Explanation Given information: The absorption law for + states that for all elements a and b in a Boolean algebra a ⋅ b + a = a . 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, we have to prove the absorption law for + which is a ⋅ b + a = a , for all elements a and b . Consider the left hand side of the equation and apply the following axioms in order − LHS = a ⋅ b + a = a ⋅ b + a ⋅ 1 { ∵ x ⋅ 1 = x using Axiom 4} = a ⋅ ( b + 1 ) { ∵ ( a + b ) ⋅ c = a ⋅ c + b ⋅ c using Axiom 3} .....

5th Edition

ISBN: 9781337694193

Author: EPP, Susanna S.

Publisher: Cengage Learning,

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Chapter 6.4, Problem 13ES

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To derive the absorption law for + using other four axioms of Boolean algebra instead of using associative law for +.

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Chapter 6.4, Problem 12ES

Chapter 6.4, Problem 14ES

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Discrete Mathematics With Applications

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To derive the absorption law for + using other four axioms of Boolean algebra instead of using associative law for + .

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

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To derive the absorption law for + using other four axioms of Boolean algebra instead of using associative law for +.

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