|a| = 2
  a^2 = 4

In modulo arithmetic (numbers wrap around like days of the week, months, hours in a day, or the units column, e.g. mod 10 has numbers 0 to 9, and 9+1=0) we can count them.

There are n^n possible functions (each of the n inputs could map to any n values), but only n! invertible functions (the first input can be to any of n, the second can be to any but the first's, etc (n)(n-1)(n-2)...(1)).

n!/n^n < 1/2 for all n>2

Does this converge to a fraction as n->infinity (i.e. for the naturals)? As a recurrence relation, we have:

n!/n^n = (n-1)!/(n-1)^(n-1) x n(n-1)^(n-1)/n^n

We could change variables to rewrite the factor as (n+1)n^n/(n+1)^(n+1) = n^n/(n+1)^n

Since n+1 converges to n, I think this converges to 1. Meaning that the whole thing converges to something - but working out what it converges to is beyond my abiility.

EDIT Wolfram says the limit is actually 0 https://www.wolframalpha.com/input/?i=n%21%2Fn%5En

  lim_(n->∞) n^(-n) n! = 0

So n!/n^n ~ sqrt(2 pi n)e^-n.