query: Is true that if $A\subseteq B$ then $\operatorname{interior}(A)\subseteq \operatorname{interior}(B)$?
positive_passages.docid: 20335
positive_passages.text: If $A\subset B$ then $\bar{A}\subset \bar{B}$
negative_passages.docid: 89177
negative_passages.text: $\operatorname{adj}(AB) = \operatorname{adj} B \operatorname{adj} A$
negative_passages.docid: 107215
negative_passages.text: Is $\emptyset \in \emptyset$ or $\emptyset \subseteq \emptyset$?
negative_passages.docid: 111538
negative_passages.text: If $S \subseteq X$ is closed, is $f(S,r)$ necessarily closed?
negative_passages.docid: 109682
negative_passages.text: Proposed proof of: If $A \subset B \subset \bar{A}$ and $A$ is connected, then $B$ is connected
negative_passages.docid: 102168
negative_passages.text: Prove that there exists disjoint open subsets $U,V$ of $X$ such that $x \in U$ and $B \subseteq V$.
negative_passages.docid: 132487
negative_passages.text: Proving that if $V$ is an inner product space and $U_1, U_2$ are subsets, then $U_1 \subset U_2$ if and only if $U^{\perp}_2 \subset U^{\perp}_1$
negative_passages.docid: 118239
negative_passages.text: If $A, B, C$ are sets prove that $(A\setminus B)\setminus C\subseteq A\setminus(B\setminus C)$
negative_passages.docid: 56399
negative_passages.text: Proof for $A \cup B = B$ if and only if $A \subset B$
negative_passages.docid: 76826
negative_passages.text: Are there sets $A$ and $B$ such that $A \in B$ and $B \in A$?
negative_passages.docid: 79424
negative_passages.text: Don't understand proof that interior of a set is open
negative_passages.docid: 80827
negative_passages.text: Let $A$ be a nonempty subset of $\mathbb{R}$. Prove...
negative_passages.docid: 73338
negative_passages.text: Prove that the closure of complement, is the complement of the interior
negative_passages.docid: 7733
negative_passages.text: If A is infinite, does there have to exist a subset of A that is equivalent to A?
negative_passages.docid: 49143
negative_passages.text: Showing $A \subset B \iff A\cap B=A$
negative_passages.docid: 82097
negative_passages.text: If $A \succeq B$, is it true that $B^{-1} \succeq A^{-1}$?
negative_passages.docid: 55023
negative_passages.text: if $A$ subset of $B$ then $P(B)-P(A)=P(B\setminus A)\ge 0$