query: What is bigger, $p(\mathbb{N})$ or $\mathbb{R}$?
positive_passages.docid: 141799
positive_passages.text: Is $2^{|\mathbb{N}|} = |\mathbb{R}|$?
negative_passages.docid: 778
negative_passages.text: For $x∈\mathbb{R}^n$ , let $B(x,r)$ denote the closed ball in $\mathbb{R}^n$(with Euclidean norm) of radius $r$ centered at $x$
negative_passages.docid: 71767
negative_passages.text: What is the difference between $\Bbb{R^n}\times\Bbb{R^m}$ and $\Bbb{R^{m+n}}$?
negative_passages.docid: 61375
negative_passages.text: Meaning of $\mathbb{R}[x]$
negative_passages.docid: 56728
negative_passages.text: Question about limit points of a Subset of $\mathbb{R}$
negative_passages.docid: 102972
negative_passages.text: Cardinality of $\mathbb{R}^\mathbb{R}$ by diagonal argument
negative_passages.docid: 33309
negative_passages.text: Subgroup of $\mathbb{R}$ either dense or has a least positive element?
negative_passages.docid: 73927
negative_passages.text: Is $\mathbb{R}$ a subset of $\mathbb{R}^2$?
negative_passages.docid: 158011
negative_passages.text: $C_c^{\infty}(\mathbb R^n)$ is dense in $W^{k,p}(\mathbb R^n)$
negative_passages.docid: 10911
negative_passages.text: $A \in M_{n}(\mathbb{R})$ has inverse if and only if it's rank is n.
negative_passages.docid: 41846
negative_passages.text: $(\mathbb{R}^n,\|\cdot\|_{p})$ is isometrically isomorphic to $(\mathbb{R}^n,\|\cdot\|_{q})$ iff $p=q,$
negative_passages.docid: 147047
negative_passages.text: Does there exist a polynomial $p(x) \in \mathbb C[x]$ such that $p(x) \notin \mathbb R[x]$ and $p(x)p(-x)=p(x^2)$?
negative_passages.docid: 30484
negative_passages.text: Radius of convergence and uniformly convergence
negative_passages.docid: 78191
negative_passages.text: Define $\|\cdot\| \colon V \rightarrow \mathbb{R} $ by $\|a\| =\sum 2^n |a_n| $, then choose the correct option
negative_passages.docid: 138653
negative_passages.text: Is $\mathbb{R}$ a subspace of $\mathbb{R}^2$?
negative_passages.docid: 2575
negative_passages.text: Showing that $\mathbb{R}$ is connected
negative_passages.docid: 124562
negative_passages.text: $f:\mathbb{R}^n \to \mathbb{R}$ has expansion $\sum_i g_i(x)x^i$