We have used ℕ as the quintessentially denumerable set. What about some of our apparently bigger sets of numbers?

Consider ℤ, the set of integers. We can easily find a bijection

So ℤ is denumerable.

Perhaps it's easier to see what's going on here if we simply make a list of the numbers

0, 1, -1, 2, -2, 3, -3, ....

More generally, all we need in order to show that an infinite set is denumerable is a scheme for listing the elements of that set, in a sequence, such that every element of the set is guaranteed to appear somewhere in the list exactly once.

Consider ℤ, the set of integers. We can easily find a bijection

*f*: ℕ → ℤ defined by*f*(1) = 0;*f*(2*n*) =*n*for every*n*∈ ℕ;*f*(2*n*+ 1) = -*n*for every*n*∈ ℕ.So ℤ is denumerable.

Perhaps it's easier to see what's going on here if we simply make a list of the numbers

*f*(1),*f*(2), etc.:0, 1, -1, 2, -2, 3, -3, ....

More generally, all we need in order to show that an infinite set is denumerable is a scheme for listing the elements of that set, in a sequence, such that every element of the set is guaranteed to appear somewhere in the list exactly once.