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Discrete Mathematics With Applications5th Edition

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Chapter 6.4, Problem 15ES
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Discrete Mathematics With Applicat...

5th Edition
EPP + 1 other
ISBN: 9781337694193

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Discrete Mathematics With Applicat...

5th Edition
EPP + 1 other
ISBN: 9781337694193
Textbook Problem
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Exercises 12-15 provide an outline for a proof that the associative laws, which were included as axiom for a Bolllem algebre, can be derived from the other four axioms. The algebra, can be derived from the four axioms. The outline is from Introduction to Boolean Algebra by S. Givant and P. Halmos, Springer, 2009. In order to avoid unneeded parentheses, assume that takes precedence over+.

The assiociative kaw for + states that for all elements a, b, and c. Show that this law , as from the other four axioms in the definition and axioms for a Boolean algebra. Then explain how to your work to obtain to obtain a dervation for the associative law for.

To determine

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

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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Sect-6.1 P-1TYSect-6.1 P-2TYSect-6.1 P-3TYSect-6.1 P-4TYSect-6.1 P-5TYSect-6.1 P-6TYSect-6.1 P-7TYSect-6.1 P-8TYSect-6.1 P-9TYSect-6.1 P-10TY
Sect-6.1 P-11TYSect-6.1 P-1ESSect-6.1 P-2ESSect-6.1 P-3ESSect-6.1 P-4ESSect-6.1 P-5ESSect-6.1 P-6ESSect-6.1 P-7ESSect-6.1 P-8ESSect-6.1 P-9ESSect-6.1 P-10ESSect-6.1 P-11ESSect-6.1 P-12ESSect-6.1 P-13ESSect-6.1 P-14ESSect-6.1 P-15ESSect-6.1 P-16ESSect-6.1 P-17ESSect-6.1 P-18ESSect-6.1 P-19ESSect-6.1 P-20ESSect-6.1 P-21ESSect-6.1 P-22ESSect-6.1 P-23ESSect-6.1 P-24ESSect-6.1 P-25ESSect-6.1 P-26ESSect-6.1 P-27ESSect-6.1 P-28ESSect-6.1 P-29ESSect-6.1 P-30ESSect-6.1 P-31ESSect-6.1 P-32ESSect-6.1 P-33ESSect-6.1 P-34ESSect-6.1 P-35ESSect-6.1 P-36ESSect-6.1 P-37ESSect-6.1 P-38ESSect-6.2 P-1TYSect-6.2 P-2TYSect-6.2 P-3TYSect-6.2 P-4TYSect-6.2 P-5TYSect-6.2 P-6TYSect-6.2 P-1ESSect-6.2 P-2ESSect-6.2 P-3ESSect-6.2 P-4ESSect-6.2 P-5ESSect-6.2 P-6ESSect-6.2 P-7ESSect-6.2 P-8ESSect-6.2 P-9ESSect-6.2 P-10ESSect-6.2 P-11ESSect-6.2 P-12ESSect-6.2 P-13ESSect-6.2 P-14ESSect-6.2 P-15ESSect-6.2 P-16ESSect-6.2 P-17ESSect-6.2 P-18ESSect-6.2 P-19ESSect-6.2 P-20ESSect-6.2 P-21ESSect-6.2 P-22ESSect-6.2 P-23ESSect-6.2 P-24ESSect-6.2 P-25ESSect-6.2 P-26ESSect-6.2 P-27ESSect-6.2 P-28ESSect-6.2 P-29ESSect-6.2 P-30ESSect-6.2 P-31ESSect-6.2 P-32ESSect-6.2 P-33ESSect-6.2 P-34ESSect-6.2 P-35ESSect-6.2 P-36ESSect-6.2 P-37ESSect-6.2 P-38ESSect-6.2 P-39ESSect-6.2 P-40ESSect-6.2 P-41ESSect-6.2 P-42ESSect-6.2 P-43ESSect-6.2 P-44ESSect-6.3 P-1TYSect-6.3 P-2TYSect-6.3 P-3TYSect-6.3 P-1ESSect-6.3 P-2ESSect-6.3 P-3ESSect-6.3 P-4ESSect-6.3 P-5ESSect-6.3 P-6ESSect-6.3 P-7ESSect-6.3 P-8ESSect-6.3 P-9ESSect-6.3 P-10ESSect-6.3 P-11ESSect-6.3 P-12ESSect-6.3 P-13ESSect-6.3 P-14ESSect-6.3 P-15ESSect-6.3 P-16ESSect-6.3 P-17ESSect-6.3 P-18ESSect-6.3 P-19ESSect-6.3 P-20ESSect-6.3 P-21ESSect-6.3 P-22ESSect-6.3 P-23ESSect-6.3 P-24ESSect-6.3 P-25ESSect-6.3 P-26ESSect-6.3 P-27ESSect-6.3 P-28ESSect-6.3 P-29ESSect-6.3 P-30ESSect-6.3 P-31ESSect-6.3 P-32ESSect-6.3 P-33ESSect-6.3 P-34ESSect-6.3 P-35ESSect-6.3 P-36ESSect-6.3 P-37ESSect-6.3 P-38ESSect-6.3 P-39ESSect-6.3 P-40ESSect-6.3 P-41ESSect-6.3 P-42ESSect-6.3 P-43ESSect-6.3 P-44ESSect-6.3 P-45ESSect-6.3 P-46ESSect-6.3 P-47ESSect-6.3 P-48ESSect-6.3 P-49ESSect-6.3 P-50ESSect-6.3 P-51ESSect-6.3 P-52ESSect-6.3 P-53ESSect-6.3 P-54ESSect-6.4 P-1TYSect-6.4 P-2TYSect-6.4 P-3TYSect-6.4 P-1ESSect-6.4 P-2ESSect-6.4 P-3ESSect-6.4 P-4ESSect-6.4 P-5ESSect-6.4 P-6ESSect-6.4 P-7ESSect-6.4 P-8ESSect-6.4 P-9ESSect-6.4 P-10ESSect-6.4 P-11ESSect-6.4 P-12ESSect-6.4 P-13ESSect-6.4 P-14ESSect-6.4 P-15ESSect-6.4 P-16ESSect-6.4 P-17ESSect-6.4 P-18ESSect-6.4 P-19ESSect-6.4 P-20ESSect-6.4 P-21ESSect-6.4 P-22ESSect-6.4 P-23ESSect-6.4 P-24ESSect-6.4 P-25ESSect-6.4 P-26ESSect-6.4 P-27ESSect-6.4 P-28ESSect-6.4 P-29ES
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