query: Is it possible that $\mathcal{X} = \mathcal{Y}$, yet $\mathcal{X} \in \mathcal{Y}$?
positive_passages.docid: 14512
positive_passages.text: Why cannot a set be its own element?
negative_passages.docid: 143735
negative_passages.text: Prove/Disprove: For any sets $X$ and $Y$, $\overline{X\cap Y} = \bar{X}\cup\bar{Y}$
negative_passages.docid: 47577
negative_passages.text: Are Variable Operators Possible?
negative_passages.docid: 70552
negative_passages.text: Prove that $\mathcal{P}(A)⊆ \mathcal{P}(B)$ if and only if $A⊆B$.
negative_passages.docid: 65827
negative_passages.text: When $\min_{x \in X,y \in Y} f(x,y) = \min_{x \in X} \min_{y \in Y} f(x,y)$?
negative_passages.docid: 45222
negative_passages.text: When $f(x) = g(y)$ for almost every $(x,y)$, must $f$ and $g$ be constant almost everywhere?
negative_passages.docid: 10444
negative_passages.text: If $\{x,\{x,y\}\} = \{z, \{z,t\}\}$, then must it be $x = z$ and $y = t$?
negative_passages.docid: 128673
negative_passages.text: Given $A : \mathcal{C} \rightarrow \mathcal{E}$, when does $\operatorname{colim} A = A(1)$?
negative_passages.docid: 132113
negative_passages.text: Does $f(x) = f(2x)$ for all real $x$, imply that $f(x)$ is a constant function?
negative_passages.docid: 113391
negative_passages.text: Prove that if $A$ is transitive, then $\mathcal{P}(A)$ is transitive too.
negative_passages.docid: 17542
negative_passages.text: Why is $\sqrt{-x}*\sqrt{-x}=-x?$
negative_passages.docid: 98271
negative_passages.text: Why does $(x \to a) ≠ (x = a)$ , But $f(x \to a) = f(x = a)$
negative_passages.docid: 80892
negative_passages.text: Does $\wp(A \cap B) = \wp(A) \cap \wp(B)$ hold? How to prove it?
negative_passages.docid: 110661
negative_passages.text: When does $\sqrt{a b} = \sqrt{a} \sqrt{b}$?
negative_passages.docid: 31846
negative_passages.text: Does $\neg(x > y)$ imply that $y \geq x$?
negative_passages.docid: 81873
negative_passages.text: Is $(X,\mathcal T)$ necessarily an indiscrete space?
negative_passages.docid: 3360
negative_passages.text: Is $\frac{\textrm{d}y}{\textrm{d}x}$ not a ratio?