I study model theory and I have questions about relations which are definable in a structure or not. I found three examples from exercises and i want to do them:
Is the relation $<$ on $\Bbb{Q}$ definable in the structure $(\Bbb{Q},+,\cdot,0,1)$ that is does there exists a formula $\phi=\phi(x_0,x_1)$ sucht that for all $p,q$ in $\Bbb{Q}$, $p if and only if $(\Bbb{Q},+,\cdot,0,1)$ realized $\phi[p,q]$ ?
Is the relation $<$ on $\Bbb{Q}$ definable in the structure $(\Bbb{Q},+,0,1)$ ?
Is the relation $+$ on $\Bbb{Q}$ definable in the structure $(\Bbb{Q},<,0,1)$ ?
I have done this already for the integers with the successor function, but I don't know how to do this in this three cases. I think the first relation is definable, but the other two not. Can someone help me? Thank you :)