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2. Let f : A ! B. DeÖne a relation R on A by xRy i§ f (x) = f (y). a. Prove that R is an equivalence relation on A. b. Let Ex = fy 2 A : xRyg be the equivalence class of x 2 A. DeÖne E = fEx : x 2 Ag to be the collection of all equivalence classes. Prove that the function g : A ! E deÖned by g (x) = Ex is surjective. c. Prove that the function h : E ! B deÖned by h (Ex) = f (x) is injective. d. Show that f = h g.

2. Let f : A B. Define a relation R on A by rRy iff f (x) -f () a. Prove that R is an equivalence relation on A b. Let E, = {

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2. Let f : A ! B. DeÖne a relation R on A by xRy i§ f (x) = f (y). a. Prove that R is an equivalence relation on A. b. Let Ex = fy 2 A : xRyg be the equivalence class of x 2 A. DeÖne E = fEx : x 2 Ag...
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