C

Please include reasons for each step.
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!
Transcribed Image Text:1.6.2 (2) [Used in Example 1. 6. 3] Let T be the set defined in Equation 1. 6. 1. (2)Prove that if T = D, for somer € Q, then r? = 2.

Question

C Please include reasons for each step. 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 numberthat is not rational; the problem is how to describe such a set without making use ofanything other than the rational numbers. In the case of the numberr = v2, there turnsout 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&#34; means, nor proved that there issuch a real number, though we will do so in Theorem 2.6.9 and Definition 2.6.10. Wehave also not yet proved that “/2&#34; is not rational, a fact with which the reader is, atleast informally, familiar; we will see a proof of this fact in Theorem 2.6.11. Moreprecisely, it will be seen in that example that there is no rational number x such thatx = 2, and this last statement makes use only of rational numbers, so it is suited toour purpose at present. Nothing in our subsequent treatment of “V2&#34; in Section 2.6makes use of the current ex ample, so it will not be circular reasoning for us to makeuse of these subsequently proved facts here.LetT= {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 thatexercise 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, andit 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 cutD, = {x€Q |x >r}.An irrational cut is a Dedekind cut that is not a rational cut at any rational number.ABefore using Dedekind cuts to form the set of real numbers in Section 1.7, we willtake the remainder of the present section to prove some technically useful properties ofDedekind cuts, starting with the following simple lemma that will be used frequently! 1.6.2 (2)[Used in Example 1. 6. 3] Let T be the set defined in Equation 1. 6. 1.(2)Prove that if T = D, for somer € Q, then r? = 2.

Accepted Answer

Given:
The set 
                                                                 T=x∈ℚ | x>0 and…

[Used in Example 1. 6. 3] Let T be the set defined in Equation 1. 6. 1. (2)Prove that if T = D, for somer € Q, then r² = 2.