Questions tagged [solution-verification]

For posts looking for feedback or verification of a proposed solution. This should not be the only tag for a question.

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Show that if A is an m x n matrix having linearly independent columns, then the entries of the reduced echelon form of A consist of 1s and 0s.

I got the following question on an exam - Show that if A is an m x n matrix having linearly independent columns, then the entries of the reduced echelon form of A consist of 1s and 0s. Now looking ...
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3answers
38 views

want to check a proof i wrote

i'm taking a non-credit real analysis course online and this is a proof i wrote for one of my answers in my first problem set. my solution is different from the instructor's, but he said in the video ...
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1answer
30 views

how to give elements of $A × B$ and $A ∩ B$

$A$ = {$3n+1: n∈Z$} and $B$ = {$12m+7: m∈Z$} I have proven that $A⊄B$ by showing that $3n+1$ $⇒$ $3n+1+7-7$ ⇒ $3n-6+7$ ⇒ $3(n-2)+7$ ⇒ $4((3(n-2))+7)/4$ ⇒ $12(n-2)(1/4) +7$ ⇒ $12m-7$, where $m = (n-2)(...
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Surjective linear transformation of finitely generated free modules and their determinants

I kind of got stuck on Aluffi Algebra Exercise 6.4 of chapter 6 which is: Let $F$ be a finitely generated free $R$-module and let $\alpha$ be a linear transformation of $F$. Give an example of an ...
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1answer
35 views

Exercise 2.6 (General form of the Chinese remainder theorem) from Eisenbud's Commutative Algebra

Exercise 2.6 (General form of the Chinese remainder theorem): Let $R$ be a ring, and let $Q_1,\dots,Q_n$ be ideals of $R$ such that $Q_i+Q_j = R$ for all $i\neq j$. Show that $R/(\bigcap_i Q_i) \cong \...
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1answer
57 views

Find all real numbers $x$ such that $\sqrt{x+2\sqrt{x}-1}+\sqrt{x-2\sqrt{x}-1}$ is a real number

I want to find all values of $x\in \mathbb R$ such that the value of $\sqrt{x+2\sqrt{x}-1}+\sqrt{x-2\sqrt{x}-1}$ is a real number. I solved it as follows: $x+2\sqrt{x}-1\ge 0$ $(\sqrt{x}+1)^2-2\ge 0$ $...
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20 views

Given a function $f:N\rightarrow N$ it is true that $f(1)=43$, $f(2)=142$ and $f(n+1)=3f(n)+f(n-1)$, $n\ge 2$. Prove that $(f(n+1), f(n))=1$

Given a function $f:N\rightarrow N$ it is true that $f(1)=43$, $f(2)=142$ and $f(n+1)=3f(n)+f(n-1)$, $n\ge 2$. Prove that $\frac{f(n+1)}{f(n)}$ is irreducible. I proved it as follows: If $n=1$ it ...
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Subcomplex is closed

Hatcher, p. 520 in the appendix: A subcomplex of a CW complex $X$ is subspace $A \subset X$ which is a union of cells of $X$ such that the closure of each cell in $A$ is contained in $A$ ... It is ...
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I am stuck with integration and differentiation that implies continuity

I am not sure about continuity of a differentiable function after integration. Let's assume, $f$ is differentiable at $[0, \infty)$ $$\int f(x)dx = F(x)$$ Where $f$ is the derivative of $F$ Now the ...
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1answer
38 views

$X$th Permutations of $1,2,3,4,5$ is $25314$. Find X

The permutations of $1,2,3,4,5$ are lexicographically ordered. $X$th permutation is $25314$. Find $X$. I am getting $1*4! + 3*3! + 1*2! + 1=45$. Is it correct? Reasoning: There are $1*4!$ numbers of ...
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A possible solution for a specific case of the Beal's Conjecture?

According to wikipedia's page on Beal's Conjecture, it states: $A^x + B^y = C^z$, where $A, B, C, x, y$ and $z$ are positive integers and $x, y$ and $z$ are all greater than $2$, then $A, B$ and $C$ ...
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Doubt in Continuity chapter of Rudin(Question - 17)

Let $f$ be real function defined on $(a,b)$. Prove that the set of points at which $f$ has a simple discontinuity is at most countable. Proof: Let $E$ be the set of points at which $f$ has a ...
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1answer
11 views

Condensed SVD decomposition of an outer product

Let $A = uv^{T} \in \mathbb{R}^{m \times n}$. Find the (condensed) SVD decomposition of $A$. Theorem (Condensed SVD decomposition) Let $A \in \mathbb{R}^{n \times m}$ be a non-zero matrix of rank $r$....
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2answers
30 views

Does this proof make sense(The well-ordering principle for N)?

I need to explain the proof given by Des McHale in his "The well-ordering principle for N". I was hoping you could tell me if it makes sense and what else I could add to it. This is what I ...
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1answer
39 views

Proof by counterexample

If $n$ is prime, then $\sqrt{n}$ is irrational. Prove this statement. If I were to prove this using proof by contradiction, I would do: Suppose $n$ is prime and $\sqrt{n}$ is rational. Let $\sqrt{n}=\...
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14 views

Show the columns of a lower triangular matrix are linearly independent if and only if the diagonal entries are non-zero.

I should note that I can't use determinants for this proof. This is what I came up with: Let $A$ be a lower triangular matrix and suppose $A^1, ..., A^n$ are linearly independent. Clearly if the ...
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1answer
49 views

Let $(x_n)$ and $(y_n)$ be sequences such that $(x_n)$ is a subsequence of $(y_n)$ and $(y_n)$ is a subsequence of $(x_n)$. $x_n=y_n$ for all $n$?

Let $(x_n)$ and $(y_n)$ be sequences such that $(x_n)$ is a subsequence of $(y_n)$ and $(y_n)$ is a subsequence of $(x_n)$. Given that $x_n$ converges, does it follow that $x_n=y_n$ for all $n$? I ...
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1answer
37 views

Conjecture: $(\frac{a}{c})^2+(\frac{b}{d})^2=e^2$ has no solutions in distinct positive integers if the fractions are fully simplified non-integers

Given that $a,b,c,d,e,f$ are all distinct positive integers, prove that there are no solutions for the following knowing their fractions are non-integers and completely simplified: $$\left(\frac{a}{c}\...
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2answers
25 views

Truth value of statements about empty set

I have a problem: Let P be the statement " x $\in$ A and x $\in$ $\mathbb{Z}$ " Determine the truth value of statement: ($\forall$x)P $\Longrightarrow$ ($\exists$x)P Is there a set A for ...
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1answer
32 views

What we need to show, $ab=1\bmod k$ or $(ab)\bmod n=1\bmod k$? For proving $U_k(n)≤U(n)$

I need to show that, For each divisor $k$ of $n$, $U_k(n)$ is subgroup of $U(n)$ where, $U_k(n)=\{x\in U(n) : x=1\bmod k\}$ My attempt: as $U(n)$ is finite group for each $n\in\mathbb{Z}^+$. Hence ...
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30 views

System of equations wrong solution $(x, y, z) = (3, 3, 2)$

Find the real solutions for the following system of 2 equations: $z^2 - 4z = -9 - 2xy$ $x(x - 6 - y) + y(y - 6 - x) = -13$ (A book of problems for 8th graders) The book says that $(x, y, z) = (3, 3,...
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69 views

Finding convergent subsequence of $(\sin n)$ in $\mathbb R$

Convergent subsequence of $(\sin n)$ exists by Bolzanno Weierstrass theorem. I am trying to construct a convergent subsequence of the sequence $(\sin n)$. My attempt: I note that the sequence $(x_n)$,...
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25 views

Deriving the definition of a simplex

This posted emerged out of the following question posted in "Introduction to Linear Algebra by Gilbert Strang (5th edition)": : Figure $1.5$b is given as follows: "$c+d+e \leq 1$" ...
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1answer
37 views

Set Theory element argument proof help please

Let $S$ be any finite set and suppose $x \notin S$. Let $K=S \cup \{x\}$. $X=\{T \subseteq K : x \in T\}$. Prove that every element of $X$ is the union of a subset of $S$ with $\{x\}$, and that if ...
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3answers
51 views

Unique questions about pentagon

Consider the (non-regular) pentagon with consecutive vertices at (-1,-1), (-1,1), (0,2), (1,1), and (1,-1). a) Prove that there is no circle that is tangent to all 5 sides of the pentagon b) Is there ...
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1answer
20 views

Showing that a set is open with metric

Let $(A, d_1)$ and be a metric. Suppose $d_2: A \times A \rightarrow \mathbb{R}$ be defined in this way: For $a, b \in A$, $d_2(a, b) = \frac{d_1(a, b)}{1 + d_1(a, b)}$. Show that $E \subset A$ is ...
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25 views

Supremum of the sum of two subsets equals to the sum of two suprema

Let $A$ and $B$ be upper bounded subsets, and let $A+B$ be the subset of numbers $x+y$ with $x \in A$ and $y \in B$. Prove that $\sup(A+B)=\sup(A)+ \sup(B)$. Let $a =\sup A$ and $b =\sup B$, it is ...
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2answers
67 views

I need a limit definition for the Hessian, does this work?

Let $f: \mathbb{R}^n \to \mathbb{R}$ be of class $C^2$. Let $x$ be a non-degenerate critical point of $f$. Prove that there is an open neighborhood of $x$ which contains no other critical points of $f$...
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1answer
63 views

Eisenbud Exercise 2.4 (Practice with $\text{Hom}$ and $\otimes$)

I finished doing Exercise 2.4 from Eisenbud: Commutative Algebra, with a View Toward Algebraic Geometry, and I was wondering if I could have someone look over my solutions to check if everything is ...
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1answer
28 views

What's wrong with my Surface Area of a solid of revolution formula?

When I learnt about the derivation for the formula $$V=\pi\int_{x_1}^{x_2} y^2~dx$$ where $V$ is volume of the solid generated when $y=f(x)$ is rotated about the $x$ axis by $2\pi$ radians between $...
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1answer
27 views

Is cocountable topology generally separable.What else is wrong with this table?

Is the cocountable complement topology always separable? Also are there Really any other issues with this table I created? $$\begin{array}{cc|c} & \mbox{T1} & \mbox{Hausdorff}&\mbox{...
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1answer
52 views

Which $N$ are multiples of $2020$ with all distinct digits where swapping any two of them makes $N$ not a multiple of $2020$?

The question I have attempted problem 3 from the Senior O-level Tournament of Towns paper, Fall 2020. The question is as follows: A positive integer number N is divisible by 2020. All its digits are ...
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5answers
95 views

Avoiding brute force: determining when a specific polynomial in $\mathbb{Q}[x]$ is an integer for any integer $x$

I have to prove that $\frac{1}{5}n^5+\frac{1}{3}n^3+\frac{7}{15}n$ is an integer for any $n$. I solved this by brute-force, exhausting all the possibilities methods. I was wondering if there was a way ...
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2answers
36 views

Counting problem assigning student to groups

Say I want to assign $4$ students $a,b,c,d$, to three teams, and each team needs at least one student in it. Only one issue, $a$ and $b$ cannot be in the same team. How many ways are there to assign ...
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26 views

Counting Questions about Position

Lets say we have a student organization of 8000 students. 5000 are undergrads and 3000 are master students. There is an organization that has a president position, and three VP positions. a) How many ...
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45 views

Proving the Lebesgue outer of a set is equal to its volume

I want to prove the following statement if $A$ has volume, then $$ \left( \operatorname{vol}(A) = \int_A \chi_A \right) = \inf\left\{ \sum_{i=1}^\infty \operatorname{vol}(S_i) \ | S_1, S_2,\cdots \...
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OCR A Level Projectile motion question: Solution verification

Question taken from the book is in the image above. $$$$ The solution in the back of the book is Solution $2$. However, I believe that the partcile is on the way down when $\theta = 37^\circ.$ $\tan \...
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3answers
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Let $f:(0,\infty)\to \mathbf{R}$ and $L,L' \in \mathbf{R}$. If $\lim_{x\to\infty}f(x)=L$ and $\lim_{x\to\infty}f'(x)=L'$, show that $L'=0$

Let $f:(0,\infty)\to \mathbf{R}$ be differentiable on $(0,\infty)$ and $L,L' \in \mathbf{R}$. Suppose that $\lim_{x\to\infty}f(x)=L$ and $\lim_{x\to\infty}f'(x)=L'$. Prove that $L'=0$. I know that ...
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1answer
17 views

Prove basis for product topology is collection of all sets with finitely many factors restricted to open sets

Prove a basis for the product topology on $\prod_{a \in \lambda}X_{a}$ is the collection of all sets of the form $\prod_{a \in \lambda}U_a$ where $U_a$ is open in $X_a$ for each $a$ and $U_a=X_a$ for ...
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1answer
64 views

Math Olympiad question: Quadratic Equations.

Math Olympiad question: If the equation in $x$ has real roots, then find the value of $a$ and $b$. $x^2 + 2(1 + a)x + (3a^2 + 4ab + 4b^2 + 2) = 0$ Approach: For at least one real root: $$b^2 -4ac \...
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1answer
52 views

If $\lim_{x\to \infty} f(x) = c\neq 0$ and $\lim_{x\to \infty} g(x) = 0$, then $\lim_{x\to \infty} \frac{f(x)}{g(x)}$ does not exist.

This is just a minor part of another problem and intuitively clear, but I think that it is necessary to have a proof of this assertion. If $\lim_{x\to \infty} f(x) = c\neq 0$ and $\lim_{x\to \infty} ...
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2answers
40 views

Prove that if $\lim_{n\to\infty} \frac{x_n}{y_n} = 0$, then $\lim_{n\to\infty} x_n \div \frac{x_n + y_n}{2} = 0$

Let $\{x_n\}$ and $\{y_n\}$ be positive sequences. Assume $\lim_{n\to\infty} \frac{x_n}{y_n} = 0$. I have to prove if the claim $\lim_{n\to\infty} x_n \div \frac{x_n + y_n}{2} = 0$ is always true, ...
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0answers
13 views

Does Edwards need to explicitly state $a\in\mathcal{W}$ in his Inverse Mapping Theorem?

In view of the answer to my question: A function is continuously differentiable in an open neighborhood of $x_o$ and differentiable at $x_o$, is it continuously differentiable at $x_o$? it seems to me ...
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1answer
49 views

Find $f_{xy}(0,0)$ and $f_{yx}(0,0)$ for the following function

Find $f_{xy}(0,0)$ and $f_{yx}(0,0)$ for the following function: $$f(x,y) = \begin{cases} \frac{xy}{x^{2}+y^{2}} & (x,y) \ne (0,0) \\ 0 & (x,y) = (0,0) ...
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1answer
83 views

$\lim_{x\to\infty}\frac{1-\cos x}{1-\sin x}$

Evaluate $$S=\lim_{x\to\infty}\frac{1-\cos x}{1-\sin x}$$ Its quite simple to conclude that this limit does not exist. Now here comes the interesting part. Consider the limit $$L=\lim_{x\to\infty}\...
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2answers
34 views

Proving an inverse function is continuous

Let $f: (a,b) \to \mathbb{R}$ be a continuous function with $f'(x) \neq 0$ for all $x$. Prove that $g: f(a,b) \to (a,b)$ is continuous. Here is my attempt. Let $\epsilon > 0$, $x \in (a,b)$. and ...
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0answers
9 views

Unipotent upper triangular matrices with integer entries is Zariski dense

Let $N$ be the group of matrices $\begin{bmatrix} 1 & z \\ 0 & 1 \end{bmatrix}$ for $z \in \mathbb{C}$, let $\Gamma$ be the subgroup of $N$ with $z \in \mathbb{Z}$. I wish to show that $\Gamma$...
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1answer
10 views

Showing that two random variables are standard normal but are not bivariate normal

Question: If $X\sim \mathcal{N}(0,1)$ and we define $Y$ such that $$ Y = \begin{cases} X& \text{ if }|X|<a \\ -X& \text{ if }|X|\geq a. \end{cases} $$ Show that $Y\sim \mathcal{N}(0,1)$ ...
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0answers
32 views

Evaluate $I(a,0)$

Let $I(a,b)=\int_{0}^{1}x^a(1-x)^bdx$, where $a \geq 1, b \geq 1$ and $a, b$ are integers. If $b=0$ and $(1-x)^0=1$ (this is to confirm the property $x^0=1$) for x in $[0,1]$, then what is $I(a,0)$? I ...
2
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1answer
30 views

Criterion of map $M_f\to Z$ to be continuous

I was about to prove the following proposition A map $M_f\to Z$ is continuous if and only if the induced maps $X\times I\to Z$ and $Y\to Z$ are both continuous. Here, $M_f$ is a mapping cylinder of $...

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