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e) the set of nonnegative rational numbers My solution to le)....(Check whether I'm correct)

e) the set of nonnegative rational numbers My solution to le)....(Check whether I'm correct)

Elements Of Modern Algebra
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter2: The Integers
Section2.6: Congruence Classes
Problem 25E
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3:48 ← Is my proof in chapter 1.1 #le of "Elementary Number Theory & It's Application" correct? Question 1: Determine whether each of the following sets is well-ordered. Either give a proof using the well-ordering property of the set of positive integers, or give an example of the subset of the set that has no smallest element. a) the set of integers greater than 3 b) the set of even positive integers c) the set of positive rational numbers d) the set of positive rational numbers that can be written in the form a/2, where a is a positive integer. e) the set of nonnegative rational numbers My solution to 1e)....(Check whether I'm correct) If the set of non-negative real numbers can be written as p/q, where p and q are integers, from the def. of le, we get p/q>=0. Therefore, multiplying both sides of p/ q>=0, gives p>=0. Therefore, since p can equal zero, p/q can equal zero. Hence zero is the smallest element of p/q. Because, there is √x 8
Transcribed Image Text:3:48 ← Is my proof in chapter 1.1 #le of "Elementary Number Theory & It's Application" correct? Question 1: Determine whether each of the following sets is well-ordered. Either give a proof using the well-ordering property of the set of positive integers, or give an example of the subset of the set that has no smallest element. a) the set of integers greater than 3 b) the set of even positive integers c) the set of positive rational numbers d) the set of positive rational numbers that can be written in the form a/2, where a is a positive integer. e) the set of nonnegative rational numbers My solution to 1e)....(Check whether I'm correct) If the set of non-negative real numbers can be written as p/q, where p and q are integers, from the def. of le, we get p/q>=0. Therefore, multiplying both sides of p/ q>=0, gives p>=0. Therefore, since p can equal zero, p/q can equal zero. Hence zero is the smallest element of p/q. Because, there is √x 8
smallest element in the set, using the well- ordering property-where every non-empty set of positive integers has a least smallest element -we can say the set in le) is well-ordered? Question Regarding Solution: If my proof is flawed, how do we improve the solution?
Transcribed Image Text:smallest element in the set, using the well- ordering property-where every non-empty set of positive integers has a least smallest element -we can say the set in le) is well-ordered? Question Regarding Solution: If my proof is flawed, how do we improve the solution?
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