# Category:Boolean Algebras

This category contains results about Boolean Algebras.
Definitions specific to this category can be found in Definitions/Boolean Algebras.

### Definition 1

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:

 $(\text {BA}_1 0)$ $:$ $S$ 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

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: $\ds \forall a, b \in S:$ $\ds a \vee b \in S$ $\ds a \wedge b \in S$ $\ds \neg a \in S$ $(\text {BA}_2 1)$ $:$ 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 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.

## Source of Name

This entry was named for George Boole.

## Subcategories

This category has the following 2 subcategories, out of 2 total.

## Pages in category "Boolean Algebras"

The following 25 pages are in this category, out of 25 total.