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

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3
votes
1answer
28 views

Find the number of days in which the job would be finished

$P_1$(person) can complete a job in $1^2$ day, $P_2$ can complete the same job in $2^2$ days. In general $P_n$ can complete the job in $n^2$ days. In how many days the job would be finished if an ...
0
votes
1answer
34 views

Proof about linear systems of equations

If $X_1$,$X_2$ are solutions of $AX=B \neq 0 $ then $aX_1 + bX_2$ is never a solution. I tryed this way: From the hypotesis we have $AX_1=B$ and $AX_2=B$ with $B \neq 0$. Then: $A(aX_1 + ...
26
votes
2answers
1k views

Lamport claims there is an error in Kelley's proof of the Schroeder-Bernstein theorem. What is it?

In section 4.1 of his note How to write a proof, Leslie Lamport mentions an error in Kelley's exposition of the Schroeder-Bernstein theorem: Some twenty years ago, I decided to write a proof of ...
0
votes
1answer
355 views

Proving the Nested Interval Property using Axiom of Completeness

I'm self-studying real analysis using Abbott's text "Understanding Analysis." I'm trying to think out/prove as much on my own as I can, so I am working on proving the Nested Interval Property (Theorem ...
0
votes
1answer
65 views

My proof is wrong, can anyone tell me why?

$$\forall x \in \mathbb{Z}, \forall y \in \mathbb{Z}, [x(x+1) = y(y+1)] \Leftrightarrow [x = y]$$ $$\forall x \in \mathbb{Z} , \forall y \in \mathbb{Z}, [x(x+1)=y(y+1)]\Leftrightarrow [x=y]$$ ...
1
vote
0answers
37 views

The only fixed-point free automorphism of order $2$ is $\phi(a)=a^{-1}$(in a finite group)

I got the problem in Dummit and Foote's Algebra book to prove if $G$ is a finite group that has an automorphism $\phi$ in which if $a=\phi(a)$ then $a=1$. And which satisfies $\phi(\phi(a))=a$ for ...
3
votes
1answer
124 views

Changing one point does not change the Riemann integral

I tried to prove the following. Please could somebody tell me if my proof is correct? Let $f: [a,b]\to \mathbb R$ be Riemann integrable. Then changing one value of $f$ then $f$ is still ...
2
votes
2answers
60 views

For all $1 \leq i < j \leq k$, the subtrees $T_i$ and $T_j$ have a vertex in common. Show that $T$ has a vertex which is in all of the $T_i$.

Can someone please verify my proof or offer suggestions for improvement? I am aware that there is a similar question elsewhere, but I want help with my proof in particular. Let $T_1, \ldots, T_k$ ...
0
votes
1answer
73 views

Integrability condition on the Fourier transform implies that the function is infinitely differentiable

This is a problem from the book from Stephane Mallat "A wavelet Tour of signal processing: a sparse way". A function $f$ is bounded and $p$ times continuously differentiable with bounded derivatives ...
2
votes
1answer
31 views

Question on Induction (Very Simple)

I've just started a course in mathematics at university, and our current topic is mathematical induction. I've been given the following question: $$1+4+4^2+....+4^{n-1}=\frac{4^{n}-1}{3}.$$ I get ...
2
votes
1answer
33 views

Prove that there is an edge $e' \in E(T')-E(T)$ such that $T'+e-e'$ and $T-e+e'$ are both spanning trees of $G$.

Can someone please verify my proof or offer suggestions for improvement? Let $T, T'$ be two spanning trees of a connected graph $G$. For $e \in E(T)-E(T')$, prove that there is an edge $e' \in ...
2
votes
1answer
49 views

Proof Verification for Discrete Math Class

Prove that $n^2$ is even iff $n$ is even. I proved it like this: Case I: $n$ is even 1) $n = 2a$ $(a\in Z)$ 2) $n^2 = 4a^2 = 2(2a^2)$ 3) $2a^2 = K$ $(K \in Z)$ 4) $n^2 = 2K$ Case II: $n$ is ...
-1
votes
1answer
55 views

Existence of certain uncountable closed sets in the order topology

This is a proof-verification request. Let $\Omega$ be the set of countable ordinals, $\omega_1$ the first uncountable ordinal, and $\Omega^*=\Omega\cup\{\omega_1\}$. Remarkable properties of these ...
1
vote
1answer
42 views

Proposed proof Lebesgue integration question

I just want to confirm the following proof: Consider a function $u: \Omega \rightarrow \mathbb{R}$ where $\Omega \subset \mathbb{R}^{n}$ and $u \in C^{2}(\bar{\Omega})$. Let $a_{jk}$ be smooth ...
1
vote
1answer
47 views

Constructing a specific normalised test function.

Given the following class of normalised test functions: \begin{equation} \mathcal{T}=\{\phi\in C_c^{\infty}(\mathbb{R}^n)\ |\ \text{supp }\phi \subseteq B(0, 1),\ \|D\phi\|_{\infty}\leq 1\} ...
0
votes
1answer
359 views

Show that a finite regular bipartite graph has a perfect matching

Some preliminaries: A matching in a bipartite graph with vertex set $X \cup Y$ is a subset $E_1$ of the edge set such that no vertex is incident with more than one edge in $E_1$ A complete matching ...
5
votes
2answers
156 views

Prove that $[0,1]$ is not isometric to $[0,2]$.

Prove that $[0,1]$ is not isometric to $[0,2]$. Suppose there is an isometry $f:[0,1]\to[0,2]$. Since f is continuous and surjective, the only values for $f(0)$ and $f(1)$ are $f(0)=0$ and ...
0
votes
0answers
34 views

Follow up on star algebra (proof verification)

I previously asked this question about a proof of the following claim: If $A$ is a commutative non-unital non-zero $C^\ast$ algebra then $\Omega (A)$ is not empty. In the meantime I believe to ...
0
votes
2answers
36 views

Vanishing moments and integrability

Is this correct? $\int_\mathbb{R}x^m f(x) dx=0 \iff \int_\mathbb{R}x^m \overline{f(x)}\,dx =0$. If yes then please tell the conditions under which this holds.
2
votes
1answer
97 views

Alternative definition of complex number, showing it is equivalent to the tradidional one.

The author of a book makes an alternative definition of the complex numbers, later he shows that this definition is equivalent to the ordinary definition where we define $i^2=-1$. Here is his ...
1
vote
1answer
31 views

General convergence of Sums

This is to be proven or disproven: Be $(a_n)_{n\in\mathbb{N}}$ a real sequence with $a_n \geq 0$ $ \forall n \in\mathbb{N}$. Then, if $\sum_{k=0}^{n} a_k$ converges for n$\to \infty$ also ...
2
votes
3answers
77 views

Proof of $A \subseteq B \Leftrightarrow A \cap B = A$ (Check chain of implications)

Prove $A \subseteq B \Leftrightarrow A \cap B = A$. My attempt: Case $\Rightarrow$: $$\begin{align} A \subseteq B & \Rightarrow & [x\in A \Rightarrow x\in B] \\ &\Rightarrow &[x ...
1
vote
0answers
151 views

Proving the 3-d pythagorean theorem on surface areas of oblique triangular pyramid

I would like suggestions if possible, other than the really sloppy picture, I'll edit that once my dad gets me Microsoft office. I got a snip of the shape, and edited it as best as I could. The ...
3
votes
2answers
105 views

If $f'(z_0)\neq 0$ then $f$ has an holomorphic inverse.

Problem: Let $U\subset\mathbb{C}$ be an open set, $f:U\to\mathbb{C}$ an holomorphic function of class $C^1$ and $z_0\in U$. Prove that if $f'(z_0)\neq 0$ then there exists a neighborhood $V$ of $z_0$ ...
1
vote
0answers
47 views

Prove that $X_nY_n\overset{\mathcal D}\rightarrow Xc$.

Let $X_n$ converge in distribution to $X$ and let $Y_n$ converge in probability to a constant $c$. Show that $X_nY_n\overset{\mathcal D}\rightarrow Xc$ and $\frac{X_n}{Y_n}\overset{\mathcal ...
0
votes
5answers
63 views

Trigonometry proof

Question: $$2\cos x -\cos 3x - \cos 5x = 16\cos^3 x\sin^2 x$$ What I have tried: Using the identities, I have converted all the cos and sin so that the angle inside is only $x$. However, I couldn't ...
0
votes
1answer
48 views

Trigonometry - proving an inequality

I came across this question while doing trigonometry. I have tried everything that I could possibly think of, AM/GM, converting it into quadratic equation, conditional identities, solving from RHS, ...
2
votes
5answers
68 views

Prove $(a,b,c)=((a,b),(a,c))$

The notation is for the greatest common divisor. I know that $$(a,b,c)=((a,b),c)=((a,c),b)=(a,(b,c))$$ Suppose $g=(a,b,c)$. Then $g\mid a,b,c$. Also, $g\mid(a,b),c$ and $g\mid(a,c),b$. Thus ...
3
votes
0answers
30 views

Prove that $\operatorname{ran} f \subseteq \operatorname{dom} g \implies\operatorname{dom} (g \circ f)=\operatorname{dom} f$

Some preliminaries: A function $f$ is a binary relation such that $(x,y_1) \in f$ and $(x, y_2) \in f$ implies $y_1 = y_2$. $\operatorname{ran} f = \{y: \exists x$ such that $(x,y) \in f\}$ ...
0
votes
1answer
36 views

Proof Writing Help: $P_UT=TP_U \Leftrightarrow U$ and $U^{\perp}$ are $T$-Invariant

I'm studying linear algebra using Axler's book on my own and this is also my first rigorous encounter with proofs would greatly appreciate suggestions to improve the writing of the first part of my ...
2
votes
3answers
72 views

Proving $k\binom{n}{k} = n\binom{n-1}{k-1}$

Prove that $$k\binom{n}{k} = n\binom{n-1}{k-1}$$ is true for all integers $n, k$ with $0 \leq k \leq n$. Would this be enough to prove this? $$\binom nk=\frac{n!}{k! ...
1
vote
1answer
27 views

Prove that $R[A \cup B] = R[A] \cup R[B]$, where $R$ is a binary relation.

Can someone please verify this? Prove that $R[A \cup B] = R[A] \cup R[B]$, where $R$ is a binary relation. Here, $R[C] = \{y: \exists x \in C $ such that $(x,y) \in R\}$ Let $z \in R[A \cup ...
0
votes
2answers
100 views

How to prove this map is injective

Let $A$ be a non-unital $C^\ast$ algebra and let $M(A)$ denote the multiplier algebra and let $\widetilde{A}$ denote the unitisation of $A$. Consider the map $\varphi : \widetilde{A}\to M(A)$ ...
2
votes
3answers
115 views

Center of $GL_n(\mathbb R)$ is the set of matrices $\lambda I$

I determined the set of all matrices $A$ such that $AB = BA$ for all $B$ in $GL_n(\mathbb R)$ to be the set of $\lambda I$. Now I'm not sure this is true. But quite sure. So I tried to prove it and it ...
4
votes
3answers
77 views

Show that $A \subseteq B \iff A \subseteq B-(B-A)$

Can someone please verify this? Show that $A \subseteq B \iff A \subseteq B-(B-A)$ $(\Rightarrow)$ Let $x \in A$. Then, $x \notin B-A$. Also, $x \in B$. Therefore, $x \in B-(B-A)$ So, $A ...
1
vote
2answers
63 views

Whats wrong with my proof

I am trying to find the angle $BCB'$ Here's my solution: But doesn't match with the answer given in the book. Its got to be $27.5^\circ$. Whats wrong with my solution?
2
votes
2answers
73 views

Prove $\dim W \ge 2$

Let $U_1, U_2, W$ subspaces of a finite dimensional vector space, such that: $U_1 \cap U_2 = \{0\}$ $U_1 \cap W \ne \{0\}$ $U_2 \cap W \ne \{0\}$ Show that $\dim W \ge 2$. ...
2
votes
2answers
101 views

Proof that the limit of $\frac{1}{x}$ as $x$ approaches $0$ does not exist

Hello I was hoping that someone might be able to verify that the following proof that $\lim_{x\to 0} {1\over x}$ does not exist is correct. First assume that $\lim_{x\to 0} {1\over x} = L$. This ...
0
votes
0answers
25 views

Limit and uniform convergence related proof

Can someone please verify this Let $f$ be a real valued function on (0, 1). Define a sequence of functions as $$f_n(x) = \left\{ \begin{array}{ll} \alpha & x < \frac{1}{n} \\ ...
0
votes
0answers
55 views

Verification of a Combinatorial Identity

I was given a question and would like to see if I made any errors in my answer. The Question: My Answer: I noticed the following identity is very useful here: $\dbinom{n+1}{r}$ = $\dbinom{n}{r}$ ...
1
vote
1answer
14 views

Clarification about equality regarding integrals

I'm reading Brezis ch. 8 and got stuck in a passage of lemma 8.2 pag. 205. Let $I=(a,b)$, let $g \in L_{loc}^1(I)$, for a fixed $y_0 \in I$, set $$ v(x) = \int_{y_0}^x g(t)dt \ \ \ \ \ \ \ \ \ x \in ...
7
votes
4answers
116 views

Herstein Question: $G^{i}$ normal in $G$?

I just wanted to ask a quick question. I'm going over the second edition of I.N. Herstein's topics in algebra and one of his exercises asks the reader to prove that each $G^{i} $ is a normal subgroup ...
0
votes
0answers
46 views

Matrix of a linear application

Given the vectors $ v_1=(-1,2,-3)$,$v_2=(0,1,1)$,$ v_3 = (0,1,-1)$,$v_4=(1,1,4)$ and $ w_2 = (3,-1,2)$,$w_3= (1,-1,0)$ , $ w_4 = (t,-3,4)$ a) Find for which value of $t$ there is a linear application ...
1
vote
1answer
51 views

Dilation proof verification

I am wanting to verify the proof below; can someone please tell me if they agree with the way I have argued this or if I have made any incorrect assumptions (and where they are). Thanks!! A ...
0
votes
2answers
31 views

Proof of conjugate verification.

Prove $\frac{z_1}{z_2} = \frac{\overline{z_1}}{\overline{z_2}}$ if $z_2\neq0$. Proof: Let $z_1=a_1+ib_1$ and let $z_2=a_2+ib_2$, where $a_1, b_1, a_2, b_2 \in \Re$. $$\frac{z_1}{z_2} = ...
1
vote
0answers
37 views

“Bypass” Operations and Groups

So I recently stumbled on this (pdf) collection of group theory related Putnam problems. Problem 1978 A-4 defines a "bypass" operation to be a mapping $\circ:S\times S\mapsto S$ such that $$(w\circ ...
3
votes
1answer
72 views

Show that if $B \subseteq C$, then $\mathcal{P}(B) \subseteq \mathcal{P}(C)$ [duplicate]

Can someone please verify this? Show that if $B \subseteq C$, then $\mathcal{P}(B) \subseteq \mathcal{P}(C)$ let $x \in \mathcal{P}(B)$. Then, $x \subseteq B$ This implies that $$\forall a \in ...
1
vote
1answer
84 views

Proof of isometries and inverses on the plane

I am taking a course on Intuitive Geometry. I am quite new to intuitive proofs however feel I've done pretty well thus far. Here is my theorem: Prove: That every isometry has an inverse. $Proof.$ ...
1
vote
2answers
70 views

Proof Verification for Homework

If $n$ is odd, then $n^2$ is odd. $1$) $n = 2k + 1$ (Definition of an odd number) $2$) $n^2 = (2k+1)^2 = (2k+1)(2k+1) = 4k^2 + 4k + 1$ (Distributive Property) $3$) $4k^2 + 4k + 1 = 2(2k^2 + 2k) + ...
2
votes
1answer
116 views

Proving $\displaystyle \int_0^{1} \sin\left(x + \frac{1}{x}\right)\, dx$ Exists

As the title says, I need to show that $$\int_0^{1} \sin\left(x + \frac{1}{x}\right)\, dx$$ exists. After performing the substitution $x = 1/u, dx = -1/u^2 du$, the integral becomes ...