   Chapter 6.4, Problem 10ES ### 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 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. 10. Cancellation law: For all x, y, and z in B, if x + y = x + z and x ⋅ y = x ⋅ z then y = z .

To determine

To Prove:

For all x,y and z in B, if x+y=x+z and xy=xz then yz.

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 operation addition "+" and multiplication "".

Suppose x,y and z are any element of B.

Consider the statement.

x+y=x+z

As B be the Boolean algebra and for all a and b in B,(a+b)a=a.

By using the above fact that y can be written as.

y=(y+x)y

=(x+y).y By the commutative law for addition "+".

=y(x+y) By the commutative law for multiplication ""

=y.(x+z) By hypothesis x+y=x+z.

=(yx)+(yz) By the Distributive law for multiplication "", over addition "+"

### 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 