For questions concerning a specific proof, asking for verification, identifying errors, suggestions for improvement, etc.

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1
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1answer
45 views

Is my proof right for this divisibility proof?

Prove For all integers $x$ if for all natural numbers $y$, $x$ does not divide $y$, then $x = 0$. I start by saying that $x\neq 0$ then $x\mid y$ there is exists an integer $d$ such that $xd=y$ if ...
34
votes
3answers
2k views

Fascinating Lampshade Geometry

Today, I encountered a rather fascinating problem in a waiting room: Notice how the light is being cast on the wall? There is a curve that defines the boundary between light and shadow. In my ...
2
votes
1answer
28 views

Proving that $\mid Gal(E/F)\mid\leq [E:F]$ by induction

I am looking for help understanding the proof that $\mid Gal(E/F)\mid\leq [E:F]$ by induction on the degree of the extension. $E/F$ is a finite algebraic extension. I'll mark them by [1], [2], [3]. I ...
2
votes
1answer
216 views

How to Prove it 4.1 ex.10

Prove that for any sets A, B, C, and D, if A × B and C × D are disjoint, then either A and C are disjoint or B and D are disjoint. Proof(someones). Suppose (A X B) and (C X D) are disjoint. Let (x,y) ...
4
votes
1answer
99 views

Uniform continuity proof verification

I wrote a proof. Can someone please verify it? I have no idea what I just wrote. Suppose $f$ is a real-valued function continuous on $[a, b]$. Show that the function defined by $f^*(x) = $ ...
2
votes
0answers
147 views

Prove that if $f$ is uniformly continuous on a bounded set $S$, then $f$ is bounded on S

Prove that if $f$ is uniformly continuous on a bounded set $S$, then $f$ is bounded on $S$. Here's my proof. Can someone please verify it? Suppose $f$ is not bounded on $S$. Then, $\forall n \in ...
1
vote
1answer
144 views

Proving uniform continuity using $\epsilon$-$\delta$ definitions

I've written some proofs. Can someone please verify them? 1) Prove that $f(x) = \displaystyle{\frac{1}{x}}$ is uniformly continuous on $\displaystyle{\left[\frac{1}{2}, \infty\right)}$ Let ...
1
vote
2answers
100 views

Causal character of a surface (Lorentz-Minkowski space $\mathbb{L}^3$)

I'm trying to analyze the causal character of the surface $x^2 + y^2 - z^2 = -1$ in Lorentz-Minkowski space $\mathbb{L}^3$, with the convention $\mathrm{diag[1,1,-1]}$, that is $$\langle \left(x_1, ...
3
votes
2answers
48 views

Check: For the integers $a,b,c$ show that $\gcd(a,bc)=\gcd(a,\gcd(a,b)\cdot c)$

For the integers $a,b,c$ show that $\gcd(a,bc)=\gcd(a,\gcd(a,b)\cdot c)$ Proof: Let $u$ and $v$ be integers. Then $\gcd(a,b)=au+bv$. Then $c\cdot \gcd(a,b)=c\cdot(au+bv)=acu+bcv$ Let $x$ and $y$ be ...
0
votes
2answers
87 views

CHECK: Let a and b be relatively prime integers. Show that $\gcd(a^2+b^2,a+b)=$1 or 2 [duplicate]

Let a and b be relatively prime integers. Show that $\gcd(a^2+b^2,a+b)=$1 or 2 Proof: $s|a^2+b^2$ and $s|a+b$ implies $s|a^2+b^2$ and $s|(a+b)^2=a^2+b^2+2ab$ implies $s|a^2+b^2-(a+b)^2=2ab$ implies ...
0
votes
1answer
82 views

Any $2\times 2$ complex matrix A is similar to one of these three: (See first line of the question)

(i) : $\left(\begin{array}{ll} \lambda_{1} & 0\\ 0 & \lambda_{2} \end{array}\right)$, (ii) : $\left(\begin{array}{ll} \lambda & 0\\ 0 & \lambda \end{array}\right)$, (iii) : ...
1
vote
1answer
60 views

About interior of the frontier (proof-checking)

Let $M$ be a metric space, and $A \subset M$ an open set. Show that $\stackrel{o}{\widehat{\partial A}} = \emptyset$. ($\stackrel{o}{\widehat{\partial A}}$ is the interior of the frontier) I ...
0
votes
0answers
33 views

Proof that whether some arbitrary Turing machine on some input outputs $5$ is undecidable

Consider the language $L = \{<M, w> \mid w \, \text{run on } M \, \text{evaluates to} \, 5\}$, ie the problem of deciding whether, for a TM $M$ and input $w$, if you run $w$ on $M$ then $M$ will ...
1
vote
1answer
18 views

Limit of a function proof verification

My proof: By Bernoulli Equation $(a^n+b^n)^{1/n}=b(1+(na)/b)^{1/n}$ By definition of a limit, fix $\epsilon > 0$ and $N>(b\epsilon^n)/a$ Then, $|a_n - b | = ...
1
vote
1answer
30 views

Inequalities involving x and y.

I am asked to prove: $(x-y)^3 \ge x^3-3x^2y$ where $x,y$ are real and $0 < y < x$ I am told Bernoulli's inequality may help. I have however reduced this to $3xy^2 - y^3 \ge 0$. I have ...
0
votes
1answer
63 views

$MN/M \cap N \cong (MN/M) \times (MN/ N )$

I want to prove the following exercise from Dummit & Foote's Abstract Algebra: Let $M$ and $N$ be normal subgroups of $G$ such that $G=MN$. Prove that $G/M \cap N \cong (G/M) \times (G/N).$ ...
1
vote
0answers
68 views

Prove/disprove questions on equivalence relations and ordered sets

If $R$ is an equivalence relation and a partial order over $A \neq \emptyset$ then every equivalence class contain at least one element. If $(A,\le)$ an ordered set, and $a\in A$ is a single ...
0
votes
0answers
41 views

Proof Verification: If 650 points in a circle of radius 16, prove that some 10 must lie in a ring of inner radius 2 and outer radius 3.

If 650 points in a circle of radius 16, prove that some 10 must lie in a ring of inner radius 2 and outer radius 3. The area of any such ring is $5\pi$ and the area covered by the union of all rings ...
6
votes
2answers
215 views

Determining the derivation of a determinant

Let $\Phi\colon E\to M$ with $E\subset \mathbb{R}\times M$ and $M\subset\mathbb{R}^n$ open. Consider the function given by $x\mapsto \Phi(t,x)$ for fixed $t\in\mathbb{R}$. (1) Determine $$ ...
3
votes
0answers
83 views

Uncountable set has uncountably many limit points. (Proof Checking Request.)

Show that any uncountable subset of the reals has uncountably many limit points. Let $S\subseteq \mathbb R$ be uncountable and let $L$ be the set of all the limit points of $S$. Assume ...
2
votes
5answers
149 views

Is it true that $f: S\to S$ be a function: $(f \circ f)$ is bijejective if, and only if $f$ is bijective?

Let $f :S\rightarrow S$ be a function. Show that $f\circ f$ is bijective if, and only if, $f$ is bijective. My solution. If $f\circ f$ is bijective if, and only if, $f$ is ...
1
vote
1answer
153 views

Proof or find a counterexample:For all sets $A;B;C$ if $A\subseteq B,\ B\subseteq C,$ and $C\subseteq A,$ then $A=B=C.$

Proof or find a counterexample:For all sets $A;B;C$ if $A\subseteq B,\ B\subseteq C,$ and $C\subseteq A,$ then $A=B=C.$ My solution: True. Let $x\in A$, and since $A\subseteq B$ this implies that ...
0
votes
1answer
24 views

How do I work out the last sentence in this section of a proof of the Unique Factorization Theorem?

The last sentence states that the number of possibilities is $2\log_2 n$ (see the below image to follow the proof). I don't understand how to get $2\log_2 n$ but I understand everything that comes ...
1
vote
3answers
790 views

Proof of Drinker paradox [duplicate]

I searched all over the internet but didn't find a formal proof for this paradox, so here is my attempt: $\exists x[P(x)\implies \forall yP(y)]$ Let $x=x_0$. Thus $P(x_0)$ is given. Let $y$ be ...
2
votes
2answers
73 views

Let $f :[0,1] \rightarrow [0,1]$ be continuous with $f(0) = 0$ and $f(1) = 1$. Prove that f is onto.

Let $f :[0,1] \rightarrow[0,1]$ be continuous with $f(0) = 0$ and $f(1) = 1$. Prove that f is onto. Suppose, for contradiction, that $y \in [0,1]$ is not in the image of $f$. Since $f$ is ...
1
vote
0answers
116 views

Fourier Series of $f(x)=e^x$ on $[0,\pi)$ as a function of period $\pi$

Can you tell me what you get? I've tried computing it, I've got some result but I don't think it's right since I need to use it for something else and it doesn't work at all... What exactly I'm trying ...
0
votes
0answers
29 views

Matrix rank and linear independence

$\mathbf{a}_i,\mathbf{b}_j$ are $n$ dimensional vectors. Consider the matrix $\mathbf{M}$ defined by: $$\mathbf{M}_{ij}=\mathbf{a}_i\cdot\mathbf{b}_j$$ Prove/disprove that ...
3
votes
2answers
74 views

Find a chance that intersection of power set entries is an empty set

We are given set $A = \{1,2, ...n\}$. $k$ entries picked from the power set of $A$. Task is to find probability that $A_1 \cap A_2 \space \cap \space... \cap \space A_k = \emptyset$. I came up with ...
2
votes
1answer
50 views

A question about Matrices and Linear Transfromations

Let $v_1,...,v_n$ be a basis of a vector space $V$ over a field $K$. Let $M(T)$ denote the matrix of a linear map $T:V \rightarrow V$ with respect to our basis. Prove $$M(ST)=M(S)M(T)$$ for all ...
0
votes
1answer
99 views

If $g:V \rightarrow V$ is an injective linear transformation. Prove if $V$ is finite dimensional then $g$ is surjective.

I am asked to prove this without the rank nullity theorem My Attempt at a Proof For the $\implies$; If $g:V \rightarrow V$ is injective then the dimension of the kernel is 0, and so as ($im$) ...
1
vote
1answer
36 views

Is my proof correct regarding the non primality of $2\cdot 17^a +1$?

Today I need your help to know if the proof I have provided below is correct or not. I want to prove that there is no prime of the form $2\cdot 17^a+1$ where $a\in \mathbb N$. Now, first of all, I ...
0
votes
0answers
64 views

Question about Proof of Cauchy's Theorem for Star Domain

I am trying to prove the following statement: Suppose $f(z)$ is analytic on the domain $\Omega'$ which is obtained by deleting a finite number of points $\zeta_1,\ldots, \zeta_n\in \Omega$ from a ...
2
votes
1answer
73 views

If first 1 by 1 upper left submatrix (principal minor) = 0, conclude straightaway saddle point ? - Question 8

Find all local extremal points for the function $f(x,y) = x^3 - 3xy+y^3 $ and classify their type. For $H(f)(0,0),$ I see that $D_1 = \det [0] = 0$. So according to the criteria that I already posted ...
1
vote
1answer
56 views

For $f(x,y,z,\ w)=x^{5}+xy^{2}-zw$, how is this stationary point $\;$ a saddle point? - Question 14

14. a$)$ Find all stationary points of $f(x,y,z,\ w)=x^{5}+xy^{2}-zw$. $b)$ Classify the stationary points of $f$ as local maxima, local minima or saddle points. Provided Solution a $)$ We compute ...
0
votes
0answers
76 views

Area of solid of known cross-section

I was looking at surface areas of solids of known cross-section (that is, you take a region in the xy plane, set up cross sections perpendicular to the region that are defined as a function of x, and ...
1
vote
1answer
262 views

For this 2 by 2 locally linear system, how to determine that this “indeterminate” critical point is a centre? Boyce, p516, Question 9.3.12

$12.$ (a) Determine all critical points of $\dfrac{dx}{dt}=(1+x)\sin y$ , $\dfrac{dy}{dt}=1−x−\cos y$ . (b) Find the corresponding linear system near each critical point. (c) Find the eigenvalues of ...
2
votes
1answer
143 views

Check if the following gradient is correct

This question regards the verification of the gradient of a given function. Notation. Let $N, K \in \mathbb{N}_0$ be given (nonzero) integers, with $K > N$. Let $\mathbf{x} = [x_b \ y_b \ z_b]^T ...
1
vote
1answer
76 views

$\epsilon-N$ sequence convergence proof verification

Prove that the sequence $\displaystyle{\frac{n^3+4}{3n^3-2n^2+1}}$ converges to $\displaystyle{\frac{1}{3}}$ Here's my (somewhat terse) proof. Can someone please verify it? Let $\epsilon > ...
1
vote
1answer
68 views

Find matrices complying to given constraints

We are given linear mapping of $n$-dimensional vector space, such as: It has $n+1$ eigenvectors Any $n$ of them are linearly independent Find all matrices which could define such a linear ...
0
votes
0answers
21 views

Limit superior proof verification

Let $(s_n)$ be a bounded sequence in $\mathbb{R}$. Prove that lim sup $|s_n| \geq $ lim sup $s_n$. Here's my proof (can someone pleasy verify it?): Let $N \in \mathbb{N}$. Then, $\forall n > ...
2
votes
0answers
55 views

Proof verification-density of smooth compactly supported functions

I am trying to show that $C_{c}^{\infty}(\mathbb{R})$ (smooth compactly supported functions) is dense in $C_{c}(\mathbb{R})$ (in the $L^{p}$ sense). Can anyone check if my proof is correct? Let $f ...
1
vote
0answers
23 views

$f:(0,1)\to\mathbb{R}$ st $|f'(x)| \le 5$ , show that $\langle f(\frac{1}{n+1})$ converges.

Let $f:(0,1)\to\mathbb{R}$ be a differentiable function st $|f'(x)| \le 5 \forall x \in (0,1)$ , show that $\langle f(\frac{1}{n+1})\rangle$ converges in $\mathbb{R}$. Derivative is bounded ...
1
vote
1answer
72 views

Uniform continuity of $\arctan x$

Check if $\arctan x$ is uniformly continuous on $\mathbb R$ If I'll show that it's contious on $[0,\pi/2]$ then because it's periodic it would be continuous on $\mathbb R$. So by the definition ...
1
vote
1answer
391 views

Critcal values of coefficient matrix with parameter where the phrase portrait changes - Boyce, p410, Question 7.6.19

The coefficient matrix contains a parameter $\alpha$. Determine the eigenvalues in terms of $\alpha$. Find the critical value(s) of $\alpha$ where the qualitative nature of the phase portrait for the ...
1
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0answers
35 views

2 by 2 Linear Homogenous System with Complex Eigenvalue. Boyce, p409, Question 7.6.4

I don't know how to align multiple equations, hence I post this screenshot. If someone can show me how, thanks. I think understand everything above the red line, but please inform me about any ...
1
vote
1answer
58 views

Unbounded sequence proof

Let $S \subseteq \mathbb{R}$, and suppose there exists a sequence $(x_n)$ in $S$ converging to a number $x_0 \notin S$. Show that there exists an unbounded continuous function on S. Here's my ...
0
votes
1answer
58 views

Less than or equal summations

Hi,I want to prove above unequal that consist of two summation both of this sides.It a formula in Computer Network to control Congestion.The way to prove it is not important, but because I weak in ...
5
votes
0answers
101 views

Jech, “Set theory” exercises 12.11 - Is my proof right?

I try to prove the Jech's "Set theory", exercises 12.11: 12.11. If $\kappa$ is an inaccessible cardinal, then $V_\kappa\models \text{there is a countable model of ZFC}$. My attempt. Since ...
1
vote
3answers
50 views

If $(B \cap C) \subset A$, then $(C\setminus A) \cap (B\setminus A) = \emptyset$

Question: Prove/disprove: For all sets $A,B,C$, if $B \cap C \subset A$, then $(C \backslash A) \cap (B \backslash A) = \emptyset$ I'm a bit confused about the question, or where to start. When we ...
6
votes
1answer
247 views

Rudin's 'Principle of Mathematical Analysis' Exercise 3.14

Since I'm studying real analysis using this book by myself, I'm not sure whether or not my method to prove convergence of sequence is right. I'm working on the above question's (d), and my solution ...