Satisfiable formula but false in any structure

Question

Exhibit a formula with no free variables that is satisfiable, but false in any structure whose universe has fewer than three elements.

I've thought about this for a while and I can't think of anything, any hint? :(

Answer 1

The following formula is satisfiable in any structure (assuming the logical identity predicate $=$ is in the language) whose universe has fewer than three elements:

$\exists x\,\exists y\,\forall z\,(z = x \vee z = y)$

Then, its negation is false when the universe has fewer than three elements:

$\neg\exists x\,\exists y\,\forall z\,(z = x \vee z = y)$