query: If $\mathcal{P}(A)=\mathcal{P}(B)$, then $A=B$?
positive_passages.docid: 28223
positive_passages.text: Prove/Disprove that if two sets have the same power set then they are the same set
negative_passages.docid: 55023
negative_passages.text: if $A$ subset of $B$ then $P(B)-P(A)=P(B\setminus A)\ge 0$
negative_passages.docid: 25187
negative_passages.text: Why $∀x(P(x)→Q)$ is equivalent to $∃xP(x)→Q$
negative_passages.docid: 35242
negative_passages.text: If $a$, $b$, $c$, $\frac{a}{b}+\frac{b}{c}+\frac{c}{a}$, and $\frac{a}{c}+\frac{b}{a}+\frac{c}{b}$ are all integers, then $|a|=|b|=|c|$
negative_passages.docid: 142470
negative_passages.text: $\mathbb{Q}(\sqrt{p},\sqrt[3]{q})= \mathbb{Q}(\sqrt{p}\cdot \sqrt[3]{q})$ ??
negative_passages.docid: 57979
negative_passages.text: If $P(A \cup B \cup C) = 1$, $P(B) = 2P(A) $, $P(C) = 3P(A) $, $P(A \cap B) = P(A \cap C) = P(B \cap C) $, then $P(A) \le \frac14$
negative_passages.docid: 145507
negative_passages.text: Is $K[a,b]=K(a,b)$ for algebraic $a, b$?
negative_passages.docid: 153355
negative_passages.text: Given $S \in B(Y^{*}, X^{*})$, does there exist $T\in B(X,Y)$ such that $S=T^{*}$?
negative_passages.docid: 83291
negative_passages.text: Prove that $p(x)=(x-a)q(x)$ for some $q(x)\in R[x]$ of degree= (degree of $p)-1$.
negative_passages.docid: 130998
negative_passages.text: Is $\exists x \in A ~:~ P(x)$ the same as $\exists x ~:~ x \in A \implies P(x)$
negative_passages.docid: 834
negative_passages.text: If $p$ is a prime and $p \mid ab$, then $p \mid a$ or $p \mid b$.
negative_passages.docid: 109054
negative_passages.text: Given an infinite set of cardinality $X$, is there a chain in $\mathcal P(X)$ of the same cardinalty of $\mathcal P(X)$?
negative_passages.docid: 79165
negative_passages.text: How to show that $P(A\cup B) = P(A) + P(B) - P(A\cap B)$
negative_passages.docid: 67394
negative_passages.text: How to interpret $\mathbb{P}(B|A)$ when $\mathbb{P}(A) = 0$
negative_passages.docid: 139934
negative_passages.text: In $\mathbb{Z}/(n)$, does $(a) = (b)$ imply that $a$ and $b$ are associates?
negative_passages.docid: 52249
negative_passages.text: Is it true that $||a|^p - |b|^p| \le C_p|a - b|^p$ for $1 < p < \infty$?
negative_passages.docid: 111003
negative_passages.text: Prove that if $A\triangle B = C\triangle B$, then $A = C$