query: Does a mathematical definition have necessary and sufficient condition hidden in it?
positive_passages.docid: 7871
positive_passages.text: Are "if" and "iff" interchangeable in definitions?
negative_passages.docid: 3902
negative_passages.text: Does $k+\aleph_0=\mathfrak{c}$ imply $k=\mathfrak{c}$ without the Axiom of Choice?
negative_passages.docid: 31898
negative_passages.text: Sufficient conditions for $x, Ax, A^2x, ...$ being linearly indepedent
negative_passages.docid: 82487
negative_passages.text: Define a mathematical set
negative_passages.docid: 113861
negative_passages.text: Exist vs exists in mathematics
negative_passages.docid: 30219
negative_passages.text: Sufficient condition(s) for $f'$ to be Riemann integrable
negative_passages.docid: 130620
negative_passages.text: Identity type between A and B being non-empty implies non-emptiness of A and B?
negative_passages.docid: 98559
negative_passages.text: subtlety of self-defined function and its evaluation
negative_passages.docid: 116527
negative_passages.text: Validating a proof for necessary and sufficient conditions on sets A & B such that $P(A) \cup P(B) = P(A\cup B)$
negative_passages.docid: 115068
negative_passages.text: If the dimension of a sufficient statistic $T(X)$ equals the dimension of parameter space, $T(X)$ is minimal sufficient?
negative_passages.docid: 34674
negative_passages.text: Does the proof of Bolzano-Weierstrass theorem require axiom of choice?
negative_passages.docid: 11405
negative_passages.text: If a nonempty set of real numbers is open and closed, is it $\mathbb{R}$? Why/Why not?
negative_passages.docid: 58285
negative_passages.text: Is it true that if $X$ is connected, then for every nonempty proper subset $A$ of $X$, we have $\mathbf{Bd} \ne \emptyset$
negative_passages.docid: 24478
negative_passages.text: What is the name of this theorem, and are there any caveats?
negative_passages.docid: 109159
negative_passages.text: What's the intuition behind $(\exists y)(\mathscr{C}(y)\supset\mathscr{D})\supset((\forall x)\mathscr{C}(x)\supset\mathscr{D})$
negative_passages.docid: 84076
negative_passages.text: Basic intuition about minimal sufficient statistic
negative_passages.docid: 86248
negative_passages.text: Is $\exists x(x=a)$ a valid formula in first order logic with equality?