query: Prove $a|b \wedge b|a \implies a=\pm b$
positive_passages.docid: 21737
positive_passages.text: If $A|B$ and $B|A$ then prove $A=\pm B$
negative_passages.docid: 115606
negative_passages.text: Proofs involving $ (A\setminus B) \cup (A \cap B)$
negative_passages.docid: 135986
negative_passages.text: Show if $a \equiv b \pmod{m}$ then $ac \equiv bc\pmod{mc}$
negative_passages.docid: 123890
negative_passages.text: Show $A\setminus (A\cap B)=(A\cup B)\setminus B$
negative_passages.docid: 152817
negative_passages.text: Proving $\sup B=\inf A$
negative_passages.docid: 67709
negative_passages.text: Stuck with proof for $\forall A\forall B(\mathcal{P}(A)\cup\mathcal{P}(B)=\mathcal{P}(A\cup B)\rightarrow A\subseteq B \vee B\subseteq A)$
negative_passages.docid: 61103
negative_passages.text: Prove that $\mathbb{|Q| = |Q\times Q|}$
negative_passages.docid: 63535
negative_passages.text: Prove that if $A – C = B – C$ and $A\cap C = B\cap C$ then $A = B$
negative_passages.docid: 47434
negative_passages.text: Proof that $A \subseteq B \Leftrightarrow A \cup B = B$
negative_passages.docid: 88982
negative_passages.text: Prove that for any two sets $A$ and $B$, $A\notin B$ or $B\notin A$
negative_passages.docid: 56399
negative_passages.text: Proof for $A \cup B = B$ if and only if $A \subset B$
negative_passages.docid: 81618
negative_passages.text: Prove $||a|-|b| | \leq |a-b|$
negative_passages.docid: 116527
negative_passages.text: Validating a proof for necessary and sufficient conditions on sets A & B such that $P(A) \cup P(B) = P(A\cup B)$
negative_passages.docid: 77164
negative_passages.text: If $a \mid m$ and $(a + 1) \mid m$, prove $a(a + 1) | m$.
negative_passages.docid: 111003
negative_passages.text: Prove that if $A\triangle B = C\triangle B$, then $A = C$
negative_passages.docid: 10107
negative_passages.text: Prove $A\cup (A\cap B) = A$
negative_passages.docid: 36602
negative_passages.text: $a^m=b^m$ and $a^n=b^n$ imply $a=b$