Question: I am currently using Rosen's "Discrete Mathematics and Its Applications" (7th ed.) for my discrete mathematics course. We recently talked about quantifiers, more specifically the universal and existential quantifier. What confuses me is when we started talking about restricting the domain (i.e. instead of all real numbers, only all real positive numbers), where does the conditional statement come from when restricting the domain of a universal quantifier and where does the conjunction come from when restricting the domain of a existential quantifier?

This link did help: [Universal and Existential quantifier in Propositional logic](https://math.stackexchange.com/questions/1845630/universal-and-existential-quantifier-in-propositional-logic)

However, the part that confuses here is how are the domains restricted in the answer given in the link above?

Answer: 
What about this proof ?

Assume $x=\frac{p}{q}$ with coprime $p$ and $q$ (If $x$ is no integer, we can always find such a representation, otherwise nothing is to prove). Assume $x^2=\frac{p^2}{q^2}$ is a natural number. 

If $p^2$ and $q^2$ were not coprime, there would be a prime $s$ with $s|p^2$ and $s|q^2$. 

We could include $s|p$ and $s|q$, hence $p$ and $q$ would not be coprime , but this contradicts our assumption.

So, $p^2$ and $q^2$ are coprime, hence we can conclude $q^2=1$. 

This completes the proof.
Let $x^2 = n \in \mathbb{N}\,$, then $x$ is a root of $x^2-n=0$. Since $x^2-n$ is a monic polynomial, any rational root must be an integer by the [rational root theorem](https://en.wikipedia.org/wiki/Rational_root_theorem), so $x$ is an integer.