query: if $U\cup W=V$ then $U=V$ or $W=V$
positive_passages.docid: 28819
positive_passages.text: A union of two subspaces not equal to the vector space.
negative_passages.docid: 142028
negative_passages.text: If $\vec{a},\vec{b},\vec{c},\vec{d}$are unit vectors, $(\vec{a}\times\vec{b}).(\vec{c}\times\vec{d})=1$ and $\vec{a}.\vec{c}=\frac{1}{2}$,then
negative_passages.docid: 107383
negative_passages.text: Derived sets - prove $(A \cup B)' = A' \cup B'$
negative_passages.docid: 94421
negative_passages.text: Why $P(A) \cup P(B)$ is not equivalent to $P(A \cup B)$
negative_passages.docid: 41520
negative_passages.text: Prove that $L(V,W)$ forms a vector space
negative_passages.docid: 133087
negative_passages.text: Prove that, if $\{u,v,w\}$ is a basis for a vector space $V$, then so is $\{u+v, v+w, u+v+w\}$.
negative_passages.docid: 123890
negative_passages.text: Show $A\setminus (A\cap B)=(A\cup B)\setminus B$
negative_passages.docid: 141114
negative_passages.text: Question about a bijection between $\mathbb R$ and $\mathbb R \cup \{\infty\}$
negative_passages.docid: 70307
negative_passages.text: $(A\cap B)\cup C = A \cap (B\cup C)$ if and only if $C \subset A$
negative_passages.docid: 104578
negative_passages.text: Suppose that $V$ is a vector space, and $W$ is a subspace of $V$. If $V$ is finite dimensional, then prove $W$ too must be finite dimensional.
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: 74187
negative_passages.text: $\operatorname{ord}(u)=r,\,\operatorname{ord}(v)=s$ $\,\Rightarrow\,\operatorname{ord}(uv)=rs\,$ if $\,r,s\,$ (co)primes
negative_passages.docid: 138212
negative_passages.text: Please help me prove: $v(a+b)\leq v(a)+v(b)$, and $v(ab)\leq v(a)v(b)$ where $v(x)=\inf{\{\vert x^n \vert}^{1/n}: n\in\mathbb{N}\}$
negative_passages.docid: 75604
negative_passages.text: Proof that $\partial(A\times B) = ((\partial A)\times B)\cup (A\times(\partial B))$
negative_passages.docid: 151852
negative_passages.text: How to prove $A \subset B \iff A \cup B = B$
negative_passages.docid: 53364
negative_passages.text: dim$(V)$ = $n$, dim$(W)$ = $m$ $\implies$ dim($L(V,W)$) = $nm$
negative_passages.docid: 117131
negative_passages.text: Canonical Isomorphism Between $\mathbf{V}$ and $(\mathbf{V}^*)^*$