Axiom:Boolean Algebra/Axioms/Formulation 2

Axiom

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 $

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