Language $\mathcal L$ is defined as follows: $$\mathcal L = \{\{ f\}, \{P \}, \{ d\} \}$$ where $f$ is a binary function, $P$ is a binary predicate and $d$ is a constant. Consider this sentence $\phi$: $$(\forall x)(\exists y)((f(x,y),d)\in P)$$
Write two models $\mathbb M_1, \mathbb M_2$ such that $\mathbb M_1 \models \phi $ and $\mathbb M_2 \models \neg\phi$
I have managed to solve this but I am not sure whether or not my solution is correct - this problem appears to be trivial.
$$\mathbb M_1 = \{\mathbb N, \{+\}, \{\ge\}, \{0\} \}$$
And so we get $$(\forall x)(\exists y)(x+y \ge 0)$$
Which is of course true.
Now,
$$\mathbb M_2 = \{\mathbb N, \{+\}, \{\le\}, \{0\} \}$$
And now we get this:
$$(\forall x)(\exists y)(x+y \le 0)$$
Which does not work in natural numbers.
