Is There a Counterexample to the Function Composition Property?

[foxjwill]

Homework Statement

Let f: A\to B. I'm trying to find a function g: B\to C such that g is not 1-1 but g\circ f is.

The original assignment (which I've completed) was to prove that for all functions f: A\to B and g: B\to C, if g\circ f is 1-1, then so is f. However, in the process of completing the assignment, I tried (out of curiosity) to find g's that weren't 1-1. But I couldn't.

 

[Preno]

f: n -> (n,n)
g: (m,n) -> m
gf = identity, neither f nor g is one-to-one

 

[VeeEight]

Have you thought about why it may not be possible? If the composite is defined then g\circ f = g(f(x)) where x is in A. If f(x) is bijective then g(f(x)) = g(B)

 

[foxjwill]

f: n -> (n,n)
g: (m,n) -> m
gf = identity, neither f nor g is one-to-one

First of all, when specifying a function (and worrying about its properties), without the domain and codomain, you can't really tell much.

But assuming you have f: R -> R^2, and g: R^2 -> R, then f actually is one to one.

edit:
But since g isn't one-to-one, you have answered my question. So, thanks!

 

[Preno]

The domain is any arbitrary set A with more than 1 element, the codomain is A x A.