Question

Show that the set S is a countable set if there is a function f from S to the positive integers such that$f^{-1}\left(j\right)$ is countable whenever j is a positive integer.

Short answer

The set S is a countable

Step-by-step solution

Step: 1

We know that countable union of countable set is countable.

Step: 2

If each $f^{-1}\left(j\right)$is countable, then$S=f^{-1}\left(1\right)\cup f^{-1}\left(2\right)\cup \dots$ is the countable union of countable sets and is therefore countable.