Definition:Boolean Algebra

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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 $\left({S, \vee, \wedge, \neg}\right)$, where $\vee$ and $\wedge$ are binary, and $\neg$ is a unary operation.

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

\((BA_2 \ 0)\)   $:$   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:      \(\displaystyle \forall a, b, c \in S:\) \(\displaystyle a \vee \left({b \vee c}\right) = \left({a \vee b}\right) \vee c \)             
\(\displaystyle a \wedge \left({b \wedge c}\right) = \left({a \wedge b}\right) \wedge c \)             
\((BA_2 \ 3)\)   $:$   Absorption Laws:      \(\displaystyle \forall a, b \in S:\) \(\displaystyle \left({a \wedge b}\right) \vee b = b \)             
\(\displaystyle \left({a \vee b}\right) \wedge b = b \)             
\((BA_2 \ 4)\)   $:$   Distributivity:      \(\displaystyle \forall a, b, c \in S:\) \(\displaystyle a \wedge \left({b \vee c}\right) = \left({a \wedge b}\right) \vee \left({a \wedge c}\right) \)             
\(\displaystyle a \vee \left({b \wedge c}\right) = \left({a \vee b}\right) \wedge \left({a \vee c}\right) \)             
\((BA_2 \ 5)\)   $:$   Identity Elements:      \(\displaystyle \forall a, b \in S:\) \(\displaystyle \left({a \wedge \neg a}\right) \vee b = b \)             
\(\displaystyle \left({a \vee \neg a}\right) \wedge b = b \)             


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


Source of Name

This entry was named for George Boole.