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Exercise 17.5.20. (a) Show that there is a well-defined function f: Z12 → Z₁, given by f([a]12) = [a]4. That is, show that if [a]12 = [b]12, then [a] = [b]4. 614CHAPTER 17 EQUIVALENCE RELATIONS AND EQUIVALENCE CLASSES (b) Generalize part (a) by showing that if m divides n, then there is a well- defined function f: Zn → Zm, given by f([a]n) = [a]m. That is, show that if [a]n[b]n, then [a]m = [b]m-

Exercise 17.5.20. (a) Show that there is a well-defined function f: Z12 → Z₁, given by f([a]12) = [a]4. That is, show that if [a]12 = [b]12, then [a] = [b]4. 614CHAPTER 17 EQUIVALENCE RELATIONS AND EQUIVALENCE CLASSES (b) Generalize part (a) by showing that if m divides n, then there is a well- defined function f: Zn → Zm, given by f([a]n) = [a]m. That is, show that if [a]n[b]n, then [a]m = [b]m-

Elements Of Modern Algebra
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter3: Groups
Section3.3: Subgroups
Problem 23E: 23. Let be the equivalence relation on defined by if and only if there exists an element in ...
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Please do Exercise 17.5.20 part A and B and please show step by step and explain

Exercise 17.5.20. (a) Show that there is a well-defined function f: Z12 → Z₁, given by f([a]12) = [a]4. That is, show that if [a]12 = [b]12, then [a] = [b]4. 614CHAPTER 17 EQUIVALENCE RELATIONS AND EQUIVALENCE CLASSES (b) Generalize part (a) by showing that if m divides n, then there is a well- defined function f: Zn → Zm, given by f([a]n) = [a]m. That is, show that if [a]n[b]n, then [a]m = [b]m-
Transcribed Image Text:Exercise 17.5.20. (a) Show that there is a well-defined function f: Z12 → Z₁, given by f([a]12) = [a]4. That is, show that if [a]12 = [b]12, then [a] = [b]4. 614CHAPTER 17 EQUIVALENCE RELATIONS AND EQUIVALENCE CLASSES (b) Generalize part (a) by showing that if m divides n, then there is a well- defined function f: Zn → Zm, given by f([a]n) = [a]m. That is, show that if [a]n[b]n, then [a]m = [b]m-
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