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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 :)

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    @Brian: How does your formula show that $0 < 2$ in $\langle \mathbb{Q} , + , \cdot \rangle$?2012-12-01
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    @Arthur: Isn’t $\sqrt 2$ rational in your world? :-) Clearly my subconscious knew what it was doing when it had me write *appears to define*!2012-12-01
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    @Brian: $\sqrt{2}$ might be irrational in my world, but after spending five days enjoying _pivo_ in Prague my world is far from rational.2012-12-01
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    @Arthur: But such a splendid irrationality!2012-12-01
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    Well, in $(\Bbb Z,+,\cdot)$ it is definable by using the fact that every positive integer can be written as the sum of $4$ squares. This can be extended to the rationals, too ($\exists$ pos.integer $b$: $\ x\cdot b$ is pos.integer) $\iff x>0$).2012-12-01

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