Problem 7.4.17

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

This preview shows page 1. Sign up to view the full content.

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.
This is the end of the preview. Sign up to access the rest of the document.

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....
View Full Document

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern