Prove that $|\Bbb N^{J_n}|=|\Bbb N|$.

Question

** Definition: ** Let $ J_n = \{1,2,3, \ldots, n \} $. We say that a set is finite if its cardinality is equal to that of $ J_n $.

To demonstrate that I gave the following function. Let $ \varphi: \Bbb N ^ {J_n} \to \Bbb N ^ n $ defined as follows: for $ f \in \Bbb N ^ {J_n} $, let $$ \varphi (f) = (f (1), f (2), \ldots, f (n)). $$ I already proved that this function is bijective. But I'm getting a bit out of the exercise, and I want to know what the inverse of this function is, and how to get it out. Can someone help me with that?

Answer 1

It depends on how you write the elements of $\mathbb{N}^n$. Based on how you've defined $\varphi$, though, the inverse would be defined pointwise as $$(\varphi^{-1}(a_1, a_2, \ldots, a_n))(i)=a_i$$