Rewriting a statement using Quantifiers

Question

I need to write the statement

The product of a rational number and an irrational number is irrational.

into a statement using quantifiers. I am also supposed to state the domain of discourse for both x and y. Here's what I have so far:

∀ xϵQ yϵR∖Q (irrational numbers) P((x*y)→y))

I doubt the last section with (x*y)=>y is correct. Is this statement accurate?

Answer 1

Let $\mathbb{R}$ be the domain of discourse.

Let $Q(t)$ be the statement that $t$ is rational.

Then the given statement can be fomulated as $$\forall x,y \;(Q(x) \land \lnot Q(y)) \rightarrow (\lnot Q(xy))$$

As a point of interest, the statement is actually false, since if $x=0$, then for any irrational $y$, the hypothesis will be true, but the conclusion will be false. However, the question was just to model the statement, not to analyze its truth value.

Answer 2

I suggest moving in smaller steps.

First, write an intermediate statement with quantifiers and some english language remaining:

• $\forall x \in Q$, $y \in R \setminus Q$, the product of $x$ and $y$ is irrational.

Next, translate into a little more math and a little less English:

• $\forall x \in Q$, $y \in R \setminus Q$, $x*y$ is irrational.

Next, translate into all math and no English:

• $\forall x \in Q$, $y \in R \setminus Q$, $x*y \in R \setminus Q$

So your doubt is on the mark.