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Math
Discrete Mathematics With Applications5th Edition

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Chapter 6.4, Problem 12ES
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Discrete Mathematics With Applicat...

5th Edition
EPP + 1 other
ISBN: 9781337694193

Solutions

Chapter
Section
Problem 1TY
Problem 2TY
Problem 3TY
Problem 1ES
Problem 2ES
Problem 3ES
Problem 4ES
Problem 5ES
Problem 6ES
Problem 7ES
Problem 8ES
Problem 9ES
Problem 10ES
Problem 11ES
Problem 12ES
Problem 13ES
Problem 14ES
Problem 15ES
Problem 16ES
Problem 17ES
Problem 18ES
Problem 19ES
Problem 20ES
Problem 21ES
Problem 22ES
Problem 23ES
Problem 24ES
Problem 25ES
Problem 26ES
Problem 27ES
Problem 28ES
Problem 29ES
BuyFindarrow_forward

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 an axiom for a Boolean algebra, can be derived from the other 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 universal bound law for + states that for every element a in a Boolean algebra, a + 1 = 1 The proof shown in exercise 2 used the associative law for + . Rederive the law without using the associative law and using only the other four axioms for a Boolean algebra.

To determine

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

Explanation

Given information:

The universal bound law for + states that for every element a in a Boolean algebra a+1=1.

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 universal bound law for + which is a+1=1, for every element a

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Chapter 6 Solutions

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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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