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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$n!$ divides the product of $n$ consecutive integers if and only if $\binom{n}{k}\in\mathbb{N}$

For all $n\in\mathbb{N}$, $n!$ divides the product of $n$ consecutive integers if and only if for all $n,k\in\mathbb{N}$ if $0\leq k\leq n$ then $\binom{n}{k}\in\mathbb{N}.$ Firstly, I want to know ...
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Is this method correct for this transportation problem?

I've attatched a picture of the question I am working on currently, note that this is a past exam paper with no mark scheme. I am only interested in solving part c, can anyone verify that this method ...
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-3 votes
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How is this proof of i = -i wrong

I ran into this proof some time ago and tried to find what made it wrong but i couldn't find anything. Since we have $$\frac{-1}{1} = \frac{1}{-1}$$ Then we can write $$\sqrt{\frac{-1}{1}} = \sqrt{\...
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Showing that $f((x_1,x_2),(y_1,y_2)):=x_1-y_1$ is continuous.

Define $f:\mathbb{R}^2\times\mathbb{R}^2\rightarrow\mathbb{R}$ by $$ f((x_1,x_2),(y_1,y_2)):=x_1-y_1 $$ I want to show that $f$ is continuous. I already know that the function $g(x,y):=x-y$ is ...
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Proof verification for $\rm dim range \,ST \le \min (dim range\, S, dim range\, T)$.

$U,V$ are finite dimensional vector spaces. $W$ is a given vector space. Let $T\in L(U,V), S\in L(V,W)$ be linear transformations then it is to be shown that $\rm dimrange \,ST\le\min (dimrange\, T, ...
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Proof verification: the harmonic sequence in the finite complement topology [duplicate]

I'm trying to determine what the sequence $p_n = 1/n$ converges to in the finite complement topology on $\mathbb{R}$. With $X = \mathbb{R}$, we define $U \subset X$ as open if $X - U$ is finite or $U$ ...
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Linear Transformation mapping linearly independent vectors onto a linearly dependent set

Let $T:\mathbb{R}^n \rightarrow \mathbb{R}^m$ be a linear transformation. Show that if $T$ maps two linearly independent vectors onto a linearly dependent set, then the equation $T($x$)=$0 has a ...
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Proof Verification: Effectivenss of Tarski-Vaught Test

Here is the definition of the Tarski-Vaught test from Marker's Model Theory: An Introduction page 44. Proposition 2.3.5 (Tarski-Vaught Test) Suppose that $M$ is a substructure of $N$, then $M$ is an ...
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Proof for Cauchy Schwarz using the trig definition of the dot product

The Cauchy Schwarz inequality for the dot product of $n$ dimensional vectors states: $$|\textbf{u}\cdot \textbf{v}|^2\leq|\textbf{u}|^2|\textbf{v}|^2 $$ The Wikipedia proof at point 4.4.1 of makes ...
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If $a_n$ is a positive sequence and $\lim \limits_{n \to \infty}a_n=0$ then there exists $N>0$ such that $(a_{N+n})$ is decreasing

If $a_n$ is a positive sequence and $\lim \limits_{n \to \infty}a_n=0$ then there exists $N>0$ such that $(a_{N+n})$ is decreasing The first way is by contradiction with an example: let $ a_n= \...
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How many non-empty subsets of $\{ 1 , 2, 3, 4, 5, 6, 7, 8 \}$ consist entirely of odd numbers?

How many non-empty subsets of $\{ 1 , 2, 3, 4, 5, 6, 7, 8 \}$ consist entirely of odd numbers? Is my reasoning correct if I say that I have $2^4$ choices either to pick one of the $4$ odd integers in ...
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Proof verification of exercise 7(a), section 31 of Munkres’ topology

Let $p \colon X \rightarrow Y$ be a closed continuous surjective map such that $p^{-1}\big(\{ y \} \big)$ is compact for each $y \in Y$. (Such a map is called a perfect map.) (a) Show that if $X$ is ...
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Proof check inequality $|$ln$((1+x_1)(1+x_2))-x_1-x_2|\leq 2||x||^2$

Given that $||x|| \leq \frac{1}{2}$, show that for all $x\in \mathbb{R}^2$ the following estimation holds: $|$ln$((1+x_1)(1+x_2))-x_1-x_2|\leq 2||x||^2$ What I did: $|$ln$((1+x_1)(1+x_2))-x_1-x_2|$ $=|...
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$\sigma$-algebra generated by uncountable random variables

Let $(X_1,\mathcal{A}_1)$, $(X_2,\mathcal{A}_2)$, and $(X_3,\mathcal{A}_3)$ denote three measurable spaces. Let $\mathcal{G}$ denote a set of (pontentially uncountable) measurable functions $g:X_2\to ...
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How to express a function into powers of $(x-1)$ and $(y-2)$ using Taylor's formula?

Use Taylor's formula to express the following in powers of $(x-1)$ and $(y-2)$: $f(x,y)=x^3 + y^3 + xy^2$ Solution: $f(1,2)=1 +8 + 4=13$ $f_x (1,2) = 3 + 4=7$ $f_y (1,2) = 12 + 4=16$ $f_{xx} (1,2) = 6$...
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If $f:X\to Y$ is an injective-continuous function on $X$, show that every $U \subseteq X$ is open.

Let $(X,d)$ be any metric space and $(Y,p)$ be the discrete metric space. If $f:X\to Y$ is an injective-continuous function on $X$, show that every $U \subseteq X$ is open. Definition. Let $(X,d)$ be ...
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Finding the expected value of $\dfrac{X}{Y}$

Below is a problem I did. I believe I did it correctly and I am hoping that somebody here can either confirm that I did it right or tell me where I went wrong. Problem: Let $X$ be a random variable ...
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Mapping the common part of the disk $|z|<1$ and $|z-1|<1$ on the inside of the unit circle

Question: Map the common part of the disk $|z|<1$ and $|z-1|<1$ on the inside of the unit circle. This is question 1 on page 96 of Ahlfors book. This question is also asked (and answered) here:...
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$\mathbb{P}\left(X\geqslant\frac{2\alpha}{\lambda}\right)\leqslant \left(\frac{2}{e}\right)^{\alpha}.$

Using $$\mathbb{P}(X\geqslant x)\leqslant e^{-tx}M_{X}(t),\text{ }t\geqslant0,$$ show that in the particular case that $X\overset{\underset{d}{}}{=}\Gamma(\alpha,\lambda)$, $$\mathbb{P}\left(X\...
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Validation: creating sequences given certain subsequences (one going to infinity and zero simultaneously)

I have problem with few exercises. I want to find a sequence that can create sequence that has different terms and has three subsequences: one going to minus inifnity, one to inifnity and other to ...
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2 answers
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Find a pair of linear transformations that do not commute

The problem statement is as follows: Suppose $\mathbb{F}$ is any field. Find a pair of linear transformations $S,T \in \mathcal{L}(\mathbb{F^{2}}, \mathbb{F^2})$ such that $ST \neq TS$ My attempt ...
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Spivak's Calculus, Ch. 11, **69b: $f$ increasing at every $a \in [0,1]$. Prove $f$ increasing on $[0,1]$.

A function $f$ is increasing at $a$ if there is some number $\delta>0$ such that $$f(x)>f(a) \text{ if } a<x<a+\delta$$ and $$f(x)<f(a) \text{ if } a-\delta<x<a$$ (a) Suppose ...
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1 answer
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Proving that the following equation does not have integer solutions

I want to prove that the following equation has no integer solutions $a,b,c$: $$-a^3 - b^3 - c^3 + ab^2 - ac^2 + bc^2 - 2a^2c + 3abc = 0$$ apart from the naive solution $a=b=c=0$. The context, in case ...
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2 answers
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Solve $(x^2-5x+5)^{x^2+4x-60}=1$-Missing Solution

Problem: $$(x^2-5x+5)^{x^2+4x-60}=1$$ Attempt: Taking logs on both sides: $$(x^2+4x-60)\ln(x^2-5x+5)=0$$ Yields 4 solutions: $$[1]: x^2+4x-60=(x+10)(x-6)=0 \implies x=-10,6$$ $$[2]: x^2-5x+5=1 \...
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1 vote
1 answer
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Fundamental group of the torus minus a disk

Consider the torus $\mathbb{T}$ given by $aba^{-1}b^{-1}$ with vertex $p$ in the square of identifications. Now remove a disk $D$ from it (I based its boundary $c$ on $p$). Then, I get the space given ...
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Proof: $S$ a subset of $l^2(\mathbb{N})$ is a closed subset

I am doing exercice on a book and sometimes or i haven't the solution to the question or i didn't understand their solution. Question: Proove that the subset $S$ that countain all of the sequences of ...
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1 answer
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Spivak's Calculus, Ch. 11, **69a: $f$ continuous and increasing at every $a \in [0,1]$. Prove $f$ increasing on $[0,1]$.

**69. A function $f$ is increasing at $a$ if there is some number $\delta>0$ such that $$f(x)>f(a) \text{ if } a<x<a+\delta$$ and $$f(x)<f(a) \text{ if } a-\delta<x<a$$ (a) ...
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If Sigma(xn) and Sigma(yn) are convergent, show that Sigma(xn +yn) is convergent.

this is from exercise 3.7 introduction to real analysis bartle fourth edition number 4 i already tried to proof the Xn+Yn as a convergent but somehow it doesnt turn to proofed and still got some stuck
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1 answer
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If $2 (a\cos x - \cos 2x) = 1$

If $2 (a \cos x - \cos 2x) = 1$ for $x\in \mathbb R$ then find all possible real values of $a$. I first expanded $\cos 2x$ then used quadratic equation inequalities for the domain $\cos x \in (-1,1)$ ...
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1 answer
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Flux of a vector field across the upper unit hemisphere

I want to compute the flux of the vector field $F(x,y,z)=(y,x,x^3)$ across the hemisphere $S\ldots x^2+y^2+z^2=1.$ My thoughts: Since the orientation isn't mentioned, I took it to mean the unit ...
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Grasshopper problem regarding jumps across the real number line

"2021 grasshoppers are placed on the real line (not necessarily in integer points). At every step, one of the grashoppers hops over another one, landing on the opposite side at the same distance. ...
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1 vote
0 answers
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Proof check: Version of IVT from Spivak's Calculus

Theorem 7-1: If $f$ is continuous on $[a,b]$ and $f(a) < 0 < f(b)$, then there is some number $x \in [a.b]$ such that $f(x) = 0$. I have rewritten Spivak's proof using my own understanding of ...
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Prove that $D+(-D)=\tilde{0}$ for every Dedekind cut $D$ ("Supplement for Measure, Integration & Real Analysis" by Sheldon Axler).

I am reading "Supplement for Measure, Integration & Real Analysis" by Sheldon Axler. I proved 0.24(f). But I am not sure that my proof is ok or not. Is my proof ok or not? Even if my ...
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Properties of sum of Poisson processes

Let $N_1$ and $N_2$ be a independent Poisson processes with intensities $\lambda_1=1$ and $\lambda_2=4$. Let $N=N_1+N_2$ and $S_n$ be a moment of $n$ event. I need to calculate the following: $P(N_1(...
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2 votes
0 answers
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Prove a nonempty set $G$ with an associative operation $\ast$ is a group iff the following equations are satisfied $y \ast g = h$ and $g \ast x = h$ [duplicate]

I'm currently working on an exercise and the body of the text for the exercise is as follows. I have a first draft of the proof but am missing some things and am unsure about some things as well so ...
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1 vote
1 answer
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$\mathbb{P}(-1\leqslant X\leqslant\frac{1}{2})$ from $\varphi_{X}(t)=\frac{1}{7}\left(2+e^{-it}+e^{it}+3e^{2it}\right).$

Let $X$ be a random variable with characteristic function given by $$\varphi_{X}(t)=\frac{1}{7}\left(2+e^{-it}+e^{it}+3e^{2it}\right).$$ Determine $\mathbb{P}(-1\leqslant X\leqslant\frac{1}{2})$. ...
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Counting measure has no Lebesgue decomposition proof verification

I am working through exercises in Rudin RCA but I have some questions about whether my justification is valid as it differs from other posts on the site. Let $\mu$ be the Lebesgue measure on $(0,1)$ ...
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1 vote
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Uniqueness of the ternary Golay codes

In [Van Lint - Introduction to Coding Theory] the uniqueness of the binary Golay codes is shown quite easily. In essence, the proof boils down to the fact that there is only one 2-$(11,5,2)$-design up ...
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1 vote
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Find $gf(x)$ in terms of $h$ and $k$ for the functions $f(x) = 5x - 1$, $g(x) = hx + k$.

Find $gf(x)$ in terms of $h$ and $k$. \begin{align*} f(x) & = 5x - 1\\ g(x) & = hx + 2k \end{align*} What I've tried: \begin{align*} g(5x-1) & = h(5x-1)+2k\\ & = 5hx-h+2k \end{...
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2 votes
1 answer
72 views

Prove that a "set of all sets" does not exist.

Axiom I used for the proof: The Axiom Schema of Comprehension: Let P$(x)$ be a property of $x$. For any set $A$, there is a set $B$ such that $x\in B$ if and only if $x\in A$ and P$(x)$. Here is my ...
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3 votes
2 answers
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Rectangle have area inside of another - Proof [closed]

I was trying forever to write a simple program that showed if one rectangle has any area inside of another. It turns out it's very simple. Can somebody show me a proof to this mathematically? If not, ...
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2 votes
1 answer
34 views

Prove that the direct product of $2$ subgroups is a subgroup

I have an exercise where I am tasked to prove that for $2$ subgroups $K < G$ and $J < H$ of $2$ groups $G,H$ the following is a subgroup: $$K \times J \subset G \times H$$ I believe I have done ...
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4 votes
3 answers
152 views

Proof: How many continuous/bounded functions on $[0,1]$ verify $f(x)=f(x/2)\frac{1}{\sqrt{2}}$?

Question: How many continuous/bounded functions on $[0,1]$ verify $f(x)=f(x/2)\frac{1}{\sqrt{2}}$? Answer: Thank to @TonyK @Ryszard Szwarc. I think that i found an ever stronger demonstration that ...
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0 votes
0 answers
23 views

Length and sign of transposition

I have solved the following problem and I would like to know if my solution is correct and/or/how it could be improved, thanks. Define a transposition as a permutation of the form $(1,2,\dots,i-1,j,\...
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1 vote
0 answers
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Bound for $L^p$ Norm of a partial sum of a stationary sequence

I could need some help on stationary sequences. Assume that $X_1,X_2,\dots$ is a stationary sequence of real-valued random variables on $(\Omega, \mathcal{A},\mathbb{P})$. ($X_1,\dots,X_{t+s}$ and $...
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0 answers
28 views

Criterion for uniform continuity multivariable calculus

Let $D\subset \Bbb{R}^n$ be an open set (non empty) and convex. Let $f:D\to \Bbb{R}$ be a $C^1(D)$ function s.t $\exists C \in \Bbb{R}$ s.t $\|\nabla f(x)\|\leq C \ \forall x \in D$. Show that $f$ is ...
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1 vote
1 answer
172 views

Three consecutive powerful integers do not exist

I (who is not a professional mathematician), ended up in the following on which I would like to have your comment, because this is overly simple solution. I know this certainly should not be easy and ...
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1 vote
1 answer
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If two functions are close apart can I prove the difference of their empirical loss is also small?

I am trying to understand the proof of Theorem 3 in the paper "A Universal Law of Robustness via isoperimetry" by Bubeck and Sellke. Basically there exist atleast one $w_{L,e}$ in $\...
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1 vote
1 answer
72 views

Prob. $14$, Sec. $30$, in Munkres' TOPOLOGY, 2nd ed: The product of a Lindelof space and a compact space is a Lindelof space

Here is Prob. $14$, Sec. $30$, in the book Topology by James R. Munkres, 2nd edition: Show that if $X$ is Lindelof and $Y$ is compact, then $X \times Y$ is Lindelof. Here is my Math Stack Exchange ...
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2 votes
0 answers
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In this simplex algorithm tableau, what are the basic variables?

At some point while running the simplex algorithm, we find this tableau: Would I be correct in saying that at this stage, our basic variables are $x_4,x_3,x_6$ as they are the only ones not equal to ...
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