Homework 7 Solutions

Homework 7 Solutions - ⇒ Assume that f X → Y is...

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

View Full Document Right Arrow Icon
MATH 74: HOMEWORK 7 COMMENTS (1) Let x 1 , x 2 X , and suppose that g f ( x 1 ) = g f ( x 2 ), i.e., g ( f ( x 1 )) = g ( f ( x 2 )). Since g : Y Z is injective, it follows that f ( x 1 ) = f ( x 2 ). Since f : X Y is injective, it follows that x 1 = x 2 . Hence g f : X Z is injective. (2) ( ) Assume that f : X Y is surjective. We must show that for all y Y the set G f X × { y } is not empty. Let y Y . Since f is surjective there exists x X such that f ( x ) = y . Thus the point ( x, y ) X × Y is an element of G f . Since ( x, y ) X × { y } as well, we have ( x, y ) G f X × { y } . Thus the set G f X × { y } is not empty (we just exhibited an element of it!). ( ) Now suppose that for all y Y , the set G f X × { y } is not empty. Let y Y . Since G f X × { y } 6 = we know ( x, y ) G f X × { y } . In particular ( x, y ) G f , which means y = f ( x ). Hence x is a preimage of y . Book Problem # 18 Let z Z . Since g : Y Z is surjective there exists y Y such that g ( y ) = z . Since f : X Y is surjective there exists x X such that f ( x ) = y . Hence g ( f ( x )) = z , which shows that x is a preimage of z . Book Problem # 19
Image of page 1
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: ( ⇒ ) Assume that f : X → Y is surjective. Define a function g : Y → X by sending y to any preimage x of y under f . Note that x exists because f is surjective. We claim that g is a right inverse of f , i.e., f ◦ g = I Y . We must check that f ◦ g ( y ) = I Y ( y ) for all y ∈ Y . Now g ( y ) = x , where x satisfies f ( x ) = y (by definition of g ). Hence f ◦ g ( y ) = f ( g ( y )) = f ( x ) = y = I Y ( y ) . ( ⇐ ) Assume there exists g : Y → X such that f ◦ g = I Y . We must show that for all y ∈ Y there exists x ∈ X such that f ( x ) = y . Let y ∈ Y , and set x = g ( y ). Then f ( x ) = f ( g ( y )) = f ◦ g ( y ) = I Y ( y ) = y, and hence x is a preimage for y . Date : March 13th, 2008. 1...
View Full Document

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    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.

    Student Picture

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

  • Left Quote Icon

    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.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    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.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern