query: Which set is more dense: set of irrational numbers or set of rational numbers?
positive_passages.docid: 152512
positive_passages.text: What is the ratio of rational to irrational real numbers?
negative_passages.docid: 77129
negative_passages.text: Density of irrationals
negative_passages.docid: 142583
negative_passages.text: Subsets of the Cantor set
negative_passages.docid: 68490
negative_passages.text: For $x\in\mathbb R\setminus\mathbb Q$, the set $\{nx-\lfloor nx\rfloor: n\in \mathbb{N}\}$ is dense on $[0,1)$
negative_passages.docid: 13550
negative_passages.text: For an irrational number $\alpha$, prove that the set $\{a+b\alpha: a,b\in \mathbb{Z}\}$ is dense in $\mathbb R$
negative_passages.docid: 111898
negative_passages.text: Is $\{ (e^{2\pi i a n},e^{2\pi i b n }) : n \in \mathbb{Z} \}$ dense in the torus, where $a,b$ irrationals such that $a/b$ irrational?
negative_passages.docid: 137030
negative_passages.text: Why are there more Irrationals than Rationals given the density of $Q$ in $R$?
negative_passages.docid: 142577
negative_passages.text: Algebraic proof that a set generated by irrational rotations is dense in $S^1$.
negative_passages.docid: 64885
negative_passages.text: Irrational and rational sequence proof
negative_passages.docid: 92740
negative_passages.text: prove that $\mathbb{Q}^n$is dense subset of $\mathbb{R^n}$
negative_passages.docid: 98666
negative_passages.text: Continuity defined for rational and irrational numbers
negative_passages.docid: 120712
negative_passages.text: Is there a way to define the "size" of an infinite set that takes into account "intuitive" differences between sets?
negative_passages.docid: 32848
negative_passages.text: Prove that the product of a non-zero rational and irrational number is irrational.
negative_passages.docid: 4946
negative_passages.text: Non-existence of irrational numbers?
negative_passages.docid: 97064
negative_passages.text: Dense set in an infinite cofinite space
negative_passages.docid: 108922
negative_passages.text: Cardinality and infinite sets: naturals, integers, rationals, bijections
negative_passages.docid: 78915
negative_passages.text: Density of the set $S=\{m/2^n| n\in\mathbb{N}, m\in\mathbb{Z}\}$ on $\mathbb{R}$?