I've been given this first order sentence with a binary relation symbol $R$:
$\forall x \exists y (R(x, y) \land \forall z(R(x, z) \implies (R(y, z) \land (y=z)) ) $
We are then given two structures of this sentence and asked if the structure models the sentence:
$S$ has domain the natural numbers and $R = \{(u, v) : u < v\}$
$S$ has domain the rational numbers and $R = \{(u, v) : u < v\}$
As these are two separate questions, I would expect the answer to change. But it appears that neither of these structures model the equation, as $R(y, z) \land (y=z)$ cannot be true for any z and y. Am I missing something obvious here?
Apparently, the formula tries to express the existence of a supreme and it's unicity (hence the $y=z$ consequent). Since that kind of number does not exist neither in $\mathbb{N}$ nor in $\mathbb{Q}$, then you can be sure that neither $S_1$ nor $S_2$ model the formula.