query: prove that $|{\{X\subseteq\mathbb{N}|\ |\mathrm{X|={\aleph_{0}}}\}}| = 2^{\aleph_{0}}$
positive_passages.docid: 85099
positive_passages.text: How many subsets of $\mathbb{N}$ have the same cardinality as $\mathbb{N}$?
negative_passages.docid: 75373
negative_passages.text: Show that $2^n>n$ for $n\in \mathbb{N}$
negative_passages.docid: 80827
negative_passages.text: Let $A$ be a nonempty subset of $\mathbb{R}$. Prove...
negative_passages.docid: 67291
negative_passages.text: Prove that there exists $n\in\mathbb{N}$ such that $f^{(n)}$ has at least n+1 zeros on $(-1,1)$
negative_passages.docid: 92740
negative_passages.text: prove that $\mathbb{Q}^n$is dense subset of $\mathbb{R^n}$
negative_passages.docid: 21250
negative_passages.text: Prove $x \equiv a \pmod{p}$ and $x \equiv a \pmod{q}$ then $x \equiv a\pmod{pq}$
negative_passages.docid: 145397
negative_passages.text: How to prove that for every $i$, $a_i = i$
negative_passages.docid: 106767
negative_passages.text: $|p(z)| \leq M$ for $|z| \leq 1$ Show that $|p(z)| \leq M|z|^n$ for $|z| \geq 1$
negative_passages.docid: 157924
negative_passages.text: $\forall x\in\mathbb{N}[(\exists y\in\mathbb{N}(2\leq y\land y<x\land y\ |\ x))\to(\exists z\in\mathbb{N}(2\leq z\land z\leq\sqrt x\land z\ |\ x))]$
negative_passages.docid: 92639
negative_passages.text: Prove that the family of open sets in $\mathbb{R}$ has cardinality equal to $2^{\aleph_0}$
negative_passages.docid: 149287
negative_passages.text: Prove that $A,B>0$ and $A_{ij}0$
negative_passages.docid: 34693
negative_passages.text: Prove $E^{\mathbb Q}[Y]=E^{\mathbb P}[XY]$ if $E^{\mathbb P}[X]=\mathbb P(X>0)=1$ and $ \mathbb Q(A)=E^{\mathbb Q}[X1_A] $
negative_passages.docid: 84021
negative_passages.text: How can we show that $\varphi(a) \neq a +1$ for every $a \in A$?
negative_passages.docid: 148582
negative_passages.text: Proof: $\|X\|_2\leq \sqrt{n}\|X\|_\infty$
negative_passages.docid: 144787
negative_passages.text: Show that the set of all infinite subsets of $\mathbb{N}$ has cardinality greater than $\mathbb{N}$
negative_passages.docid: 59459
negative_passages.text: Binary sequences and ${2}^{\mathbb{N}}$ have the same cardinality
negative_passages.docid: 42292
negative_passages.text: Show that $a_n<\sqrt{2}$ for every $n\in\mathbb{N}$.