** 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?
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 $$(\varphi^{-1}(a_1, a_2, \ldots, a_n))(i)=a_i$$