BuyFindarrow_forward

Discrete Mathematics With Applicat...

5th Edition
EPP + 1 other
ISBN: 9781337694193

Solutions

Chapter
Section
BuyFindarrow_forward

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
Sect-7.1 P-2ESSect-7.1 P-3ESSect-7.1 P-4ESSect-7.1 P-5ESSect-7.1 P-6ESSect-7.1 P-7ESSect-7.1 P-8ESSect-7.1 P-9ESSect-7.1 P-10ESSect-7.1 P-11ESSect-7.1 P-12ESSect-7.1 P-13ESSect-7.1 P-14ESSect-7.1 P-15ESSect-7.1 P-16ESSect-7.1 P-17ESSect-7.1 P-18ESSect-7.1 P-19ESSect-7.1 P-20ESSect-7.1 P-21ESSect-7.1 P-22ESSect-7.1 P-23ESSect-7.1 P-24ESSect-7.1 P-25ESSect-7.1 P-26ESSect-7.1 P-27ESSect-7.1 P-28ESSect-7.1 P-29ESSect-7.1 P-30ESSect-7.1 P-31ESSect-7.1 P-32ESSect-7.1 P-33ESSect-7.1 P-34ESSect-7.1 P-35ESSect-7.1 P-36ESSect-7.1 P-37ESSect-7.1 P-38ESSect-7.1 P-39ESSect-7.1 P-40ESSect-7.1 P-41ESSect-7.1 P-42ESSect-7.1 P-43ESSect-7.1 P-44ESSect-7.1 P-45ESSect-7.1 P-46ESSect-7.1 P-47ESSect-7.1 P-48ESSect-7.1 P-49ESSect-7.1 P-50ESSect-7.1 P-51ESSect-7.1 P-52ESSect-7.1 P-53ESSect-7.2 P-1TYSect-7.2 P-2TYSect-7.2 P-3TYSect-7.2 P-4TYSect-7.2 P-5TYSect-7.2 P-6TYSect-7.2 P-7TYSect-7.2 P-8TYSect-7.2 P-9TYSect-7.2 P-10TYSect-7.2 P-11TYSect-7.2 P-1ESSect-7.2 P-2ESSect-7.2 P-3ESSect-7.2 P-4ESSect-7.2 P-5ESSect-7.2 P-6ESSect-7.2 P-7ESSect-7.2 P-8ESSect-7.2 P-9ESSect-7.2 P-10ESSect-7.2 P-11ESSect-7.2 P-12ESSect-7.2 P-13ESSect-7.2 P-14ESSect-7.2 P-15ESSect-7.2 P-16ESSect-7.2 P-17ESSect-7.2 P-18ESSect-7.2 P-19ESSect-7.2 P-20ESSect-7.2 P-21ESSect-7.2 P-22ESSect-7.2 P-23ESSect-7.2 P-24ESSect-7.2 P-25ESSect-7.2 P-26ESSect-7.2 P-27ESSect-7.2 P-28ESSect-7.2 P-29ESSect-7.2 P-30ESSect-7.2 P-31ESSect-7.2 P-32ESSect-7.2 P-33ESSect-7.2 P-34ESSect-7.2 P-35ESSect-7.2 P-36ESSect-7.2 P-37ESSect-7.2 P-38ESSect-7.2 P-39ESSect-7.2 P-40ESSect-7.2 P-41ESSect-7.2 P-42ESSect-7.2 P-43ESSect-7.2 P-44ESSect-7.2 P-45ESSect-7.2 P-46ESSect-7.2 P-47ESSect-7.2 P-48ESSect-7.2 P-49ESSect-7.2 P-50ESSect-7.2 P-51ESSect-7.2 P-52ESSect-7.2 P-53ESSect-7.2 P-54ESSect-7.2 P-55ESSect-7.2 P-56ESSect-7.2 P-57ESSect-7.2 P-58ESSect-7.3 P-1TYSect-7.3 P-2TYSect-7.3 P-3TYSect-7.3 P-4TYSect-7.3 P-5TYSect-7.3 P-1ESSect-7.3 P-2ESSect-7.3 P-3ESSect-7.3 P-4ESSect-7.3 P-5ESSect-7.3 P-6ESSect-7.3 P-7ESSect-7.3 P-8ESSect-7.3 P-9ESSect-7.3 P-10ESSect-7.3 P-11ESSect-7.3 P-12ESSect-7.3 P-13ESSect-7.3 P-14ESSect-7.3 P-15ESSect-7.3 P-16ESSect-7.3 P-17ESSect-7.3 P-18ESSect-7.3 P-19ESSect-7.3 P-20ESSect-7.3 P-21ESSect-7.3 P-22ESSect-7.3 P-23ESSect-7.3 P-24ESSect-7.3 P-25ESSect-7.3 P-26ESSect-7.3 P-27ESSect-7.3 P-28ESSect-7.3 P-29ESSect-7.3 P-30ESSect-7.4 P-1TYSect-7.4 P-2TYSect-7.4 P-3TYSect-7.4 P-4TYSect-7.4 P-5TYSect-7.4 P-6TYSect-7.4 P-7TYSect-7.4 P-8TYSect-7.4 P-9TYSect-7.4 P-1ESSect-7.4 P-2ESSect-7.4 P-3ESSect-7.4 P-4ESSect-7.4 P-5ESSect-7.4 P-6ESSect-7.4 P-7ESSect-7.4 P-8ESSect-7.4 P-9ESSect-7.4 P-10ESSect-7.4 P-11ESSect-7.4 P-12ESSect-7.4 P-13ESSect-7.4 P-14ESSect-7.4 P-15ESSect-7.4 P-16ESSect-7.4 P-17ESSect-7.4 P-18ESSect-7.4 P-19ESSect-7.4 P-20ESSect-7.4 P-21ESSect-7.4 P-22ESSect-7.4 P-23ESSect-7.4 P-24ESSect-7.4 P-25ESSect-7.4 P-26ESSect-7.4 P-27ESSect-7.4 P-28ESSect-7.4 P-29ESSect-7.4 P-30ESSect-7.4 P-31ESSect-7.4 P-32ESSect-7.4 P-33ESSect-7.4 P-34ESSect-7.4 P-35ESSect-7.4 P-36ESSect-7.4 P-37ESSect-7.4 P-38ES