Axiom:Boolean Algebra/Axioms

Axiom

Let $\struct {S, \vee, \wedge, \neg}$ be an algebraic system such that $\vee$ and $\wedge$ are binary operations, and $\neg$ is a unary operation.

The axioms that define a Boolean algebra can be specified as follows:

Formulation 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
\((\text {BA}_1 4)\)	$:$							$\forall a \in S: a \vee \neg a = \top, a \wedge \neg a = \bot$

Formulation 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:	\(\ds \forall a, b \in S:\)	\(\ds \paren {a \wedge \neg a} \vee b = b \)
					\(\ds \paren {a \vee \neg a} \wedge b = b \)