Suppose that $S$ is a set of functions such that $f \in S$ if and only if $f$ is either multiplicative or completely multiplicative and every $f \in S$ is defined on the set of natural numbers and takes natural numbers as its values ($f: \mathbb N \to \mathbb N$).
Recall that the function $f$ is completely multiplicative if, for every integers $a$ and $b$ we have $f(ab)=f(a)f(b)$ and $f(1)=1$. Such functions exist, one example is Liouville function.
Recall that the function $f$ is multiplicative if $f(1)=1$ and $f(ab)=f(a)f(b)$ for every coprime integers $a$ and $b$. An example of such a function is Euler`s totient function.
Only curiosity led me to the question of "how big" is the set of either multiplicative or completely multiplicative functions defined on the set $\mathbb N$ and which take values in the set $\mathbb N$.
It is clear that the set $S$ is at least countable because for every $n \in \mathbb N$ the function $f(a)=a^n$ is completely multiplicative.
So the real question is:
Is $S$ uncountable?