1. ## Surjective/Injective Proofs

I am having difficulties understanding how to build proofs and my instructor only puts some of them on the board, but never explains how he knows what to do to work proofs. I have an idea of how to build the proofs I am asking a question on, but am unsure how much information needs to be included. Please let me know how to build the following proof.

Let f: A--->B and g: B--->A be functions where A & B are given sets. Prove that:

a) f is injective if (g o f)(a)=a for all a elements in set A.
b) f is surjective if (f o g)(b)=b for all b elements in set B.

2. Originally Posted by s-sleedom2
I am having difficulties understanding how to build proofs and my instructor only puts some of them on the board, but never explains how he knows what to do to work proofs. I have an idea of how to build the proofs I am asking a question on, but am unsure how much information needs to be included. Please let me know how to build the following proof.

Let f: A--->B and g: B--->A be functions where A & B are given sets. Prove that:

a) f is injective if (g o f)(a)=a for all a elements in set A.
b) f is surjective if (f o g)(b)=b for all b elements in set B.
I start you out on (a), to show f is injective we need to show if $f(a) = f(b)$ then $a=b$ for all $a,b\in A$.
Say $f(a) = f(b) \implies g(f(a)) = g(f(b)) \implies a = b$.
Thus, $f$ is injective.