query: If $\epsilon=\{\}$ then what is $\bigcup \epsilon$ and $\bigcap \epsilon$?
positive_passages.docid: 143003
positive_passages.text: $\bigcup \emptyset$ is defined but $\bigcap \emptyset$ is not. Why?
negative_passages.docid: 127488
negative_passages.text: If $L:\mathbb{R}^n \rightarrow \mathbb{R}^m$ and $m<n$ linear trans. What does this tell about injectivity and surjectivity?
negative_passages.docid: 107701
negative_passages.text: Show that a connected metric space is $\epsilon$-chainable for $\epsilon>0$
negative_passages.docid: 158189
negative_passages.text: If $\{E_\alpha\}$ is connected, $\bigcap\limits_{\alpha\in A}E \neq \emptyset$, then $\bigcup\limits_{\alpha\in A}E$ is connected
negative_passages.docid: 75009
negative_passages.text: $A = \bigcap_{\mathfrak{p} \in \text{Spec(A)}} A_{\mathfrak{p}} = \bigcap_{\mathfrak{m} \in \text{MaxSpec(A)}} A_{\mathfrak{m}}$
negative_passages.docid: 51129
negative_passages.text: Proving $n^\epsilon > \log(n)$ for sufficiently large $n$
negative_passages.docid: 52323
negative_passages.text: Maps - question about $f(A \cup B)=f(A) \cup f(B)$ and $ f(A \cap B)=f(A) \cap f(B)$
negative_passages.docid: 67635
negative_passages.text: Why does $\epsilon_0=1$ in CGS units?
negative_passages.docid: 125008
negative_passages.text: Define a bijection between $Z^{X \cup Y}$ and $Z^X \times Z^Y$, where $X, Y \in \mathcal{P}(\Omega) $ and $X \cap Y = \emptyset$
negative_passages.docid: 157215
negative_passages.text: Order of statements in the Delta-Epsilon Limit definition
negative_passages.docid: 46056
negative_passages.text: If $\mathcal{E}$ are subsets of $X$ and $A \subset X$. Show that $\mathcal{A}\mathcal{(E \cap}$ A)=$\mathcal{A}\mathcal{(E)} \cap$ A
negative_passages.docid: 32965
negative_passages.text: Proving that for infinite $\kappa$, $|[\kappa]^\lambda|=\kappa^\lambda$
negative_passages.docid: 51922
negative_passages.text: Epsilon Delta Proof: How do I use the definitions to prove
negative_passages.docid: 14215
negative_passages.text: Show that the union $\bigcup A_n$ has empty interior in $X$
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: 55461
negative_passages.text: How to prove that if $a$ belongs to $\mathbb R$, such that $0\leq a \leq\epsilon$, then $a = 0$
negative_passages.docid: 150882
negative_passages.text: If $\Gamma \cup \{ \neg \varphi \}$ is inconsistent, then $\Gamma \vdash \varphi$