If $\mathbb{P}$ is non-atomic, prove that if $0<a<\mathbb{P}(A),$ then there exists $B\subseteq A$ with $\mathbb{P}(B)=a.$
64320
Prove that if $0<a<Pr(A)$ then there exists $B\subset A$ such that $Pr(B)=a$
63148
Power Set Difference Proof P(A-B)=P(A)-P(B)
149382
Prove that $(0,1) \subseteq\mathbb R$ and $(4,10) \subseteq\mathbb R$ have the same cardinality.
95800
Prove that if $\mathcal P(A) \cup \mathcal P(B)= \mathcal P(A\cup B)$ then either $A \subseteq B$ or $B \subseteq A$.
131983
Prove that $A \subset B$ if and only if $A \setminus B = \emptyset$
70278
$A\subseteq B\to C\setminus B\subseteq C\setminus A\,$ -- how to prove this?
5293
Show that the logic $ \ \forall x \exists y \ P(x,y) \ $ is not logically equivalent to $ \ \exists x \forall y \ P(x,y) \ $
155929
Let $A, B$ be sets. Show that $\mathcal P(A ∩ B) = \mathcal P(A) ∩ \mathcal P(B)$.
144693
Show that if $B \subseteq C$, then $\mathcal{P}(B) \subseteq \mathcal{P}(C)$
90055
Show that $\mathcal P(x)\notin x$
55461
How to prove that if $a$ belongs to $\mathbb R$, such that $0\leq a \leq\epsilon$, then $a = 0$
3902
Does $k+\aleph_0=\mathfrak{c}$ imply $k=\mathfrak{c}$ without the Axiom of Choice?
56399
Proof for $A \cup B = B$ if and only if $A \subset B$
72144
Show that $\mathbb Q(\sqrt p) \not\simeq\mathbb Q(\sqrt q)$
98897
Proving that the set of $\nu$ measurable sets $M\subseteq P(X)$ is an algebra
110662
Show that $P(\mathbb{N})$ is equivalent to $2^\mathbb{N}$
23125
Prove that there is a bijection between the set of all subsets of $X$, $P(X)$, and the set of functions from $X$ to $\{0,1\}$.