Questions tagged [solution-verification]

For posts looking for feedback or verification of a proposed solution.

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Fermat' Last Theorem

Fermat‘s Last Theorem Fermat‘s last theorem (proofed by Andrew Wiles in 1994) a^(i) – b^(i) <> c^(i) a, b, c, i are elements of N with a > b, i >1, i<>2; a, b, c are coprime. I) We ...
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Any matrix has a unique decomposition of a sum of a symmetric and anti symmetric matrix proof

I am given the following task: Any $n \times n$ matrix $\ A$ can always be written,$$ A= S+ C$$ Where $ S$ is symmetric and $ C$ is antisymmetric. Prove that this decomposition is unique. My ...
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Show $f \cup g$ is a function $\iff f|C = g|C$ where $\text{dom}(f) = A, \text{dom}(g) = B \text{ and } \emptyset \ne C = A \cap B$

Assume $(x, y) \in f \cup g.$ Then $(x, y) \in f$ or $(x, y) \in g.$ Assume $(x, y) \in g.$ By definition, $(x, y) \in f|C \implies (x, y) \in f \text{ and } x \in C$. Since $(x, y) \in g$ and $x \in ...
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Prob. 4, Sec. 29, in Munkres' TOPOLOGY, 2nd ed: $[0, 1]^\omega$ with uniform topology is not locally compact

Here is Prob. 4, Sec. 29, in the book Topology by James R. Munkres, 2nd edition: Show that $[0, 1]^\omega$ is not locally compact in the uniform topology. Here is a Math Stack Exchange (MSE) post ...
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Subbasis for a topology two problems

Problem 1: $\S$ $=$ $\{$ $(-\infty,a)$ $:$ $a\in \mathbb{R}$ $\}$ $\cup$ $\{$ $(a,\infty)$ $:$ $a\in \mathbb{R}$ $\}$ is a subbasis for $\mathbb{R}$ with the standard topology. Attempt: Clearly, $\S$...
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Prove $\frac{n-1}{n^2+2} \rightarrow 0$ as $n \rightarrow \infty$

I'm doing a university real analysis course and I'm practising proving the limits of sequences. I've been tasked prove $x_n = \frac{n-1}{n^2+2}$ converges. Using algebra of limits we deduce. $x_n =\...
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Proof of $\sup(1,3) \cup (5, 8) = 8$.

Proof: Let $S = (1,3)\cup(5,8)$ Clearly $\forall x \in S,\;x<8$. Hence $8$ is an upper bound of $S$. To prove that 8 is the least upper bound suppose there exists an upper bound less than $8$. So $\...
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Stuck with induction Divisibility

I have seen many on the questions on here about induction divisibilty, but I haven't found any question that covers the doubt that I'm having. The preposition says: "For any integer n $\leq$-3, 8 ...
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Is this proof by contraposition of the implication of the disjunkt union correct?

I'm trying to prove that $X \dot\cup Y \implies X \cap Y = \emptyset$ by way of contraposition and would like to know if the following suffices as a proof: 1$$\lnot(X \cap Y = \emptyset) = X \cap Y \...
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Pinter's Abstract Algebra, Chapter 27, Exercise G2.

Let $F$ be a field, and let $c$ be transcendental over $F$. Prove that $F(c)$ is the field of quotients of $\{a(c): a(x) \in F[x]\}$, and is isomorphic to $F(x)$, the field of quotients of $F[x]$. ...
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Scalar potential vector

I am trying to find the scalar potential, $\phi(\vec r)$, of a conservative vector field $\vec a(\vec r)$. I am integrating along a straight line from $\vec r_0$ to $\vec r$ which is parametrised by $\...
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Laplace transform of Legendre's equation, differential form

I'm trying to find a differential equation involving $Y(s) = \mathcal{L}[y(t)]$ of the Legendre's equation $$ (1-t^{2})y'' -2ty' + \alpha(\alpha+1)y = 0\qquad\qquad ,y(0)=1, \quad y'(0)=1,$$ for some ...
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Let $V$ be a finite-dimensional vector space and $T:V\rightarrow V$ be linear. Suppose that $V = R(T) + N(T)$. Prove that $V = R(T)\oplus N(T)$.

Let $V$ be a finite-dimensional vector space and $T:V\rightarrow V$ be linear. (a) Suppose that $V = R(T) + N(T)$. Prove that $V = R(T)\oplus N(T)$. (b) Suppose that $R(T)\cap N(T) = \{0\}$. Prove ...
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Does $\ \sin \left(\frac{(k\pi)^2}{x+k\pi}\right)\ {\to \sin(x)}\ $ as odd integers $\ k { \to } \infty$?

I am investigating $\sin(\frac{1}{x})$ and it's properties out of interest, and this question is related to my investigation. Initially, I was trying to prove the result in the title for $0<x<2\...
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Verification of matrix chain rule

I wrote down a simple example of function composition for multivariate and vector-valued functions to see if I can apply the matrix chain rule. I would appreciate it if someone could verify that this ...
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Debunking issue in proving $ \sqrt2 $ is irrational

Proving $\sqrt{2}$ is irrational: $ \sqrt{2}= \frac{a}{b} $ in lowest terms, thus $ {2}= \frac{a^2}{b^2} $ which leads to $ {2}= \frac{2{b^2}}{b^2} $, so $ {a} $ must be even, as even times even ...
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Prove that polynomial has no integer roots.

Coeficcients of polynomial $P(x)=ax^3+bx^2+cx+d$ are integers. Numbers $P(0)$ and $P(1)$ are odd. Show polynomial $P(x)$ has no roots that are integers. My proof: $P(0)=d$ $P(1)=a+b+c+d$ is odd ...
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Extreme value theorem proof

Let $f:\mathbb{R}\to\mathbb{R}$ and $$f(x)=x^2-\cos(x)$$ Prove using the definitions that $f$ achieves a minimum value. So since $f$ is continuous by defintion it has has a minima and maxima on ...
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Solution verification in this Probability exercise.

There is a box with four drawers. The contents are respectively $GGG, GGS, GSS$, and $SSS$. You pick a drawer at random and a coin at random. The coin is $G$. What is the probability that there is ...
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Prove that a given line is a horizontal asymptote for a given function

So, here's the problem I'm trying to do: Let $f(x) = \frac{x^2+2}{x^2+1}$. Prove that $f(x)$ has a horizontal asymptote given by $y = 1$. Proof Attempt: This amounts to proving that either of the ...
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Check if the following sets are (i) open or closed, (ii) bounded and (iii) compact. Explain [closed]

Indicate whether the following sets are (i) open or closed, (ii) bounded in some manner, and (iii) compact. Briefly (using English, not math) support your responses. a. $\{x,y|x\in[0,10], y<...
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Prove that $\log(x!)$ is $O(x\log x)$

I'm trying to prove that $\log(x!) = O(x\log x)$. I have got an idea but I'm not sure it's strong enough to be significant. I don't think I made any mistakes in reasoning, but I want to know if there ...
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A problem in my solution: find a polynomial $P(x)$ such that the derivative of $e^{-x^2}P(x)$ equals $0$ for $x=0$, $x=\pm a$ and $x=\pm b$.

$\left(P(x)e^{-x^2}\right)'=0$ must be satisfied by $0$, $\pm a$, $\pm b$, and no other values of $x$. I have a solution that I can't find a problem with, but I get the wrong answer: $\left(P(x)e^{-...
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Why can't we choose a simpler bound in this proof?

A basic property of limits is: "the limit of the product is the product of the limits". More precisely, Claim: Let $D \subseteq \mathbb{R}$, $a$ be a cluster point of $D$, $f: D \to \mathbb{R}$, and $...
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Let $V$ be a vector space over $F$ such that $\dim(V) = n$. prove that $V$ is isomorphic to $F^n$.

lSo, here's what I'm trying to prove: Let $V$ be a vector space over $F$. Prove that if $\dim(V) = n$, then $V$ is isomorphic to $F^n$. Proof Attempt: We need to construct a bijective linear ...
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Proof of Fatou's Lemma.

I would like to know if this proof of the lemma is correct and full of all the details. Fatou's Lemma. Let a sequence $\{f_n\}$ of non-negative measurable function. Then $$\int_X \liminf_{n\to\...
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Bounds for the box dimension of the Koch curve with different definitions

I am working on Kenneth Falconer's book on fractal geometry. The book gives us 5 different equivalent definitions for the box dimension and these are the two needed for this question: (i) Let $N_{\...
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Big-O notation, equality

I have the following definition of the Big-O notation: $$ \begin{align*} &f(x)=O(g(x))\text{ as }x\to a\\ &\qquad\text{iff}\quad \exists m>0, \;\exists \delta>0 \Big(0<|{x-a}|<\...
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Find stability function of Runge Kutta Method starting from a 3x4 Butcher Tableau

Given the scalar IVP $\dot{y}(t) = f(t, y(t))$ , $y(0)=y_0$, and the butcher Tableau $$ \begin{array} {c|cccc} 0 & 0 & 0 & 0\\ \frac{1}{2} & \frac{1}{2} & 0 & 0\\ 1& -1&...
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79 views

Find $lim _{n\to \infty}\frac{n-1}{n-2}$

I'm taking a university real analysis course and I have been tasked with proving that the sequence $x_n = \frac{n-1}{n-2}$ converges using first principles. First fix $\epsilon >0$. Using the ...
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Prove that $|x|<|y| \iff x^2<y^2$.

My proof is below. I am not sure about the second part. Proof: First I prove $|x|<|y| \implies x^2<y^2$. Suppose $|x|<|y|$. Then, \begin{align*} |x||x| < |y||x| &\implies x^2<|y||x|\...
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Show not possible to find positive whole numbers $m,n$ such that $m^2 − n^2 = 6$.

Show not possible to find positive whole numbers $m,n$ such that $m^2 − n^2 = 6$. $m^2 − n^2 = 6\implies (m+n)(m-n) = 6; \ m,n$ can be either even or odd. If $m$ is odd, $m=2k+1, k\ge 1$; else $m=...
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Proving a quadrilateral may have at most four axes of symmetry

I've attempted to construct a proof that a non-self-intersecting quadrilateral $\square ABCD$ can have at most four axes of symmetry: There must be the same number of vertices on either side of the ...
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62 views

I must prove that $\lim_{\mu(A)\rightarrow 0}\int_{A}\lvert f \lvert{\rm d}\mu=0$.

I am attempting to prove that $$\lim_{\mu(A)\rightarrow 0}\int_{A}\lvert f \lvert {\rm d}\mu=0.$$ Given $(X,\mathcal{M},\mu)$ a measured space , we suppose that $$f\in L^{1}(X,\mathcal{M},\mu) \...
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If $X_1,\dots,X_{100}$ are iid continuous rvs, find $P(X_{12}\text{ is the smallest and }X_{20}\text{ is the largest})$

$\textbf{The Problem:}$ Let $X_1,\dots,X_{100}$ be independent absolutely continuous random variables that all have the same marginal density function. Find the probability that $X_{20}$ is the ...
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Prove that $T$ is one-to-one iff $T$ carries linearly independent subsets of $V$ onto lineraly independent subsets of $W$.

Let $V$ and $W$ be vector spaces and $T:V\rightarrow W$ be linear. (a) Prove that $T$ is one-to-one iff $T$ carries linearly independent subsets of $V$ onto linearly independent subsets of $W$. (b) ...
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Is this a valid proof that $e$ is irrational?

We will start with the known identity that $$e=\sum_{n=0}^\infty\frac{1}{n!}$$ and assume for contradiction that $e=\frac pq$. Then we have that $p=qe$, where $p$ is an integer. So then $q!e$ is ...
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Proof of the Nine Lemma using the Snake Lemma

I attempted to prove the $\it 3\times3$ Lemma (or Nine Lemma). First off, I am well aware how to proceed by means of diagram chase and I know how to prove the $\it 3\times3$ Lemma doing so. However, I ...
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Prove that $\text{span}(S_{1}\cap S_{2})\subseteq \text{span}(S_{1})\cap\text{span}(S_{2})$.

Let $S_{1}$ and $S_{2}$ be subsets of a vector space $V$. Prove that $\text{span}(S_{1}\cap S_{2})\subseteq \text{span}(S_{1})\cap\text{span}(S_{2})$. Give an example in which $\text{span}(S_{1}\cap ...
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if $\sum a_n$ is a divergent series of positive terms then so is $\sum \sqrt{a_n}$

Here's my attempt: Suppose $(a_n)$ isn't bounded. Then $(\sqrt{a_n})$ won't be bounded either. Hence, $\sum \sqrt{a_n}$ is divergent. Now, if $(a_n)$ is bounded, then for all $n$, $a_n < M$ for ...
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Elliptic Paraboloid, and Plane $y=1$, and Rate in respect to $x$ at point $P(2,1,7)$

A point moves along the intersection of the elliptic paraboloid $z=x^2+3y^2$, and the plane $y=1$. At what rate is $z$ changing with respect to $x$ when the point is at $(2,1,7)$. My Work $$\...
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Infinitely many finite maximal chains

Without cause or reason, a common absuse of the Stack, this answer was down voted. Is the proof flawed? If so, what's the mistake? Given infinitely many finite maximal chains in a poset P, ...
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Prove $\{a(c):a(x) \in F[x]\}$ is an integral domain isomorphic to $F[x]$

This question originates from Pinter's Abstract Algebra, Chapter 27, Exercise G1. Let $F$ be a field, and let $c$ be transcendental over $F$. Prove {$a(c):a(x) \in F[x]$} is an integral domain ...
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Prove that $T$ is linear and $W_{1} = \{x\in V\mid T(x) = x\}$ as well as $W_{1} = T(V)$ and $W_{2} = T^{-1}(\{0\})$.

Assume that $T:V\rightarrow V$ is the projection on $W_{1}$ along $W_{2}$. (a) Prove that $T$ is linear and $W_{1} = \{x\in V\mid T(x) = x\}$. (b) Prove that $W_{1} = T(V)$ and $W_{2} = T^{-1}(\{0\})...
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Convert $\dddot{y}(t) = (\dot{y}(t)- y(t))^2 + 3\sin{(t)}y(t)$ and $\ddot{y}(t) = \dot{y}(t) -y(t)^2$ into 1. Order IVP.

1) We're given the IVP 3. Order $$\dddot{y}(t) = (\dot{y}(t)- y(t))^2 + 3\sin{(t)}y(t)$$ with initial values $y(0)=a, \dot{y}(0)=b, \ddot{y}(0)=c$ and want to convert it into a 1. Order IVP. 2) We're ...
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Find $c(x,y), d(x,y) \in F[x,y]$ such that $a(x,y)c(x,y) + b(x,y)d(x,y)$ is a non-zero polynomial in $F[X]$

Let $F$ be a field. Let $a(x,y), b(x,y)$ be co-prime in $F[x,y]$.Prove that there exist $c(x,y), d(x,y) \in F[x,y]$ such that $a(x,y)c(x,y) + b(x,y)d(x,y)$ is a non-zero polynomial in $F[x]$. Here’s ...
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How to prove that for any set E, the Lebesgue outer measure of E is bounded by Jordan inner and outer content?

Prove that for any $E \subset \mathbb{R}^2$ $c_i (E) \leq m∗ (E) \leq c_e (E)$. Thus, when E is rectifiable we get $m∗(E)=c(E)$. where $c_i$ is Jordan's interior content and $c_e$ is Jordan's exterior ...
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Prove that any two simple extensions of $\Bbb{R}$ are isomorphic (hence isomorphic to $\Bbb{C}$)

This question originates from Pinter's Abstract Algebra, Chapter 27, Exercise F5. If $a$ and $b$ are nonsquares in $\Bbb{R}, a/b$ is a square (why?). Use the same argument as in Exercise F4 to ...
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Probability of pendulums colliding

There are two pendulums hanging from the same point, having length $l$. The pendulum bob has a diameter $d$. The planes of oscillations of the two pendulums are perpendicular to each other. One of ...

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