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:

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

• Equivalence of Definitions of Boolean Algebra

• 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