Question 7
Let $S \subseteq \mathbb{R}$ be a set of real numbers. We say that a pair $(f, g)$ of functions from $S$ into $S$ is a Spanish Couple on $S$, if they satisfy the following conditions:
(i) Both functions are strictly increasing, i.e. $f(x) \lt f(y)$ and $g(x) \lt g(y)$ for all $x, y \in S$ with $x \lt y$;
(ii) The inequality $f(g(g(x))) \lt g(f(x))$ holds for all $x \in S$.
Decide whether there exists a Spanish Couple
a. on the set $S = \mathbb{N}$ of positive integers;
b. on the set $S = \{ a - 1 / b : a, b \in \mathbb{N} \}$.
(i) Both functions are strictly increasing, i.e. $f(x) \lt f(y)$ and $g(x) \lt g(y)$ for all $x, y \in S$ with $x \lt y$;
(ii) The inequality $f(g(g(x))) \lt g(f(x))$ holds for all $x \in S$.
Decide whether there exists a Spanish Couple
a. on the set $S = \mathbb{N}$ of positive integers;
b. on the set $S = \{ a - 1 / b : a, b \in \mathbb{N} \}$.