Suppose that f : A → B. Define a relation R on A by xRy iff f(x) = f(y).
(a) Prove that R is an equivalence relation on A.
(b) For any x ∈ A, let Ex be the equivalence class of x. That is, Ex = {y ∈ A : yRx}. Let E be the collection of all equivalence classes. That is, E = { Ex : x ∈ A }. Prove that the function g: A→E defined by g(x) = Ex is surjective.
(c) Prove that the function h : E → B defined by h(Ex) = f(x) is injective.
(d) Prove that f = h ° g. [That is, f(x) = h(g(x)) for all x ∈ A.] Thus we conclude that any function can be written as the composition of a surjective function and an injective function.
(e) Let A be the set of all students in the school. Define f : A → [0, 200] by "f(x) is the age of x." Describe the functions h and g as given above.
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