I am trying to solve Exercise 3.4.3 from Marker's Model Theory: An introduction which states
a) Show that the Theory of $(\mathbb{Z},s)$ has quantifier elimination, where $s(x)=x+1$. Show that this theory is strongly minimal and that $acl(A)$ is the set of elements "reachable" from A.
b) Show that the theory of $(\mathbb{N},s)$ does not have quantifier elimination.
First off, I'm not sure I understand what the language is. Can we assume that the language includes $+$, as a function, and $1$ as a constant? In that case every element is definable.
My idea to prove this is to show that for every $\exists x\varphi(\bar{y},x) $-formula, where $\varphi$ is a conjunction of atomic formulas, is equivalent to a quantifier-free formula (modulo the theory). I think that by these formulas it is only possible to denote a specific integer, or an element of a cofinite set. If this is true, the theory is strongly minimal but I'm not sure how to show the last claim nor if my interpretation of the question is correct.