   Chapter 6.4, Problem 8ES ### Discrete Mathematics With Applicat...

5th Edition
EPP + 1 other
ISBN: 9781337694193

#### Solutions

Chapter
Section ### Discrete Mathematics With Applicat...

5th Edition
EPP + 1 other
ISBN: 9781337694193
Textbook Problem
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# In 4—10 assume that B is a Boolean algebra with operations + and •. Prove each statement using only the axioms for a Boolean algebra and statements proved in the text or in lower-numbered exercises. 8. De Morgan’s law for •: For all a and b in B, a ⋅ b ¯ = a ¯ + b ¯ . (Hint: Prove that ( a ⋅ b ) + ( a ¯ + b ¯ ) = 1 and that ( a ⋅ b ) + ( a ¯ + b ¯ ) = 0 , and use the fact that has a unique complement.)

To determine

To Prove:

For all a and b in B,a.b¯=a¯+b¯.

Explanation

Given information:

Let B be the Boolean algebra, with the operations, addition "+" and multiplication ".".

Concept used:

a¯+a=a+a¯     by the commutative law=1            by the complement law for 1

And

a¯a=aa¯     by the commutative law=0            by the complement law for 0

Calculation:

Let B be the Boolean algebra, with the operations, addition "+" and multiplication ""

Suppose a and b are any elements of B.

Then, a¯ is the complement of a and b¯ is the complement of b.

The uniqueness of the complement law tells that, for all a and x in B if,

a+x=1a.x=0x=a¯

To show that a.b¯=a¯+b¯, it is enough to show that (a.b)+(a¯+b¯)=1 and,

(a.b).(a¯+b¯)=0 Then use the uniqueness of complement law.

Prove (a.b).(a¯+b¯)=1

(a.b)+(a¯+b¯)=(a¯+b¯)+(a.b) By the Commutative law for addition "+"

=((a¯+b¯)+a).((a¯+b¯)+b) By the Distributive law for addition "+"

Over multiplication "."

=((b¯+a¯)+a).((a¯+b¯)+b) By the Commutative law for addition "+"

=(b¯+(a¯+a)).(a¯+(b¯+b)) By the Associative law for addition "+"

=(b¯+(a+a¯))

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