Let A = {n♥| n ∈ Z}. Examples of lements of A are 0♥, −10♥ and 10♥. Show that A is countable by enumerating its elements and then, give a compact definition of your enumeration function.

Answer 1

The lelements can be enumerated as

0♥, -1♥, 1♥, -2♥, 2♥, -3♥, 3♥, ...

The enumeration function f(n) defined for natural numbers n:

f(n) = n/2 ♥ if n is even

f(n) = -(n+1)/2 ♥ if n is odd