   Chapter 6.4, Problem 1ES ### 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 1-3 assume that B is a Boolean algebra with operations = and ‘ . Give the reasons needed to fill in the blanks in the proofs using only the axioms for a Boolean algebra.1. Idempotent law for’ : For every a in B, ??? Proof: Let a be any element of B. Then a = a .1                                                           ( a ) = a . ( a + a ¯ )                                         ( b ) = ( a ⋅ a ) + ( a ⋅ a ¯ )                       ( c )   = ( a ⋅ a ) + 0                                             (d)   = a . a                                                                     ( e )

To determine

a=a1                     (a)_=a(a+a¯)          (b)_=(aa)+(aa¯)    (c)_=(aa)+0             (d)_=aa                     (e)_

Explanation

Given information:

Let a be any element of B.

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 is any element of B.

Then, a¯ is the complement of a.

By tile Identity law for multiplication”

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