Prove that set $\{f \in \mathbb{N^N} \: | \:f \: $is strictly increasing $\}$ has the same cardinality as $\mathbb R$
71967
cardinality of the set of $ \varphi: \mathbb N \to \mathbb N$ such that $\varphi$ is an increasing sequence
92639
Prove that the family of open sets in $\mathbb{R}$ has cardinality equal to $2^{\aleph_0}$
97697
Countable union of sets of cardinality $c$ has cardinality $c$
28223
Prove/Disprove that if two sets have the same power set then they are the same set
110662
Show that $P(\mathbb{N})$ is equivalent to $2^\mathbb{N}$
117478
Prove that $\mathbb{R}$ and the interval $(0, \infty)$ have the same cardinality.
114262
Find the cardinality of the following set $S_n=\{(x,y)|x^2+y^2=n, \text{where }x,y,n \in \mathbb Z\text{ and }n\geq0 \}$
85099
How many subsets of $\mathbb{N}$ have the same cardinality as $\mathbb{N}$?
33157
Proof that an infinite subset of $\mathbb{N}$ is countable
138026
To show that $n!f(x) \in \mathbb{Z}[x]$
120562
If A is an infinite set and B is at most countable set, prove that A and $A \cup B$ have the same cardinality
73850
Cardinality of set of functions
141227
How to show the integers have same cardinality as the natural numbers?
44214
The number of subsets of a set of cardinality $n$
125683
Power set of a finite set is finite proof, no cardinality of power set
122814
Finite Sets, Equal Cardinality, Injective $\iff$ Surjective.
68490
For $x\in\mathbb R\setminus\mathbb Q$, the set $\{nx-\lfloor nx\rfloor: n\in \mathbb{N}\}$ is dense on $[0,1)$