• Support PF! Buy your school textbooks, materials and every day products Here!

If g o f is surjective, then is f surjective?

  • Thread starter EV33
  • Start date
  • #1
196
0

Homework Statement


Assume f:A[tex]\rightarrow[/tex]B
g:B[tex]\rightarrow[/tex]C
h=g(f(a))=c
Give a counterexample to the following statement. If h is surjective, then f is surjective.



Homework Equations


Definition ofSurjection: Assume f:A[tex]\rightarrow[/tex]B, For all b in B there is an a in A such that f(a)=b


The Attempt at a Solution


f(a)=1/a from [tex]\Re[/tex] to [tex]\Re[/tex]
g(b)=1/b from [tex]\Re[/tex] to [tex]\Re[/tex]
h(a)=a from [tex]\Re[/tex] to [tex]\Re[/tex]

h is a surjection and f is not.

Does this work?
 

Answers and Replies

  • #2
vela
Staff Emeritus
Science Advisor
Homework Helper
Education Advisor
14,631
1,267


No, h(0) isn't defined since f(0) isn't defined. You can only say h(a)=a for a≠0.
 
  • #3
Dick
Science Advisor
Homework Helper
26,258
618


Don't try and make life complicated. You can find a counterexample with finite sets. Take A={1,2}, B={1,2} and C={1}. Now define f and g.
 
  • #4
196
0


It is always little things like that which I don't see with these problems. Does this one work?

f(a)=a2 [tex]\Re[/tex] [tex]\rightarrow[/tex] [0,[tex]\infty[/tex])
g(b)=b3/2 [0,[tex]\infty[/tex]) [tex]\rightarrow[/tex] [0,[tex]\infty[/tex])
h(a)=a3 [tex]\Re[/tex] [tex]\rightarrow[/tex] [0,[tex]\infty[/tex])
 
  • #5
196
0


oh ok thanks dick I will try that
 

Related Threads on If g o f is surjective, then is f surjective?

Replies
24
Views
2K
  • Last Post
Replies
1
Views
1K
Replies
5
Views
1K
  • Last Post
Replies
5
Views
1K
Replies
2
Views
796
Replies
11
Views
6K
Replies
2
Views
3K
  • Last Post
Replies
5
Views
4K
Top