# Axiom:Boolean Algebra/Axioms/Definition 1

## Definition

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:

 $(\text {BA}_1 0)$ $:$ $S$ 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 $(\text {BA}_1 4)$ $:$ $\forall a \in S: a \vee \neg a = \top, a \wedge \neg a = \bot$