# Definition:Boolean Algebra/Axioms/Definition 2

## 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:

 $(BA_2 \ 0)$ $:$ 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: $\displaystyle \forall a, b \in S:$ $\displaystyle \paren {a \wedge \neg a} \vee b = b$ $\displaystyle \paren {a \vee \neg a} \wedge b = b$