Q91 Matemática  (IMO Longlists 1989)

Para $\varphi :\mathbb{N}\mapsto \mathbb{Z}$ vamos definir $$M_{\varphi }=\{f:\mathbb{N}\mapsto \mathbb{Z},f(x)>f(\varphi (x)),\forall x\in \mathbb{N}\}.$$ Prove que se $M_{{\varphi }_1}=M_{{\varphi }_2}\ne \varnothing ,$ então ${\varphi }_1={\varphi }_2.$ Esta propriedade permanece verdadeira se $$M_{\varphi }=\{f:\mathbb{N}\mapsto \mathbb{N},f(x)>f(\varphi (x)),\forall x\in \mathbb{N}\}?$$