Is the statement $(∀x ∈ Q)(∃y ∈ \Bbb Q)(x \cdot y ∈ \Bbb Z)$ true?

Question

If $x=3/2$ and $y= 2/3$ this is true, but if, for example, $x=7/2$, this is false $(21/4 ∉ Z)$. So this predicate sentence is not correct.

Is this method of proof good?

Answer 1

The important thing about statements like $(\forall x \in \mathbb{Q}) (\exists y \in \mathbb{Q}): x \times y \in \mathbb{Z}$ is that you can literally read them from left to right in order to understand what they say and how to prove/disprove them.

"For all $x$ in the rationals..." - so, we need to start with some arbitrary fraction $x$ which we have no control over - "... there exists $y$ in the rationals..." - next we get to pick any $y$ we choose (which may depend on $x$: we already have that available since it came earlier in the statement) - "... such that $x$ times $y$ is an integer." That is, finally we test the arbitrary $x$ we were given and the specific $y$ we chose to see if their product is an integer. If, for any possible $x$, we can pick a $y$ that makes it true, then the statement is true.

With that framework for the proof, can you show that the statement is true?