# 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 $\left({S, \vee, \wedge, \neg}\right)$, where $\vee$ and $\wedge$ are binary, and $\neg$ is a unary operation.

Furthermore, these operations are required to satisfy the following axioms:

 $(BA_1 \ 0)$ $:$ $S$ is closed under $\vee$, $\wedge$ and $\neg$ $(BA_1 \ 1)$ $:$ Both $\vee$ and $\wedge$ are commutative $(BA_1 \ 2)$ $:$ Both $\vee$ and $\wedge$ distribute over the other $(BA_1 \ 3)$ $:$ Both $\vee$ and $\wedge$ have identities $\bot$ and $\top$ respectively $(BA_1 \ 4)$ $:$ $\forall a \in S: a \vee \neg a = \top, a \wedge \neg a = \bot$

### 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:

 $(BA_2 \ 0)$ $:$ Closure: $\displaystyle \forall a, b \in S:$ $\displaystyle a \vee b \in S$ $\displaystyle a \wedge b \in S$ $\displaystyle \neg a \in S$ $(BA_2 \ 1)$ $:$ Commutativity: $\displaystyle \forall a, b \in S:$ $\displaystyle a \vee b = b \vee a$ $\displaystyle a \wedge b = b \wedge a$ $(BA_2 \ 2)$ $:$ Associativity: $\displaystyle \forall a, b, c \in S:$ $\displaystyle a \vee \paren {b \vee c} = \paren {a \vee b} \vee c$ $\displaystyle a \wedge \paren {b \wedge c} = \paren {a \wedge b} \wedge c$ $(BA_2 \ 3)$ $:$ Absorption Laws: $\displaystyle \forall a, b \in S:$ $\displaystyle \paren {a \wedge b} \vee b = b$ $\displaystyle \paren {a \vee b} \wedge b = b$ $(BA_2 \ 4)$ $:$ Distributivity: $\displaystyle \forall a, b, c \in S:$ $\displaystyle a \wedge \paren {b \vee c} = \paren {a \wedge b} \vee \paren {a \wedge c}$ $\displaystyle a \vee \paren {b \wedge c} = \paren {a \vee b} \wedge \paren {a \vee c}$ $(BA_2 \ 5)$ $:$ Identity Elements: $\displaystyle \forall a, b \in S:$ $\displaystyle \paren {a \wedge \neg a} \vee b = b$ $\displaystyle \paren {a \vee \neg a} \wedge b = b$

### Definition 3

A Boolean algebra is an algebraic structure $\left({S, \vee, \wedge}\right)$ such that:

 $(BA \ 0)$ $:$ $S$ is closed under both $\vee$ and $\wedge$ $(BA \ 1)$ $:$ Both $\vee$ and $\wedge$ are commutative $(BA \ 2)$ $:$ Both $\vee$ and $\wedge$ distribute over the other $(BA \ 3)$ $:$ Both $\vee$ and $\wedge$ have identities $\bot$ and $\top$ respectively $(BA \ 4)$ $:$ $\forall a \in S: \exists \neg a \in S: a \vee \neg a = \top, a \wedge \neg a = \bot$

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.

## Subcategories

This category has only the following subcategory.

## Pages in category "Boolean Algebras"

The following 24 pages are in this category, out of 24 total.