query: Find $\{x\in \mathbb R\colon\neg(3x>21\implies x\leq 5) \}$
positive_passages.docid: 129824
positive_passages.text: For what $a \in \Bbb R$ does $\neg(3a>21\implies a \leq 5)$ hold?
negative_passages.docid: 77026
negative_passages.text: Proof with inequalities
negative_passages.docid: 63124
negative_passages.text: Show that $ \lVert A \rVert_2^2 \leq \lVert A \rVert _1 \lVert A \rVert _ \infty $
negative_passages.docid: 126889
negative_passages.text: Proof by induction: inequality $n! > n^3$ for $n > 5$
negative_passages.docid: 115004
negative_passages.text: If $x\notin\mathbb Q$, then $\left|x-\frac{p}{q}\right|<\frac{1}{q^2}$ for infinitely many $\frac{p}{q}$?
negative_passages.docid: 50273
negative_passages.text: How to prove $\lvert \lVert x \rVert - \lVert y \rVert \rvert \overset{\heartsuit}{\leq} \lVert x-y \rVert$?
negative_passages.docid: 39953
negative_passages.text: $a\mid b,\ c\mid d\,\Rightarrow\ ac\mid bd $ $\ \, \bf\small [Divisibility\ Product\ Rule]$
negative_passages.docid: 75639
negative_passages.text: Showing that $\sqrt[3]{2}\notin\Bbb Q(\alpha_1,...,\alpha_k)$ where $\alpha_i^2\in\Bbb Q\ \forall i$
negative_passages.docid: 122944
negative_passages.text: Prove $|y''(x)| \leq 40$ for all $x \in [1,3]$.
negative_passages.docid: 90826
negative_passages.text: Constructing a subset not in $\mathcal{B}(\mathbb{R})$ explicitly
negative_passages.docid: 34960
negative_passages.text: Lower bound for $\phi(n)$: Is $n/5  1$?
negative_passages.docid: 109764
negative_passages.text: Find the cardinality of the quotient of $\mathbb R$ in respect to R
negative_passages.docid: 140759
negative_passages.text: What is exactly the difference between $\forall x \neg P(x)$ and $\neg \forall xP(x)$?
negative_passages.docid: 57647
negative_passages.text: Prove that s is finite and find an n so large that $S_n$ approximate s to three decimal places.
negative_passages.docid: 141529
negative_passages.text: Proving existence of a minimum
negative_passages.docid: 37183
negative_passages.text: Show that the set of all $x$ such that $x \in P$ and $x \notin Q$ exists
negative_passages.docid: 29679
negative_passages.text: Prove: $|x-y|\leq |x|+|y|$