Category:Boolean Algebras

From ProofWiki
Jump to navigation Jump to search

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.