Problem 7.4.17 - Question 2 Must f be onto Answer 2 No...

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INFS 501 Problem 7.4.17 Suppose f: X Y and g : Y Z are functions and g ° f is onto. Question 1 : Must g be onto? Answer 1: Yes. Proof : Choose any z Z. Since g ° f is onto, we can find x X such that g ° f x  = z. This makes f x  ∈ Y the pre-image we need under g because, g(f(x)) = z. Since z was arbitrary, we've proven g is onto.
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Unformatted text preview: Question 2 : Must f be onto? Answer 2: No. Counterexample : X = { 1 } , Y = ℝ ,Z = { 1 } . Define f: X ℝ by f x = 1 ∀ x ∈{ 1 } and define g: ℝ Z by g y = 1 ∀ y ∈ ℝ . Here g ° f x = 1, and g ° f: { 1 }{ 1 } is onto, but f: { 1 } ℝ is not onto....
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