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4. Let A, B be nonempty sets and f : A → B be a function. For x, y € A, define ¤ ~ y if and only if f(x) = f(y). a. Show that is an equivalence relation on A. b. Determine the equivalence class [x] for ¤ € A. c. Let & = {[x] :x € A} and define F : & → B by F([x]) = f(x) for all x E A. Show that F is well defined and injective.

4. Let A, B be nonempty sets and f : A → B be a function. For x, y € A, define ¤ ~ y if and only if f(x) = f(y). a. Show that is an equivalence relation on A. b. Determine the equivalence class [x] for ¤ € A. c. Let & = {[x] :x € A} and define F : & → B by F([x]) = f(x) for all x E A. Show that F is well defined and injective.

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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4. Let A, B be nonempty sets and f : A → B be a function. For , y E A, define r ~ y if and only if f (x) = f(y). a. Show that ~ is an equivalence relation on A. b. Determine the equivalence class [x] for r E A. c. Let & = {[x] : x € A} and define F : & → B by F([x]) = f(x) for all x E A. Show that F is well defined and injective.
Transcribed Image Text:4. Let A, B be nonempty sets and f : A → B be a function. For , y E A, define r ~ y if and only if f (x) = f(y). a. Show that ~ is an equivalence relation on A. b. Determine the equivalence class [x] for r E A. c. Let & = {[x] : x € A} and define F : & → B by F([x]) = f(x) for all x E A. Show that F is well defined and injective.
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