query: Does $2^{\mathfrak m}=2^{\mathfrak n}$ imply $\mathfrak m=\mathfrak n$?
positive_passages.docid: 21663
positive_passages.text: Bijection between power sets of sets implies bijection between sets?
negative_passages.docid: 29204
negative_passages.text: $a\equiv\bar a\!\pmod{\!kn}\Rightarrow a\equiv\bar a\!\pmod{\! n};\ $ $(a\bmod kn)\bmod n=a\bmod n.\ $ Congruences persist mod factors of the modulus
negative_passages.docid: 21782
negative_passages.text: Is $\aleph_0 = \mathbb{N}$?
negative_passages.docid: 144836
negative_passages.text: How to prove that $\frac{(mn)!}{m!(n!)^m}$ is an integer?
negative_passages.docid: 63952
negative_passages.text: Disproving: If $(\Gamma\vdash(\phi\,\lor\psi))$, either $(\Gamma\vdash\phi)$ or $(\Gamma\vdash\psi)$ is true
negative_passages.docid: 106584
negative_passages.text: If $A^{2n}=I$, then $A^n=\pm I$ where $A\in M_n(\mathbb{R})$
negative_passages.docid: 38549
negative_passages.text: $\mathbb{Q}(\sqrt{n}) \cong \mathbb{Q}(\sqrt{m})$ iff $n=m$
negative_passages.docid: 32965
negative_passages.text: Proving that for infinite $\kappa$, $|[\kappa]^\lambda|=\kappa^\lambda$
negative_passages.docid: 111746
negative_passages.text: $(A/\mathfrak{a}) \otimes_A M \simeq M / \mathfrak{a}M$ by tensoring the canonical exact sequence.
negative_passages.docid: 1712
negative_passages.text: $ d_1,d_2\mid n\iff {\rm lcm}(d_1,d_2)\mid n\ $ [LCM Universal Property]
negative_passages.docid: 123698
negative_passages.text: Can $2^k + 2^j$ be expressed as $2^n$?
negative_passages.docid: 48291
negative_passages.text: Proof that $\mathrm{ker}(f^*) = \mathrm{im}(f)^{\perp} $
negative_passages.docid: 145600
negative_passages.text: Is it true that $ U \oplus W_1 = U \oplus W_2 \implies W_1 = W_2 $?
negative_passages.docid: 65371
negative_passages.text: Does $2^X \cong 2^Y$ imply $X \cong Y$ without assuming the axiom of choice?
negative_passages.docid: 155750
negative_passages.text: Why does $\mathbb{C}$ have transcendence degree $\mathfrak{c}$ over $\mathbb{Q}$?
negative_passages.docid: 49294
negative_passages.text: prove that $\operatorname{lcm}(n,m) = nm/\gcd(n,m)$
negative_passages.docid: 94291
negative_passages.text: $\frac{2^{2n}}{2n} \le {2n \choose n}$?