I am trying to find image of $(x,y)\mapsto \max(x,y)$ and $(x,y)\mapsto\min(x,y)$, defined as follows:
$\Bbb R^2\to\Bbb R:\quad \max(x,y) = \begin{cases} x \text{ if } x \ge y, \\ y \text{ otherwise};\end{cases}$
$\Bbb N^2\to\Bbb R :\quad \min(x,y) = \begin{cases} y \text{ if } x \ge y, \\ x \text{ otherwise}.\end{cases}$
I am not quite sure where to start.
Basically, if I take $\max(x,y)$, I am trying to prove that $S=T$, where $S$ is the respective image, and $T$ for $\max(x,y)$ is: $z = x$ for $z\in T$, if $x \ge y$ and $z = y$ for $z\in T$, if $x < y$.
So, $S\subseteq T$. But how do I prove that all elements in $T$ are in $S$? Thanks for any pointers!