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]...

4. Let A, B be nonempty sets and f : A → B be a function. For , y E A, define r ~<br>y if and only if f (x) = f(y).<br>a. Show that ~ is an equivalence relation on A.<br>b. Determine the equivalence class [x] for r E A.<br>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<br>and injective.<br>

Extracted 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.

Jun 04, 2022

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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]...

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