Explanation Given information: The associative law states that for all elements a , b and c in a Boolean algebra ( a + b ) + c = a + ( b + c ) for + ( a ⋅ b ) ⋅ c = a ⋅ ( b ⋅ c ) for ⋅ Calculation: There are total five axioms of the Boolean algebra which are listed in the following table. Axioms Addition ( + ) Multiplication ( . ) (1) Commutative laws a + b = b + a a ⋅ b = b ⋅ a (2) Associative laws ( a + b ) + c = a + ( b + c ) ( a ⋅ b ) ⋅ c = a ⋅ ( b ⋅ c ) (3) Distributive laws ( a + b ) ⋅ c = a ⋅ c + b ⋅ c ( a ⋅ b ) + c = ( a + c ) ⋅ ( b + c ) (4) Identity laws a + 0 = a a ⋅ 1 = a (5) Complement laws a + a ' = 1 a ⋅ a ' = 0 Now, to prove the associative law for + which states that for all elements a , b and c in a Boolean algebra ( a + b ) + c = a + ( b + c ) Let us consider the left hand side of the above equation- LHS = ( a + b ) + c = a + b + c ⋅ 1 { ∵ x ⋅ 1 = x using unit theorem} = a + b + c ( b + b ' ) { ∵ x + x ' = 1 using axiom (4)} = a + ( b + c

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Author: EPP, Susanna S.

Publisher: Cengage Learning,

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Chapter 6.4, Problem 15ES

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To derive the associative law for + and ⋅ using other four axioms of Boolean algebra.

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To derive the associative law for + and ⋅ using other four axioms of Boolean algebra.

Solution Summary: The author explains how to derive the associative law for + and cdot using other four axioms of Boolean algebra.

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To derive the associative law for + and ⋅ using other four axioms of Boolean algebra.

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