1 The naturals
In this chapter we give some basic definitions and lemmas about the natural numbers.
The set of naturals, denoted \(\mathbb {N}\), is the set of numbers \(\{ 0, 1, 2, \ldots \} \).
Let \(n\) be a natural number. Then \(n = n + 0\).
Proof
Let \(n\) and \(m\) be natural numbers. Then \((n + m) + 1 = n + (m + 1)\).
Proof
Addition is commutative, i.e. \(n+m = m+n\) for all \(m, n\in \mathbb {N}\).
Proof
Let \(n\) be a natural number. Then \(2*n = n + 2\).
Proof
The proof is trivial and left as an exercise to the reader.