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Unformatted text preview: Math 230 Fall 2011 Boolean Algebra Drew Armstrong A Boolean algebra is a quadruple ( B, ∨ , ∧ , ) where B is a set with two binary operations and a unary operation, • join ∨ : B × B → B , • meet ∧ : B × B → B , and • complement : B → B , satisfying the following five axioms: (1) Associaive Property. For all a,b,c ∈ B we have • a ∧ ( b ∧ c ) = ( a ∧ b ) ∧ c • a ∨ ( b ∨ c ) = ( a ∨ b ) ∨ c (2) Commutative Property. For all a,b ∈ B we have • a ∨ b = b ∨ a • a ∧ b = b ∧ a (3) Special Elements. There exist 0 6 = 1 ∈ B such that, for all a ∈ B : • a ∨ 0 = a • a ∧ 1 = a (4) Definition of Complement. For all a ∈ B we have • a ∨ a = 1 • a ∧ a = 0 (5) Distributive Property. For all a,b,c ∈ B we have • a ∨ ( b ∧ c ) = ( a ∨ b ) ∧ ( a ∨ c ) • a ∧ ( b ∨ c ) = ( a ∧ b ) ∨ ( a ∧ c ) Given these axioms, we can begin to prove some theorems....
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This note was uploaded on 01/08/2012 for the course MATH 461 taught by Professor Armstrong during the Fall '11 term at University of Miami.
 Fall '11
 Armstrong
 Math, Algebra

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