Translating plain english to math-ish using quantifiers and modulus

Question

There is a rational number and an irrational number which are 2/3 apart from each other. So far I have :

$$\forall \delta >0.\exists x\in\mathbb{Q}. \exists y\in\left(\mathbb{R}/\mathbb{Q}. | x-y| > \frac{2}{3}\delta\right)$$

I'm not too sure on the $2/3\ \delta$ part, any help is appreciated.

Thanks in advance!

Answer 1

You don't need that initial universal quantifier. What you write says that for any $\delta$ there is a rational and an irrational more that $2/3\delta$ apart -- a much stronger claim!

∃x ε ℚ, ∃y ε (ℝ \ ℚ) s.t. |x - y| > 2/3

is what you want to render the claim that there is a rational and an irrational whose difference is more than 2/3. Or at least, that works in the sort of syntax you are using.