Definition:Boolean Algebra/Axioms/Definition 2

From ProofWiki
Jump to navigation Jump to search


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

Also see