Let $\displaystyle A,B$ be sets and let $\displaystyle f:A \rightarrow B$ and $\displaystyle g,h:B \rightarrow A$ be functions.

(1) Suppose $\displaystyle h o f$ is an injective map from $\displaystyle A$ to itself. Show that $\displaystyle f$ is injective.

(2) Suppose now that $\displaystyle f o g = 1_{B}$ and $\displaystyle hof = 1_{A}$. Show that $\displaystyle f$ is bijective and $\displaystyle g=h$.

My attempt:

(1) $\displaystyle h o f$ is 1-1 $\displaystyle \Leftrightarrow \forall a,a' \in A$ such that $\displaystyle a \neq a'$, $\displaystyle h o f(a) \neq h o f(a') $

$\displaystyle \Leftrightarrow h(f(a)) \neq h(f(a'))$

$\displaystyle \Leftrightarrow f(a) \neq f(a')$

It follows that $\displaystyle a \neq a'$. Is this proof correct? Do I need to add any more explanation?

(2) I need some help with question two, so I can get started. Btw, the $\displaystyle 1_A$ and $\displaystyle 1_B$ notation represents the identity.

Thanks in advance,