To prove whether the given set is countable or not

Question

The function $f(k) = \mathbb{N}^k$ is a one to one function but it is not an onto function. Hence the function is not a one to one correspondence function.

Does that give sufficient justification to proof that the given set in not countable? Or is there any other way to approach the same.

Answer 1

$N^k$ is a set.

You are suppose to show whether a bijection between $\mathbb{N}$ and $\mathbb{N}^k$ exists.

Guide:

You might want to first verify whether the following is true.

If $A$ is countable and $B$ is countable, then their cartesian product $A \times B$ is countable and then you might want to consider using induction.

Answer 2

Apparently $N$ here is the set of all integers, while $k$ is a particular integer. What do you mean by $f(k) = N^k$?

In any case, one particular function failing to be a one-to-one correspondence is not grounds to conclude that there is no one-to-one correspondence.