As long as the bijection exists at all, the sets have the same cardinality. To prove two sets don't have the same cardinality, the most direct way is to prove t

As long as the bijection exists at all, the sets have the same cardinality. To prove two sets don't have the same cardinality, the most direct way is to prove that such a bijection is impossible. Anyway, the bijection from counting numbers to the set of all natural numbers is easy: 1, 2, 3, 4, 5, ... 0, 1, -1, 2, -2, ...