This question is from Enderton's mathematical logic. Question 8 sec 2.5 pg 146. It says assume the language that has $\forall$ and P, where P is a two place predicate symbol. Let A be the structure with |A|=$\mathbb{Z}$ and $\langle a, b \rangle\in P^A$ iff |a-b|=1. Show there is an elementarily equivalent structure B that is not connected.
Okay so for this problem I took a strategy similar to constructing the non-standard reals so I just wanted to see how valid it was. Take TH(A)$\cup\{\langle v_1, v_2 \rangle\in P^A\wedge\forall x\langle x,v_1 \rangle\notin P^A\wedge\forall x\langle x, v_2 \rangle\notin P^A\}$. Since this is finitely satisfiable, ie take a finite interval [a,b] then there are 2 integers outside of it not connected by P in the interval. Thus by the compactness theorem, there is an infinite structure B satisfying TH(A) implying B is elementarily equivalent to A, and a variable assignment s such that $v_1,v_2$ satisfy the accompanying formula.Thus B is also not connected.