# Definition:Boolean Algebra/Definition 2

## Definition

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:

### Boolean Algebra 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: | \(\ds \forall a, b \in S:\) | \(\ds \paren {a \wedge \neg a} \vee b = b \) | |||||

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

### Join

The operation $\vee$ is called **join**.

### Meet

The operation $\wedge$ is called **meet**.

### Complement

The operation $\neg$ is called **complementation**.

Thus for $a \in S$, $\neg a$ is called the **complement** of $a$.

## 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

- 1993: Keith Devlin:
*The Joy of Sets: Fundamentals of Contemporary Set Theory*(2nd ed.) ... (previous) ... (next): $\S 1$: Naive Set Theory: $\S 1.8$: Problems: $1$ - 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next):**Boolean algebra** - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next):**Boolean algebra**