Definable relations in given structure (first order logic)

Question

I'm bothering with this problem. I'm given a first order predicate language with only one ternary predicate symbol $p$ (no equality sign). Also there is structure for the language $\mathcal{A}$ over $\mathbb{N}$ ($0 \in \mathbb{N}$), where $p(x,y,z) \leftrightarrow x+y+1 = z$. The problem is to prove that the set $M = \{ (a,b) \in \mathbb{N} \times \mathbb{N} : a \leq b\}$ is definable in $\mathcal{A}$. I feel that the set $N = \{(a, b) \in \mathbb{N} \times \mathbb{N} : a < b\}$ is definable by the formula $\psi = \exists x \ p(x,y,z)$. However I don't know how to prove it and how to switch to the set $M$. I was thinking of $\neg \psi$, but I guess it will define the set $\{(a,b) \in \mathbb{N} \times \mathbb{N} : a \geq b\}$. Any help?

Answer 1

You are correct that $\exists x: p(x,y,z)$ defines $N = \{(a,b): a < b\}$, since $0 \in \Bbb N$. In order to prove this, you demonstrate (to a satisfactory level of detail) that the conditions $\exists x: p(x, a, b)$ and $a < b$ are equivalent.

Now what do you know about $a, b \in \Bbb N$ if $a \not< b$ and $b \not< a$?