Questions tagged [solution-verification]

Question 1

Considering the space, $ X = S^{1} \times \partial D^{2} \,\cup\, \{x, y\} \times D^{2}. $ the subspace of the solid torus $ S^{1} \times D^{2} $ given by the union of the boundary of the boundary ...

Question 2

Let $n \in \mathbb{Z}_{2^q}$ such that $ n \equiv 2^r m \pmod{2^q}$ for some odd $m$ and $1\leq r<q$. Then the number of solutions to the congruence $x^2 - y^2 \equiv n \pmod{2^q}$ is $(r-1)2^q$. ...

Question 3

I sketch a proof of the following assert: If R is a commutative ring with unit an I is an ideal of R, such that R/I is a flat R-module, then I is a flat R-module. It sounds too nice to be true, and I ...

Question 4

I am working on the following problem from the L’Hôpital’s Rule section of Stewart Calculus, and I would appreciate feedback on whether my approach is sound, as well as clarification on a few things. ...

Question 5

I am trying to prove that if two events $A$ and $B$ are independent, their complements $A'$ and $B'$ are also independent. Given: $$P(A \cap B) = P(A)P(B)$$ Target: $$P(A' \cap B') = P(A')P(B')$$ Here ...

Question 6

Exercise 53.3 of Munkres's Topology asks: Let $p:E\rightarrow B$ be a covering map; let $B$ be connected. Show that if $p^{-1}(b_0)$ for some $b_0\in B$ has $k$ elements, then $p^{-1}(b)$ has $k$ ...

Question 7

Tricky question to find the domain $$f(x)=\sqrt{\log(\log x)-\log(4-\log 3)-\log 3}$$ My attempt $\log(\log x)-\log(\frac{4}{\log 3})-\log 3=\log(\log x)-(\log(\frac{4}{\log 3})+\log 3) $ $=\log(\log ...

Question 8

[A triangle ABC circumscribes a circle with center O and radius 4,the point of contact between the incircle and AB is at F and at AC it is E and at BC it is D,the lengths of BD is 6,CD=10, find AE] $$(...

Question 9

Question: Let $$p(\lambda) =\lambda^3 -2\lambda^2+\lambda. $$ Can $p(\lambda)$ be the characteristic polynomial of a linear difference equation? Justify. My answer: Yes, because there exists a ...

Question 10

I read Theorem 4.25 in Lee's Smooth Manifolds book. This theorem states that a smooth map $F:M\rightarrow N$ is a smooth immersion if and only if it is locally a smooth embedding (let's ignore in this ...

Question 11

Help in understanding a proof written by a teacher on the following theorem. Let $(X, \Sigma, \mu)$ be a finite measure space and let $(f_n)_{n\in\mathbb{N}}$ be a sequence of functions in $L^p(X)$. ...

Question 12

I was reading the solution provided for PB–Basic–007 in the DeepSeek-Math-V2 IMO-ProofBench-Basic dataset, and I am unsure whether one of its main steps is valid. I would like to confirm whether I am ...

Question 13

I am studying the following problem: Consider a family $K_\alpha$ ($\alpha<\omega_1$) of closed subsets of $[0,1]$ such that $\mu(K_\alpha)>0.22$, where $\mu$ denotes Lebesgue measure. (a) ...

Question 14

NOTE- The source of question is Advanced Calculus on real axis which doesn't cover $\textsf{Lebesgue integral}$ and $\textsf{Measure Theory}$ so therefore an approach by those wouldn't be of much ...

Question 15

I am currently self studying real analysis from the book Understanding Analysis, Stephen Abbott, 2nd edition. In page 11, exercise 1.2.2 the problem asks to show that there is no rational $r$ ...

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Questions tagged [solution-verification]

CW-structure on $ X = S^1\times\partial D^2 \;\cup\; \{x,y\}\times D^2 \subset T=S^1\times D^2. $

Number of solutions to $x^2 - y^2 = n $ in $\mathbb{Z}_m$

About the relationship between flatness as R-module of an ideal K of a commutative ring R and the flatness of the quotient R/K

Continuity and Differentiability of $|x|^x$

Proof verification: If $A$ and $B$ are independent, then $A'$ and $B'$ are independent

Preimage of a connected space in its covering space

How do I find the domain of the function $f(x)=\sqrt{\log(\log x)-\log(4-\log 3)-\log 3}$?

Is my proof wrong? The length of AE seems to be 64/11 but i got 6 [closed]

Can $p(\lambda)=\lambda^3 -2\lambda^2+\lambda$ be the characteristic polynomial of a linear difference equation?

Smooth immersion is a local embedding

$(X, \Sigma, \mu)$ finite measure space and $(f_n)_{n\in\mathbb{N}}$ is sequence in $L^p(X)$. If $f_n \to f$ uniformly on $X$, then $f \in L^p(X)$.

Is the PB–Basic–007 “solution” in DeepSeek-Math-V2 correct? Possible flaw in its convex-hull argument

Intersections of Families of Closed Sets under $MA_{\omega_1}$ and under CH

Let$\ \ f:[0,1]\to \mathbb{R}$ be a continuous function. Prove $\ \ \lim_{\lambda\to\infty}\int_0^1 f(x)\sin(\lambda x)\,dx = 0$. [duplicate]

Showing that $\nexists r \in \mathbb{Q} : 2^r=3$

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