vee \paren {b \wedge c} = \paren {a \vee b} \wedge \paren {a \vee c} \) | ||||||||

\((\text {BA}_2 5)\) | $:$ | 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 $\struct {S, \vee, \wedge}$ such that:

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

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

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

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

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

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:

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

- 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next): Entry:**Boolean algebra** - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next): Entry:**Boolean algebra** - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next): Entry:**Boolean algebra**