booleanalgebra - Math 230 Boolean Algebra Fall 2011 Drew...

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Math 230 Fall 2011 Boolean Algebra Drew Armstrong A Boolean algebra is a quadruple ( B, , , 0 ) where B is a set with two binary operations and a unary operation, join : B × B B , meet : B × B B , and complement 0 : 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 0 = 1 a a 0 = 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. (6) Theorem. (Idempotent Property) For all a B we have a a = a a a = a Proof. Note that there is a symmetry in the axioms between and , so we really only need to prove the first statement. We have:
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