query: $\mathbb{R}^3$ \ $\mathbb{Q}^3$ is union of disjoint lines. The lines are not in an axis diretion.
positive_passages.docid: 23851
positive_passages.text: Is there a set of lines in $\mathbb{R}^3$ such that their union is equal to the set of points with not all coordinates rational?
negative_passages.docid: 87682
negative_passages.text: Show that $\sqrt{2}\notin \mathbb{Q}(\sqrt[4]{3})$
negative_passages.docid: 139176
negative_passages.text: The real line is not homeomorphic to any non-trivial product space
negative_passages.docid: 54496
negative_passages.text: Showing that $\mathbb{R}$ and $\mathbb{R}\backslash\mathbb{Q}$ are equinumerous using Cantor-Bernstein
negative_passages.docid: 114142
negative_passages.text: How to show that $\mathbb{R}^3$ is the direct sum of $W_1={\rm span}\{(1,1,1)\}$ and $W_2={\rm span}\{(1,0,0),(1,1,0)\}$?
negative_passages.docid: 57037
negative_passages.text: How does $(a,b)$ where $a,b\in \mathbb{Q}$ form a basis for $\mathbb{R}$?
negative_passages.docid: 93952
negative_passages.text: Show that $X$ can be represented as a union of disjoint equivalence classes
negative_passages.docid: 76501
negative_passages.text: How can we show that $\mathbb Q$ is not a free $\mathbb Z$-module?
negative_passages.docid: 96354
negative_passages.text: Linear independence of $\sin(x)$, $\sin(2x)$, $\sin(3x)$ in Map($\mathbb{R},\mathbb{R}$)
negative_passages.docid: 69046
negative_passages.text: Proof of $\mathbb{Q}$ is not cyclic
negative_passages.docid: 15865
negative_passages.text: Show that $X = \{ (x,y) \in\mathbb{R}^2\mid x \in \mathbb{Q}\text{ or }y \in \mathbb{Q}\}$ is path connected.
negative_passages.docid: 153444
negative_passages.text: $\mathbb{Q}(\sqrt[3]{2})$ is not in any cyclotomic extension of $\mathbb{Q}$
negative_passages.docid: 126927
negative_passages.text: $\mathbb Q$ is not the direct product of two non-trivial groups.
negative_passages.docid: 42883
negative_passages.text: Cross product in $\mathbb R^n$
negative_passages.docid: 44307
negative_passages.text: Prove $\sqrt[3]{3} \notin \mathbb{Q}(\sqrt[3]{2})$
negative_passages.docid: 11301
negative_passages.text: Union of two vector subspaces not a subspace?
negative_passages.docid: 55026
negative_passages.text: $\mathbb{Q}[X,Y]/(Y^2-X^3)$ is not a UFD