Consider $(\mathbb N, +)$ as a model for the language with one symbol $+$ for a binary function. Are the following statements true?
- $(\mathbb N, +) \vDash \forall x \exists y \forall z\ x + y\neq z$
- $(\mathbb N, +) \vDash \exists x \forall y \exists z\ x + y\neq z$
I have put true for for both based mainly on intuition. I reasoned that the statements could be translated into English as (1) For any $x$ and $z$ you give me (from the natural numbers) there exists a $y$ such that $x + y\neq z$. Which I think is true. For the second one, I interpreted as for any y you give me, there exists an x and z such that $x + y\neq z$ is true, which I think is true as well. Is this the correct way to approach the problem?