(1) ∀x, y, z (x < y ∧ y < z → x < z) (transitivity)

(2) ∀x ¬(x < x) (antisymmetry)

(3) ∀x, y (x < y ∨ x = y ∨ y < x) (linearity)

I need to give an example of a (nonempty) structure with one binary relation that satisfies (1)and (2) but not (3), and in addition the formula ∀x ∃y (x < y). I was thinking naively of y = x + 1 but im certain this is wrong.

I’m essentially trying to find a structure where every X there is a bigger y.