I need help proving the following:

Let f: A--> B and g: B--> C be functions. Prove that if gf is injective, then f is injective. Also, prove that if gf is surjective, then g is surjective.

2. Originally Posted by page929
I need help proving the following:
Let f: A--> B and g: B--> C be functions. Prove that if gf is injective, then f is injective. Also, prove that if gf is surjective, then g is surjective.
I see that you have nine postings.
By now you should understand that this is not a homework service
So you need to either post some of your work on a problem or explain what you do not understand about the question.

3. I understand that g◦f is g(f(x)). So if g◦f is injective then every element of A matches to an element of C. Then every element of A has to match to B and therefore f is injective. And for the other part, since g◦f is surjective then every element in C is the image of something in A. Then g is surjective since every element in C is the image of something in B and every element in B is the image of something in A.

I don't know if this will make any since.

4. Start this way. If $\displaystyle f(a)=f(b)$ can you use the given to show that $\displaystyle a=b?$
That shows f is injective.

What does $\displaystyle g\circ f(a)=g\circ f(b)$ imply and WHY?