Show the only definable subsets of $ \langle \mathbb{Q} , < \rangle $are the empty set and $ \mathbb{Q} $

Question

I'm trying to show the above, and I know it should be quite simple, but I'm struggling to get my head around how you show subsets are not definable using automorphisms- so a detailed explanation would be appreciated- thanks!

Answer 1

Assume $A\subseteq \Bbb Q$ is definable and $a\in A, b\notin A$. Then consider $\Bbb Q\to \Bbb Q$, $x\mapsto x+b-a$.