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Give a direct proof of $(i),(ii)$ and $(iii)$. $$(i)\qquad \forall\,x\in\Bbb Q\quad\exists\,y\in\Bbb Q : x^2=y$$ Let $$x=\frac pq\,\in\Bbb Q:\,p,q\in \Bbb Z \wedge q \not = 0$$ $$x^2=\left(\frac pq\right)\left(\frac pq\right)=\frac {p^2}{q^2} =y $$

$$\therefore \quad y=\frac {p^2}{q^2}\in\Bbb Q$$

Then, $$(ii)\qquad\forall\,x,y\in\Bbb Q\quad\exists\,z\in\Bbb Q : x+y=z$$

Let $$x=\frac {p}{q}\,\in\Bbb Q:\,p,q\in \Bbb Z \wedge q \not = 0$$ $$y=\frac {r}{s}\,\in\Bbb Q:\,r,s\in \Bbb Z\wedge s \not = 0$$ $$x+y =\frac {p}{q}+ \frac {r}{s} = \frac {sp+qr}{qs}=z$$ $$\therefore \quad z= \frac {sp+qr}{qs}\in\Bbb Q \iff p, q, r, s \in \Bbb Z$$

Are these two proofs correct?

Also, By giving a proof of both of these statements, does it also prove that $$(iii) \qquad \forall\,x,y\in\Bbb Q\quad\exists\,z\in\Bbb Q : x^2+y^2=z$$ is true and is the following $$((i)\wedge (ii))\Rightarrow (iii)$$ the correct way to express such a statement in propositional logic terms?

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    The proofs are essentially correct, but you are missing specifying that $q$ and $s$ must be nonzero. – 2017-01-26
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    The implication $p^2/q^2\in\mathbb Q\implies p,q\in\mathbb Z$ does not hold. But it is not required in the proof. – 2017-01-26
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    @Yves Daoust Should I just take it off and only leave $ \therefore y=\frac {p^2}{q^2} \in \Bbb Q$ ? – 2017-01-26
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    @Hermitian: that's it. – 2017-01-26
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    In your proof of (i) you should include the statement that $p^2$ and $q^2$ are integers with $q^2\ne 0,$ in order to conclude that $p^2/q^2 $ is rational. – 2017-01-27

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I would prefer

$$x\in\mathbb Q\implies\exists p,q\in\mathbb Z:x=\frac pq\implies y=x^2=\frac{p^2}{q^2}\in\mathbb Q.$$

$$x,y\in\mathbb Q\implies\exists p,q,r,s\in\mathbb Z:x=\frac pq\land y=\frac rs\implies z=x+y=\frac{ps+qr}{qs}\in\mathbb Q.$$

$$x,y\in\mathbb Q\implies\exists p,q,r,s\in\mathbb Z:x=\frac pq\land y=\frac rs\implies z=x^2+y^2=\frac{p^2s^2+q^2r^2}{q^2s^2}\in\mathbb Q$$ (or a combination of $i$, $ii$).

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    @TheStrikeofRunningWater: state your question clearly in the post. – 2017-01-27