Prove that for a subset $A$ of $\mathbb{R}$, if $|A|=\aleph_{0}$ then $|\mathbb{R}-A|=2^{\aleph_0}$.
3902
Does $k+\aleph_0=\mathfrak{c}$ imply $k=\mathfrak{c}$ without the Axiom of Choice?
102972
Cardinality of $\mathbb{R}^\mathbb{R}$ by diagonal argument
149382
Prove that $(0,1) \subseteq\mathbb R$ and $(4,10) \subseteq\mathbb R$ have the same cardinality.
21599
$(0,1)$ is an open subset of $\mathbb{R}$ but not of $\mathbb{R}^2$, when we think of $\mathbb{R}$ as the x-axis in $\mathbb{R}^2$. Prove this.
83714
$A \in B$ vs. $A \subset B$ for proofs
13162
Prove $\alpha \in R[[x]]$ is a unit iff $a_0 \in R$ is a unit
86505
Let $A,B \subset \mathbb{R}$. Show that $\sup(A \cup B) = \max\{\sup A, \sup B\}$
13415
How to show if $A \subseteq B$, then $A \cap B = A$?
123613
Double Complement of a set proof
143105
Show that, for every real number a, the set {$x ∈ S | f(x) ≤ a$} is a closed subset of $\mathbb R^d$
64320
Prove that if $0<a<Pr(A)$ then there exists $B\subset A$ such that $Pr(B)=a$
40524
Let $A \subseteq (0,\infty$) such that $\inf A>0$. Let $\frac{1}{A}=\{\frac{1}{a},a\in A\}$. Prove that $\sup \frac{1}{A}=\frac{1}{\inf A}$.
80916
How to prove that if $A\subseteq A\times A$, then $A=\varnothing$?
151295
Proving $|S \times \mathbb{N}|=|S|$ when $S$ is an infinite set
10911
$A \in M_{n}(\mathbb{R})$ has inverse if and only if it's rank is n.
92740
prove that $\mathbb{Q}^n$is dense subset of $\mathbb{R^n}$
73927
Is $\mathbb{R}$ a subset of $\mathbb{R}^2$?