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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!

Bill Dubuque
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omnipotxnce
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1 Answers1

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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.

Peter Smith
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  • okay thank you for that. Just quickly, if a rational is 2/3 away from an irrational, wouldnt that mean that the irrational is in fact, rational? Its a true/false question so if this is the case then it would be false. I hope that makes sense. – omnipotxnce Nov 12 '20 at 13:44
  • edit 2/3 away from a rational then that number must also be rational – omnipotxnce Nov 12 '20 at 13:51
  • Yes. When a number $x$ is at distance exactly 2/3 of a rational, then $x$ is rational itself. I guess that is why Peter was talking about 'more than 2/3'. – Vincent Nov 12 '20 at 15:08