Category:Boolean Algebras

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

Definition 1

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

$(\text {BA}_2 0)$	$:$	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 $

Source of Name

This entry was named for George Boole.

Subcategories

This category has the following 4 subcategories, out of 4 total.

B

Boolean Algebra is Equivalent to Boolean Lattice‎ (2 P)
Boolean Lattices‎ (1 C, 18 P)
Boolean Prime Ideal Theorem‎ (1 C, 4 P)

I

Identities of Boolean Algebra are also Zeroes‎ (3 P)

Pages in category "Boolean Algebras"

The following 29 pages are in this category, out of 29 total.

A

B

C

D

O

P

Power Set with Union and Intersection forms Boolean Algebra

S

Stone's Representation Theorem for Boolean Algebras

T

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