The Abstract Definition Of A Boolean Algebra

The Abstract Definition of a Boolean Algebra: In this section we have focused on Boolean functions and expressions. However, the results we have established can be translated into results about propositions or results about sets. Because of this, it is useful to define Boolean algebras abstractly. Once it is shown that a particular structure is a Boolean algebra, then all results established about Boolean algebras in general apply to this particular structure.

Boolean algebras can be defined in several ways. The most common way is to specify the properties that operations must satisfy, as is done in Definition 1.

DEFINITION 1 A Boolean algebra is a set B with two binary operations ∨ and ∧, elements 0 and 1, and a unary operation such that these properties hold for all x, y, and z in B:

Using the laws given in Definition 1, it is possible to prove many other laws that hold for every Boolean algebra, such as idempotent and domination laws. (

For a latticeLto be a Boolean algebra as specified in Definition 1, it must have two properties. First, it must be complemented. For a lattice to be complemented it must have a least element 0 and a greatest element 1 and for every element x of the lattice there must exist an element x such that x ∨ x = 1 and x ∧ x = 0. Second, it must be distributive. This means that for every x, y, and z in L, x ∨ (y ∧ z) = (x ∨ y) ∧ (x ∨ z) and x ∧ (y ∨ z) = (x ∧ y) ∨ (x ∧ z).