   Chapter 7.4, Problem 14ES ### 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 10—14 S denotes the set of real numbers strictly between 0 and 1. That is, S = { x ∈ R | 0 < x < 1 } .* 14. Define a function g from the set of real numbers to S by the following formula:For each real number x, g ( x ) = 1 2 ⋅ ( x 1 + | x | ) + 1 2 .Prove that g is a one-to-one correspondence. (It is possible to prove this statement either with calculus or without it.) What conclusion can you draw from this fact?

To determine

To prove that g is a one-to-one correspondence

Explanation

Given information:

g(x)=12.(x1+|x|)+12

Concept used:

One to one: Every element in domain must be mapped with element in codomain.

Calculation:

Let S be the set of all real numbers that are between 0 and 1.

That is S={x/0<x<1}

Define the function g:RS by g(x)=12.(x1+|x|)+12, for all x in R.

The objective is to show that g is a one to one correspondence.

The function g is one to one.

Let

x1,x2S12( x 1 1+| x 1 |)+12=12( x 2 1+| x 2 |)+1212( x 1 1+| x 1 |)=12( x 2 1+| x 2 |)               Substrac 12 from both sidesx11+| x 1|=x21+| x 2|                              Divide both sides by 12x1(1+| x 2|)=(1+| x 1|)x2x1+x1|x2|=x2+|x1|x2x1x2=|x1|x2x1|x2|( x 1 x 2)2=(| x 1 | x 2 x 1| x 2 |)2       Squaring on both sidesx12+x222x1x2=x12x22+x12x222x12x22( x 1 x 2)2=0x1x2=0x1=x2

Therefore, g is one to one

### 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 