recursive definition of the relation

Question

Give a recursive definition of the relation greater than on N X N using the successor operators s?

I answered this question throw this way:

Basis: o ∈ N X N recursive step: if n ∈ N X N, then s(n) ∈ N X N

Can anyone plz help me out further with this question?

Answer 1

Hint: Let $R \subseteq \mathbb N \times \mathbb N$ be our desired relation. Then here are some examples of elements that we want to belong to $R$: \begin{align*} &(1,0),\\ &(2,0),(2,1),\\ &(3,0),(3,1),(3,2),\\ &(4,0),(4,1),(4,2),(4,3),\\ &~~~~~\vdots~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\ddots \end{align*}

This suggests that our basis is that $(1,0) \in R$. For the recursive step, you should think about how to answer the following two questions:

• How do we get to the next row?
• How do we get to the next pair in a given row, without going too far?