Axiom:Boolean Algebra/Axioms
Jump to navigation
Jump to search
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: | \(\ds \forall a, b \in S:\) | \(\ds \paren {a \wedge \neg a} \vee b = b \) | |||||
\(\ds \paren {a \vee \neg a} \wedge b = b \) |