Boolean algebra

Definition

A Boolean algebra is a set $A$ with two binary operations $\wedge$ and $\vee$, a unary operation $\neg$, and two elements $0, 1 \in A$. These axioms must hold for all $a, b, c \in A$. Associativity: $a \vee (b \vee c) = (a \vee b) \vee c$ and $a \wedge (b \wedge c) = (a \wedge b) \wedge c$. Commutativity: $a \vee b = b \vee a$ and $a \wedge b = b \wedge a$. Absorption: $a \vee (a \wedge b) = a$ and $a \wedge (a \vee b) = a$. Identity: $a \vee 0 = a$ and $a \wedge 1 = a$. Distributivity: $a \vee (b \wedge c) = (a \vee b) \wedge (a \vee c)$ and $a \wedge (b \vee c) = (a \wedge b) \vee (a \wedge c)$. Complements: $a \vee \neg a = 1$ and $a \wedge \neg a = 0$.