   Chapter 6.4, Problem 3ES ### 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
1 views

# In 1-3 assume that B is a Boolean algebra with operation s = and ‘. Give the reasons needed to fill in the blanks in the proofs using only the axioms for a Boolean algebra. Absorption law for-over +: For all a and b in B, ( a + b ) . a = a .Proof: Let a be any element of B. Then   ( a + b ) ⋅ a = a ⋅ ( a + b )                                 ( a ) = a . a + a . b                                   ( b ) = a + a . b                                   by   exercise 1 = a .1+ a . b                                           ( c )

To determine

(a+b)a=a(a+b)       (a)_=aa+ab       (b)_=a+ab            By exercise 1=a1+ab         (c)_=a(1+b)         (d)_=a(b+1)          (e)_=a1                   By exercise 2=a                      (f)_

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, by the Commutative law for multiplication "."

(a+b).a=a.(a+b)

By the Distributive law for multiplication "." over addition "+" the above expression can be writer as

(a+b).a=a.a+a.b

As B is the Boolean algebra, for all a in B, a.a=a

(a+b).a=a+a.b

By the Identity law for multiplication "."

(a+b).a=a.1+a.b

By the Distributive law for multiplication "." over addition "+"

(a+b).a=a.(1+b)

By the Commutative law for addition "+"

(a+b)

### Still sussing out bartleby?

Check out a sample textbook solution.

See a sample solution

#### The Solution to Your Study Problems

Bartleby provides explanations to thousands of textbook problems written by our experts, many with advanced degrees!

Get Started 