# Homework Help: Cancellation law with surjective functions

1. Sep 9, 2010

### annoymage

1. The problem statement, all variables and given/known data

suppose function $$f : A \to B$$, $$g: A \to B$$, $$h : B \to C$$ satisfy $$g \circ f=h \circ f$$. If is surjective then prove that $$g=h$$

2. Relevant equations

n/a

3. The attempt at a solution

so for any $$x \in A$$, $$gf(x)=hf(x)$$, and for any $$b \in B$$ there exist $$a \in A$$, such that $$f(b)=a$$

so $$g(a)=h(a)$$ so $$g=h$$ is this correct and sufficient?

i'm suppose to to show for any $$v \in B$$, $$g(v)=h(v)$$. i don't know but something's missing.

Last edited: Sep 9, 2010
2. Sep 9, 2010

### vela

Staff Emeritus
Re: function

You need to clean it up a bit. For one thing, you have g mapping A to B. You meant B to C, right? Also, the element a in in A, but h is defined on B, so when you say h(a), it's not clear that h(a) is defined.