1.6.2 (1) [Used in Example 1.6. 3] Let T be the set defined in Equation 1. 6. 1. (1)Prove that Tis a Dedekind cut.
1.6.2 (1) [Used in Example 1.6. 3] Let T be the set defined in Equation 1. 6. 1. (1)Prove that Tis a Dedekind cut.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Please include reasons for each step.
![1.6.2 (1)
[Used in Example 1.6. 3] Let T be the set defined in Equation 1. 6. 1.
(1)Prove that T is a Dedekind cut.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8975dae8-fbcb-4b49-8e3a-715e0e53990f%2Fa6eced6f-d3c5-4e9c-a3d5-98b4dea3c6f6%2Fwmky8ei_processed.png&w=3840&q=75)
Transcribed Image Text:1.6.2 (1)
[Used in Example 1.6. 3] Let T be the set defined in Equation 1. 6. 1.
(1)Prove that T is a Dedekind cut.
![Example 1.6.3. To find a Dedekind cut that is not of the form given in Lemma 1.6.2,
the simplest idea is to look at the set of all rational numbers greater than a real number
that is not rational; the problem is how to describe such a set without making use of
anything other than the rational numbers. In the case of the numberr = v2, there turns
out to be a simple solution to this problem, as we will now see. We note first, however,
that we have not yet formally defined what “V2" means, nor proved that there is
such a real number, though we will do so in Theorem 2.6.9 and Definition 2.6.10. We
have also not yet proved that “/2" is not rational, a fact with which the reader is, at
least informally, familiar; we will see a proof of this fact in Theorem 2.6.11. More
precisely, it will be seen in that example that there is no rational number x such that
x = 2, and this last statement makes use only of rational numbers, so it is suited to
our purpose at present. Nothing in our subsequent treatment of “V2" in Section 2.6
makes use of the current ex ample, so it will not be circular reasoning for us to make
use of these subsequently proved facts here.
Let
T= {x €Q |x>0 andx² > 2}.
It is seen by Exercise 1.6.2 (1) that T is a Dedekind cut, and by Part (2) of that
exercise it is seen that if T has the form {x € Q ]x >r} for some reQ, then ? = 2.
By Theorem 2.6.11 we know that there is no rational number x such that x = 2, and
it follows that T is a Dedekind cut that is not of the form given in Lemma 1.6.2. O
(1.6.1)
Example 1.6.3 explains the need for the following definition.
Definition 1.6.4. Let rE Q. The rational cut at r, denoted D,, is the Dedekind cut
D, = {x€Q |x >r}.
An irrational cut is a Dedekind cut that is not a rational cut at any rational number.
A
Before using Dedekind cuts to form the set of real numbers in Section 1.7, we will
take the remainder of the present section to prove some technically useful properties of
Dedekind cuts, starting with the following simple lemma that will be used frequently!](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8975dae8-fbcb-4b49-8e3a-715e0e53990f%2Fa6eced6f-d3c5-4e9c-a3d5-98b4dea3c6f6%2Fhpgx9if_processed.png&w=3840&q=75)
Transcribed Image Text:Example 1.6.3. To find a Dedekind cut that is not of the form given in Lemma 1.6.2,
the simplest idea is to look at the set of all rational numbers greater than a real number
that is not rational; the problem is how to describe such a set without making use of
anything other than the rational numbers. In the case of the numberr = v2, there turns
out to be a simple solution to this problem, as we will now see. We note first, however,
that we have not yet formally defined what “V2" means, nor proved that there is
such a real number, though we will do so in Theorem 2.6.9 and Definition 2.6.10. We
have also not yet proved that “/2" is not rational, a fact with which the reader is, at
least informally, familiar; we will see a proof of this fact in Theorem 2.6.11. More
precisely, it will be seen in that example that there is no rational number x such that
x = 2, and this last statement makes use only of rational numbers, so it is suited to
our purpose at present. Nothing in our subsequent treatment of “V2" in Section 2.6
makes use of the current ex ample, so it will not be circular reasoning for us to make
use of these subsequently proved facts here.
Let
T= {x €Q |x>0 andx² > 2}.
It is seen by Exercise 1.6.2 (1) that T is a Dedekind cut, and by Part (2) of that
exercise it is seen that if T has the form {x € Q ]x >r} for some reQ, then ? = 2.
By Theorem 2.6.11 we know that there is no rational number x such that x = 2, and
it follows that T is a Dedekind cut that is not of the form given in Lemma 1.6.2. O
(1.6.1)
Example 1.6.3 explains the need for the following definition.
Definition 1.6.4. Let rE Q. The rational cut at r, denoted D,, is the Dedekind cut
D, = {x€Q |x >r}.
An irrational cut is a Dedekind cut that is not a rational cut at any rational number.
A
Before using Dedekind cuts to form the set of real numbers in Section 1.7, we will
take the remainder of the present section to prove some technically useful properties of
Dedekind cuts, starting with the following simple lemma that will be used frequently!
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