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(j) is countable whenever j is a positive integer.

Step 1:

In this problem, we have to 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(j) is countable.

Step 2:

The definition of the countable set:

Now a set S is countable if there exists an injective function f: SN, (where N is the set of the natural number).

Therefore a countable set is a set with the same cardinality (number of elements) as some subset of the set of the natural numbers.