7

Boolean Algebra

Boolean algebra is a significant tool for the analysis and design of electronic computers. It has wide applications to switching theory and logical design of electronic circuits.

7.1 DEFINITIONS AND BASIC PROPERTIES

Definition 7.1

A non-empty set B with two binary operations ∨ and ∧, a unary operation ′, and two distinct elements 0 and I is called a Boolean Algebra if the following axioms holds for any elements a, b, cB:

[B1]: Commutative Laws

 

ab = ba    and    ab = ba.

[B2]: Distributive Laws

 

a ∧ (bc) = (ab) ∨ (ac)    and    a ∨ (bc) = (ab) ∧ (ac).

[B3]: Identity Laws:

 

a ∨ 0 = a    and    aI = a.

[B4]: Complement Laws:

 

aa′ = I    and    aa′ = 0.

We shall call 0 as zero element, ...

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