Model and countermodel to $\exists x.\forall y. x<y$ (with $<$ an arbitrary relation)

Question

Can someone please help me with this question. I have been struggling with it for ages and can't quite seem to work it out:

Let $<$ be a binary relation symbol that we will write infix. Let $\phi = \exists x.\forall y.x<y$. Define structures $A,B$ such that $A \models \phi$ and $B \not\models \phi$.

I would be really grateful of any help.

Answer 1

HINT: The symbol chosen to represent the relation is a big hint. Suppose that $<$ really is, as the symbol suggests, some kind of order relation. Then $\varphi$ just says that there is a smallest element with respect to $<$. Can you think of sets $A$ and $B$ and relations on $A$ and $B$ for which we normally use the symbol $<$ or $\le$ such that $A$ does have a smallest element with respect to the, and $B$ does not?

Answer 2

Hint:

• You have no constraints on the relation whatsoever.
• You have no constraints on the universe either.
• In a universe of size $1$ the meaning of quantifiers is rather straightforward.
• There are only a few relations $ \subseteq \{\spadesuit\} \times \{\spadesuit\}$.

I hope this helps $\ddot\smile$