Definition:Boolean Algebra/Definition 2

From ProofWiki
Jump to navigation Jump to search


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:

\((\text {BA}_2 0)\)   $:$   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:      \(\displaystyle \forall a, b \in S:\) \(\displaystyle \paren {a \wedge \neg a} \vee b = b \)             
\(\displaystyle \paren {a \vee \neg a} \wedge b = b \)             

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

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


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.