Hello,

Here's a relation defined on the set $\displaystyle \mathbb{N}^{ \mathbb{N}}$.

$\displaystyle f$ is in relation with $\displaystyle g$ when

$\displaystyle \exists_{c>0} \exists_{k>0} \forall_{n>k}(f(n) \le c*g(n) \wedge g(n) \le c*f(n))$

1)What's the cardinality of the quotient set for $\displaystyle f$ such that $\displaystyle f(n)=0$ for every $\displaystyle n \in \mathbb{N}$

In order for $\displaystyle g$ to be in relation with $\displaystyle f$ it has to always equal $\displaystyle 0$ after some time..

1)What's the cardinality of the quotient set for$\displaystyle f$ such that $\displaystyle f(n)=1$ for every$\displaystyle n \in \mathbb{N}$

For $\displaystyle g$ to be in relation with $\displaystyle f$ after some time it can't go upwards from some chosen $\displaystyle c$.

I have literally no idea how to even begin assigning cardinal number to those quotient sets. Could someone provide me a hint?