Prove, that predicate is inexpressible in the given signature

Question

I have a predicate $y=x+1$. I want to prove, that this predicate is inexpressible in $(\mathbb{Z}, {=}, f)$, where $f = x\mapsto(x+2)$.

I understand, that I need to come up some automorphism, in which this predicate is wrong. But I don't have succeed.

Answer 1

Let $$h(n)=\begin{cases} n, & n\text{ is odd} \\ n+2, & n\text{ is even} \end{cases}$$

Then $h$ is a structure isomorphism from $(\mathbb Z,{=},f)$ to itself.

Suppose now that you have a wff $\varphi(x,y)$ that expresses your predicate. Then, in particular $(\mathbb Z,{=},f)\vDash_{x=2,y=3} \varphi$. But due to the isomorphism we then also have that $(\mathbb Z,{=},f)\vDash_{x=h(2),y=h(3)} \varphi$ -- in other words, $\varphi(4,3)$ is true in $\mathbb Z$. So $\varphi$ doesn't in fact express your predicate, a contradiction.