query: If $a\in A$ and $A\in B$, is $a\in B$?
positive_passages.docid: 155136
positive_passages.text: $x\in \{\{\{x\}\}\}$ or not?
negative_passages.docid: 100844
negative_passages.text: How should I understand "$A$ unless $B$"?
negative_passages.docid: 139934
negative_passages.text: In $\mathbb{Z}/(n)$, does $(a) = (b)$ imply that $a$ and $b$ are associates?
negative_passages.docid: 56399
negative_passages.text: Proof for $A \cup B = B$ if and only if $A \subset B$
negative_passages.docid: 43246
negative_passages.text: Showing that if $a<b$, then $a^{1/n} < b^{1/n}$
negative_passages.docid: 81296
negative_passages.text: If A and B are ideals of a ring, show that A + B = $\{a+b|a \in A, b \in B\}$ is an ideal
negative_passages.docid: 67717
negative_passages.text: $A \oplus B = A \oplus C$ imply $B = C$?
negative_passages.docid: 134071
negative_passages.text: If A is a finite set and B is an infinite set, then B\A is an infinite set
negative_passages.docid: 14162
negative_passages.text: How can I prove $\sup(A+B)=\sup A+\sup B$ if $A+B=\{a+b\mid a\in A, b\in B\}$
negative_passages.docid: 92975
negative_passages.text: Does $a^n \mid b^n$ imply $a\mid b$?
negative_passages.docid: 58654
negative_passages.text: If $|A| > \frac{|G|}{2} $ then $AA = G $
negative_passages.docid: 98470
negative_passages.text: If $A \ne \varnothing$, then $\varnothing^A=\varnothing$
negative_passages.docid: 133699
negative_passages.text: Proving that there is a bijection between $A$ and $B$.
negative_passages.docid: 79697
negative_passages.text: If $A$ is a subset of $B$, then the closure of $A$ is contained in the closure of $B$.
negative_passages.docid: 134914
negative_passages.text: Let $A,B$ and $U$ be sets so that $A\subseteq U$ and $B\subseteq U$. Prove that $A\subseteq B$ iff $(U\setminus B)\subseteq(U\setminus A)$.
negative_passages.docid: 71012
negative_passages.text: If $a, b$ are positive integers such that $a|b^2, b^2|a^3,....$ then $a=b$.
negative_passages.docid: 82097
negative_passages.text: If $A \succeq B$, is it true that $B^{-1} \succeq A^{-1}$?