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, c ∈ B:
[B1]: Commutative Laws
a ∨ b = b ∨ a and a ∧ b = b ∧ a.
[B2]: Distributive Laws
a ∧ (b ∨ c) = (a ∧ b) ∨ (a ∧ c) and a ∨ (b ∧ c) = (a ∨ b) ∧ (a ∨ c).
[B3]: Identity Laws:
a ∨ 0 = a and a ∧ I = a.
[B4]: Complement Laws:
a ∨ a′ = I and a ∧ a′ = 0.
We shall call 0 as zero element, ...
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