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

[EV33]

Homework Statement

Assume f:A$$\rightarrow$$B
g:B$$\rightarrow$$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$$\rightarrow$$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 $$\Re$$ to $$\Re$$
g(b)=1/b from $$\Re$$ to $$\Re$$
h(a)=a from $$\Re$$ to $$\Re$$

h is a surjection and f is not.

Does this work?

 

[vela]

No, h(0) isn't defined since f(0) isn't defined. You can only say h(a)=a for a≠0.

 

[Dick]

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.

 

[EV33]

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

f(a)=a² $$\Re$$ $$\rightarrow$$ [0,$$\infty$$)
g(b)=b^3/2 [0,$$\infty$$) $$\rightarrow$$ [0,$$\infty$$)
h(a)=a³ $$\Re$$ $$\rightarrow$$ [0,$$\infty$$)

 

[EV33]

oh ok thanks dick I will try that