Let $X=\{2, 3, 4, \ldots\}$ and endow $X\times X$ with the order $$(x, y)\leq (z, t)\Leftrightarrow x|z\ \textrm{and}\ t|y.$$
I'm supposed to find the minimal and maximal elements of $X\times X$ with respect to this order.
Conjecture. There are no minimal and no maximal elements.
Proof attempt: I'm going to show that no pair $(p, q)$ can be a minimal element. In fact, notice:
$$(p, 2q)\leq (p, q)$$ since $p|p$ and $q|2q$. But, $2q\neq q$ once $q$ is a positive integer. Therefore, given $(p, q)$ there is always a pair which preceeds $(p, q)$ (namely, $(p, 2q)$).
Analogously, there are no maximal elements. In fact, no pair $(p, q)$ can be a maximal element for:
$$(p, q)\leq (2p, q)$$ and $2p\neq p$ once $p$ is a positive integer.
Are my arguments correct?