Prove that $f: \{0, \dots , n \} \mapsto \{0, \dots , n \}$ is surjective $\iff$ is injective

Question

Prove that $f: \{0, \dots , n \} \mapsto \{0, \dots , n \}$ is surjective $\iff$ is injective

I have written a solution to this problem but I am not sure whether it's correct and formally satisfactory.

1. injective $\Rightarrow$ surjective
   Assume that this function is not surjective. Then, there is $y \in img(f)$ such that there is no $x \in dom(f)$ such that $f(x) =y$. And so, there are $x_1, x_2 \in dom(f)$ such that $f(x_1) = f(x_2)$ But $f$ is injective - contradiction.
   

2.surjective $\Rightarrow$ injective
Assume that $f$ is not injective. Then, there are $x_1, x_2 \in dom(f)$ such that $f(x_1) = f(x_2)$ This implies that there is a $y \in img(f)$ such that there is no$x \in dom(f) $ satisfying $f(x) = y$. However,$f$ is injective. Contradiction.

What do you think of my proof, is it valid?

Answer 1

None of them is correct. In the first proof, you don't justify why there is a $y$ for which there is no $x$ such that $f(x)=y$. In the second one, you don't justify the assertion that begins with “This implies that…”.

Note that at no point you used the fact that you are working with the set $\{0,1,\ldots,n\}$.

Answer 2

In (1), if $\;y\in\text{Im}\,(f)\;$ , then by definition $\;f(x)=y\;$ for some $\;x\in X\;$ . Even overseeing this, why then would we get that there are $\;x_1,x_2\in X\;$ s.t. ...etc.?

The same can be said about (2): not clear at all how you deduce stuff.

Here are a few ideas: Suppose $\;f:\{0,1,...,n\}\to\{0,1,...,n\}\;$ is injective . This means $\;\left|\{0,1,...,n\}\right|=;\left|\{f(0),f(1),...,f(n)\}\right|\;$ (why?), and thus every element $\;y\in\{0,1,...,n\}\;$ must be accounted for, which means that for any $\;y\in\{0,1,...,n\}\;$ there exists $\;x\in\{0,1,...,n\}\;$ s.t. $\;f(x)=y\;$ .

Now try to do by yourself the other direction.