Originally Posted by

**MadSoulz** By my text's definition, a set $S$ is countable if there is a function $f$ which establishes a one-to-one correspondence (bijection) between $S$ and $\mathbb{N}$.

Could we say that, consequently, a set $A$ is countable if there is a function which establishes a one-to-one correspondence between $A$ and another countable set?

For example, say we want to show that $S_n = \left\{ \frac{m}{n} : m\in \mathbb{Z} \right\}$ is countable for each $n \in \mathbb{N}$. I know that the function $f : S_n \rightarrow \mathbb{N}$ defined by

$$f\left(\frac{m}{n}\right) = \begin{cases} 2m-1& m > 0 \\ -2m&m\leq 0 \end{cases}$$

is a bijection from $S_n$ to $\mathbb{N}$.

There's also a bijection between $S_n$ and $\mathbb{Z}$, the identity map (kinda). Could this be enough to deduce countability?