When can we use the equality and the inequality symbol as a relation between functions ?
Consider $f(x)=4$ and $g(x)=1$, then we can easily say $f(x)>g(x)$.
Let $F(x)=2x$ and $G(x)=x$ where both $F(x)$ and $G(x)$ have the domain
$[1, \infty)$ then $F(x) > G(x)$.
If I change the domain of $F(x)$ to $[1,10]$ and $G(x)$ to $[21,\infty)$ then I can say $G(x) > F(x)$. I can replace the functions and write $x>2x$. But this implies $1>2$ which is not true. An easy way of solving this can be to let $G(x)=y$, where $x=y$. But this looks like we are cheating.
So in order to use the inequality sign or even the equality sign in some cases as a relation between two functions, do we need both functions to have the same domain?
Besides that, all the output obtained from $G(x)$ will be greater than what we will get from $F(x)$ and hence is writing $G(x) > F(x)$ valid?
The collection of functions F = { f:(X,<=) -> (Y,<=') }
from an ordered domain into an ordered codomain is
ordered by f <= g when for all x in X, f(x) <= g(x).
Such functions are called order preserving.
Of course this order is not linear, nor do
the orders of X and Y need be linear.
Thusly f < g when f <= g and exists x with f(x) < g(y).
One may like to use
f << g when for all x in X, f(x) < g(x).