Let $\mathcal{L}=\{\leq\}$ and $\mathcal{A}=\{\mathbb{N},\leq\}$, and $\mathcal{B}=\{K,\leq\}$ where $K=\{(a,b)\in\mathbb{N}^2:a\in\mathbb{N} \text{ and } b=1 \text{ or } b=2\}$ and pairs $(a,b)$ have the natural ordering on themselves but pairs $(a,2)$ are strictly greater then all pairs of the form $(a,1)$
Show that $\mathcal{A}\not\equiv\mathcal{B}$
So I believe what I want is a sentence which is true in one structure but not true in the other structure.
I can obviously identify some things that are properties of the naturals that won't be properties of the set $K$ like every natural has only finitely many elements less then or equal to it. But not ones that can be expressed in a single sentence.
Exactly one element of $\mathcal{A}$ has no predecessors, while two distinct elements of $\mathcal{B}$ have no predecessors. See if you can write $\text{isPred}(x,y)$ and use this to distinguish the structures.
I hope this helps ^_^