query: For $A$ any subset of vector space $X$, is it true that $A+A=2A$?
positive_passages.docid: 23764
positive_passages.text: Proving that $A+A \not\subset 2A $ using a particular definition for sum of sets
negative_passages.docid: 144181
negative_passages.text: Given an arbitrary set $X$, can one always find an element outside $X$?
negative_passages.docid: 124042
negative_passages.text: Why is a vector space equal to its tangent space for any point?
negative_passages.docid: 4151
negative_passages.text: Every subspace of a vector space has a complement
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: 145980
negative_passages.text: For every matrix $A\in M_{2}( \mathbb{C}) $ there's  $X\in M_{2}( \mathbb{C})$ such that $X^2=A$?
negative_passages.docid: 77764
negative_passages.text: Is "$a + 0i$" in every way equal to just "$a$"?
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: 86325
negative_passages.text: Is an increasing union of subspaces a subspace?
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: 38516
negative_passages.text: Show a subset is subspace of a vectorspace and find the dimension
negative_passages.docid: 73927
negative_passages.text: Is $\mathbb{R}$ a subset of $\mathbb{R}^2$?
negative_passages.docid: 10472
negative_passages.text: Is every set a subset?
negative_passages.docid: 144360
negative_passages.text: For any sets $A$ and $B$, show that $(B\smallsetminus A)\cup A=B \iff A\subseteq B$.
negative_passages.docid: 134770
negative_passages.text: For sets $A,B$, $A \cap (B \setminus A) \subseteq \varnothing$
negative_passages.docid: 47856
negative_passages.text: $f(x)=ax+b$ for some $a,b\in\mathbb{Q}$ if $f(\mathbb{Q})\subset\mathbb{Q}$ and $f(\mathbb{R-Q})\subseteq \mathbb{R}-\mathbb{Q}$
negative_passages.docid: 9149
negative_passages.text: Any $B \subset \{10,11,...,99\}$, with $|B| =10$ has two subsets such that $\sum_{a \in B_1} a = \sum_{b \in B_2} b$