query: Prove that for any set $A$ there is some $x \notin A$
positive_passages.docid: 25834
positive_passages.text: Proof a set cannot contain its own powerset - Russell's Paradox
negative_passages.docid: 59724
negative_passages.text: Proving that $x_n\to L$ implies $|x_n|\to |L|$, and what about the converse?
negative_passages.docid: 132532
negative_passages.text: Things around $(\forall x)(\Psi \Rightarrow \Phi) \Leftrightarrow ((\exists z)\Psi) \Rightarrow \Phi)$
negative_passages.docid: 66948
negative_passages.text: Proving question on sets
negative_passages.docid: 83996
negative_passages.text: For any two sets $A,B$ , $|A|\leq|B|$ or $|B|\leq|A|$
negative_passages.docid: 78463
negative_passages.text: Prove that $\sup \{-x \mid x \in A\} = -\inf\{x\mid x \in A\}$
negative_passages.docid: 90590
negative_passages.text: Prove that the following set is dense
negative_passages.docid: 19856
negative_passages.text: Prove that a subset of a separable set is itself separable
negative_passages.docid: 90055
negative_passages.text: Show that $\mathcal P(x)\notin x$
negative_passages.docid: 80827
negative_passages.text: Let $A$ be a nonempty subset of $\mathbb{R}$. Prove...
negative_passages.docid: 158882
negative_passages.text: Prove the existence of some weird family of sets
negative_passages.docid: 70278
negative_passages.text: $A\subseteq B\to C\setminus B\subseteq C\setminus A\,$ -- how to prove this?
negative_passages.docid: 59535
negative_passages.text: Prove that there exists such a set of $n$ positive integers
negative_passages.docid: 92534
negative_passages.text: Prove that any finite subset of a linearly independent set is linearly independent
negative_passages.docid: 46039
negative_passages.text: Prove that if $x \in R,$ then there exists $n \in Z$ satisfying $x \leq n < x+1$
negative_passages.docid: 17461
negative_passages.text: Show that a set is dense in $[-1,1]$
negative_passages.docid: 48523
negative_passages.text: Let $A$ be any uncountable set, and let $B$ be a countable subset of $A$. Prove that the cardinality of $A = A - B $