Formula in a signature

Question

Construct a formula in a signature without equality, which has a 3-element model, but don't has 2-element models.

Have no idea how I can do it .

Answer 1

We can state (any of) the properties of equality for an arbitrary binary relation symbol $E$.
Specifically for the problem, consider the following formula: $$(\forall x: xEx)\ \land \ (\exists x,y,z: \lnot xEy\,\land\, \lnot xEz\,\land \,\lnot yEz)$$

Answer 2

Hint: see if you can formalise this: Alice has a key for the red door, but not for the green or the blue door, Bob has a key for the green door, but not for the red or the blue door, Charlie has a key for the blue door, but not for the red or the green door.