query: Counter example to $b_1^c : b_2^c \subset (b_1 : b_2)^c$
positive_passages.docid: 157967
positive_passages.text: Contraction of quotient ideal is quotient of contractions?
negative_passages.docid: 140440
negative_passages.text: Proving $(A\times C)\cup(B\times D)\subseteq(A\cup B)\times(C\cup D)$
negative_passages.docid: 114393
negative_passages.text: If $a$ is divisible by $4$, then there exist int $b$ and $c$ such that $a = b^2 - c^2$
negative_passages.docid: 135577
negative_passages.text: Prove the existence of $c$ such that $f'(c) = 2c(f(c) - f(0))$
negative_passages.docid: 13136
negative_passages.text: Element chasing proof that $(A\setminus C)\setminus(B\setminus C) = (A\setminus B)\setminus C$
negative_passages.docid: 1821
negative_passages.text: Proving $f(C) \setminus f(D) \subseteq f(C \setminus D)$ and disproving equality
negative_passages.docid: 43246
negative_passages.text: Showing that if $a<b$, then $a^{1/n} < b^{1/n}$
negative_passages.docid: 125668
negative_passages.text: A proof that BS(1,2) is not polycyclic
negative_passages.docid: 119084
negative_passages.text: How many ordered pairs $(A,B)$ they are if $A$ and $B$ are subsets of $\{ 1,2,3,4,5 \}$ such that $A \cap B = \{1\}$?
negative_passages.docid: 146664
negative_passages.text: The union of two linearly independent subsets
negative_passages.docid: 3636
negative_passages.text: Are any two uncountable sets similar to each other?
negative_passages.docid: 41099
negative_passages.text: If $A \subseteq C$ and $B \subseteq D$ then $A \times B \subseteq C \times D$
negative_passages.docid: 70164
negative_passages.text: Proof exercise that the cardinal number of a set in $\mathbb R^2$ is $\frak c$
negative_passages.docid: 48127
negative_passages.text: Are there infinitely many $A\in \mathbb{C}^{2 \times 2}$ satisfying $A^3 = A$?
negative_passages.docid: 53660
negative_passages.text: Show that $ a，b，c, \sqrt{a}+ \sqrt{b}+\sqrt{c} \in\mathbb Q  \implies  \sqrt{a},\sqrt{b},\sqrt{c} \in\mathbb Q $
negative_passages.docid: 123613
negative_passages.text: Double Complement of a set proof
negative_passages.docid: 145719
negative_passages.text: Similarity between $A$ and $B (2\times 2)$.