# 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.

### B

- Boolean Lattices (5 P)

## Pages in category "Boolean Algebras"

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

### C

- Cancellation of Join in Boolean Algebra
- Cancellation of Meet in Boolean Algebra
- Complement in Boolean Algebra is Unique
- Complement of Bottom
- Complement of Bottom (Boolean Algebras)
- Complement of Bottom/Boolean Algebra
- Complement of Complement (Boolean Algebras)
- Complement of Top
- Complement of Top (Boolean Algebras)
- Complement of Top/Boolean Algebra