Definition:Boolean Algebra

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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:

Definition 1

\((\text {BA}_1 0)\)   $:$   $S$ 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

\((\text {BA}_2 0)\)   $:$   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 operation $\vee$ is called join.


The operation $\wedge$ is called meet.


The identity element $\bot$ for meet ($\wedge$) is called bottom.


The identity element $\top$ for join ($\vee$) is called top.


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:

+ & 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


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.