Questions tagged [solution-verification]

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

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23 views

Range of the function $f:\mathbb{Z} \to (\mathbb{Z}/4\mathbb{Z},\mathbb{Z}/6\mathbb{Z})$

Let $f:\mathbb{Z} \to (\mathbb{Z}/4\mathbb{Z},\mathbb{Z}/6\mathbb{Z})$ be he function given by $f(n)=(n$ mod 4,$n $ mod $6)$.Then $(1)(0$ mod $ 4 ,3$ mod $6)$ is in the image of $f$ $(2)(a$ mod $ 4 ...
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27 views

Find the probability of receiving the unwanted present

There are $3n$ people in the anonymous gift society. They're preparing the presents for one another. It's known that exactly $n$ people would like to have a tie as a gift, $n$ people would like to ...
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17 views

About similar functions to Dirichlet Eta Function

It is known that the Dirichlet Eta Function converges for $s \in \mathbb{C}: Re(s)>0$ and it is defined to be $$\eta(s) = - \displaystyle\sum_{n=1}^\infty \underbrace{\frac{(-1)^n}{n^s}}_{a(n)}$$ ...
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1answer
14 views

Doubt on inclusion-exclusion principle result for adjacent choices

I wish to find the number of ways in which at least two adjacent points can be chosen from a line of five points with the inclusion-exclusion principle. We start with two adjacent points, of which ...
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1answer
18 views

Proving a given set is not a vector space

Let $V$ denote the set of ordered pairs of real numbers. If $(a_1,a_2)$ and $(b_1,b_2)$ are elements of $V$ and $c\in \mathbb{R}$, define $$(a_1,a_2)+(b_1,b_2)=(a_1+b_1,a_2b_2)$$ and $$c(a_1,a_2)=(...
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1answer
19 views

Describe the automorphism group of $Aut(\mathbb{Z}/9\mathbb{Z})$.

a)Describe the automorphism group of $Aut(\mathbb{Z}/9\mathbb{Z})$. b) Prove that if a Group G has the trivial center then $|Aut(G)| \geq |G|$. My attempt: a) Clearly, $\mathbb{Z}/9\mathbb{Z}$ is a ...
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1answer
42 views

Open set in product topology which is not “cylindrical”

I know that the product topology on $\prod_{i=1}^\infty\mathbb{R}$ has a basis of the form: $$ \prod_{i=1}^N U_i \times \prod_{i=N+1}^\infty \mathbb{R} \tag{*} $$ where $U_i\subseteq \mathbb{R}$ is ...
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44 views

Let A and B be sets. Prove A=B implies B=A.

In order to prove the foregoing statement we use the next lemma: For a set $A, A = A$. Let $A$ and $B$ be a couple of sets. Assume that $A = B$. Notice we must show that $B = B$ for $A = B$. ...
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An equivalent definition for the limsup $a_n$

Suppose that for $(a_n)$ the limit superior is finite. Prove the following statement: $$ L = \limsup_{n \to \infty} a_n \iff [ \forall \varepsilon>0 \exists k \in \mathbb{N} : \forall n >...
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Prove that $u(x,y) = \frac{x}{x^2+y^2}$ is harmonic in $\mathbb{R}^2\setminus\{(0,0)\}$

I want to check if I did this right. I reached the conclusion that $u$ is not harmonic. We know that a function is harmonic if $$\displaystyle\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\...
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31 views

Suppose that $x_{n}\to x\in X$ and $y_{n}\to y\in X$. Show that $\displaystyle\lim_{n\rightarrow\infty}d(x_{n},y_{n}) = d(x,y)$.

Let $x_{n}$ and $y_{n}$ be two sequences in a metric space $(X,d)$. Suppose that $x_{n}$ converges to a point $x\in X$ and $y_{n}$ converges to a point $y\in X$. Show that $\displaystyle\lim_{n\...
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Application of Burnside's Lemma on the vertices of a cube

Eight congruent equilateral triangles, each of a different color, are used to construct a regular octahedron. How many distinguishable ways are there to construct the octahedron? (Two colored ...
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1answer
15 views

Proving intersection of POsets is a POset - Reflexive

I need to prove that the intersection of 2 POsets R and S is a POset. So we basically want to prove that if $R$ and $S$ are POsets then $R \cap S$ is reflexive, transitive and anti-symmetric. ...
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33 views

If $f$ is differentiable at $x_{0}$, then $f$ is also differentiable in the direction $v$ at $x_{0}$, and $D_{v}f(x_{0}) = f'(x_{0})v$

Let $E$ be a subset of $\textbf{R}^{n}$, $f:E\to\textbf{R}^{m}$ be a function, $x_{0}$ be an interior point of $E$, and let $v$ be a vector in $\textbf{R}^{n}$. If $f$ is differentiable at $x_{0}$, ...
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Computation for the Fourier series for a function $|\sin x|$ on an interval $[-\pi, \pi]$

Here is what I've done: $$a_0 = \frac{1}{2\pi}\int^{-\pi}{\pi}|\cos x| = \frac{1}{2\pi}\int^{-\pi}_{\pi}|\cos x|dx = -\frac{1}{2\pi}\int_{-\pi}^{-\frac{\pi}{2}}\cos x dx + \frac{1}{2\pi}\int_{-\frac{\...
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1answer
35 views

Prove by contradiction that $A\cup B=B\cup A.$

For arbitrary sets $A$ and $B$, assume the opposite of our conclusion: $A\cup B \neq B \cup A$. Thus, either there exists $x\in (A \cup B)$ such that $x \notin (B\cup A)$ or there exists $x'\in (B\...
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Find $g(x)$ if $f(x)= \begin{cases} -1, & -2 \leq x \leq 0 \\ x-1, & 0 < x \leq 2 \end{cases}$ and $g(x) = |f(x)| + f(|x|)$

$$f:[-2,2] \rightarrow \Bbb R$$ $$\text {and }f(x)= \begin{cases} -1, & -2 \leq x \leq 0 \\ x-1, & 0 < x \leq 2 \end{cases}$$ And, let $g(x)$ be equal to $|f(x)|+f(|x|)$ We need to find ...
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1answer
38 views

Prove the following two statements about differentiability are equivalent

Let $E$ be a subset of $\textbf{R}$, $f:E\to\textbf{R}$ be a function, $x_{0}\in E$, and $L\in\textbf{R}$. Then the following two statements are equivalent (a) $f$ is differentiable at $x_{0}$, and $...
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2answers
52 views

Prove by contradiction that: $A∩B=B∩A$. Is my proof right?

Prove by contradiction that: $A∩B=B∩A$. My Solution: Assume to the contrary that $A∩B ≠ B∩A$ .... (i) Consequently, $B∩A≠ A∩B$ ...... (ii). Since for all sets $A$ and $B$ , $A=B → B=A$. Then, ...
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72 views

Let $T:\textbf{R}^{n}\to\textbf{R}^{m}$ be a linear transformation. Show that there exists a number $M > 0$ such that $\|Tx\|\leq M\|x\|$.

Let $T:\textbf{R}^{n}\to\textbf{R}^{m}$ be a linear transformation. Show that there exists a number $M > 0$ such that $\|Tx\|\leq M\|x\|$. Conclude in particular that every linear transformation ...
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Proof verification: the characteristic of a subdomain of an integral domain $D$ is equal to the characteristic of $D$

I attempted a proof for: the characteristic of a subdomain of an integral domain $D$ is equal to the characteristic of $D$ Proof: Assume $D$ is an integral domain with characteristic $r$. Since $D$ ...
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1answer
43 views

Proving $(\mathbb{Z}/n\mathbb{Z},+)$ is a group

I want to prove that $(\mathbb{Z}/n\mathbb{Z},+)$ is a group. I have proved it myself but, as an unexperienced student, I'd like to check with you if my reasoning is correct. I have already proved ...
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43 views

Prove that $A\cap B =B \cap A$. [closed]

Let $A$ and $B$ be arbitrary sets. Assumme to the contrary that $A\cap B \neq B \cap A$. If $A \cap B=B \cap A$ then $B \cap A = A \cap B$. That is $A \cap B = A \cap B$. Yet since $A \cap B \neq B \...
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1answer
37 views

commenting on whether $f'(x)$ is even or odd.

The question is as follows: Let $f(x)$ be a differentiable function $\forall x,y \in \Bbb R$ and $$f(x-y),f(x),f(y),f(x+y)$$ are in AP then comment whether $f'(x)$ is even or odd (given $f(0) \neq 0)...
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30 views

Find vectors $u,v,w$ so that their combination $cu+dv+ew$ only fill a plane. [closed]

For the question, "Find vectors $u,v,w$ so that their combination $cu+dv+ew$ only fill a plane", can the answer also be $u=v$ or $v=w$? The answer in the solution manual is $w=u+v$.
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1answer
16 views

Hamiltonian graph join

Assume a graph $(V,E)$ with $n$ vertices and degree sequence $d_1 \geq d_2 \geq \dots \geq d_n$. The purpose of this question is to understand when the graph join $G^{(k)}$ (defined as the union of $...
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34 views

Sheafification from Vakil and alternate methods

I would like to verify my understanding of sheafification, and so I am going to apply it to work out the sheaf associated to the presheaf $\mathcal F$ defined on a topological space $X$ by $\mathcal F(...
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1answer
33 views

Prove: $R$ is local $\iff$ $R$ has exactly one maximal ideal.

We have that a commutative ring $R$ with $1$ is called local if $R − R^×$ is an ideal of $R$. I have to proof the following: $R$ is local $\iff$ $R$ has exactly one maximal ideal. We have that every ...
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1answer
34 views

Vanishing of cohomology of affine scheme

In EGA I 5.1, more specifically the proof of 5.1.9, which states that $X$ is affine iff the closed subscheme defined by a quasi-coherent sheaf of ideals $\mathscr{I}$ such that $\mathscr{I}^n = 0$ for ...
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1answer
20 views

How to prove that two series converge simultaneously?

Do $\Sigma_{n=1}^{\infty} a_n$ and $\Sigma_{n=1}^{\infty} n(a_n -a_{n+1})$ converge simultaneously? $$\Sigma_{n=1}^{\infty} n(a_n -a_{n+1}) = a_1 - a_2 + 2a_2 -2a_3 + 3a_3-3a_4 + ... = \Sigma_{n=1}...
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1answer
39 views

Finding the no of V-shaped Permutations?

Consider $n$ distinct real numbers: $a_{1}, a_{2}, \ldots, a_{n},$ A permutation $([1],[2], \ldots,[n])$ of the indices $\{1,2, \ldots, n\}$ is said to $V$ -shaped if there exists an integer $r \quad(...
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1answer
32 views

How to translate this statement into a mathematical one(using appropriate quantifiers)?

The statement I'd like to translate into a mathematical one is "Every American has a dream". Let $A$ and $D$ denote the set of all Americans and the set of all dreams, respectively, and $P(a,d)$ ...
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2answers
48 views

The Axiom of Choice: Proof Validity

Synopsis In Enderton's Element's of Set Theory, he introduces several forms of the Axiom of Choice. Currently, I've gotten through the first and second forms. Mainly: (1) For any relation $R$, ...
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22 views

Prove that $f$ has a power series expansion around any point in its disc of convergence

${\bf Exercise:}$ Let $f$ be a power series centered at the origin. Prove that $f$ has a power series expansion around any point in its disc of convergence. Proof: We are given that $f = \sum_{n \...
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1answer
56 views

Spivak Calculus Chapter 1 Problem 5 (ii)

Prove : If $a < b$ then $-b < -a$ My proof : $a + (-b) < b + (-b) $ $a - b < 0$ $a - b + (-a) < 0 + (-a)$ $a + (-a) -b < -a$ $-b < -a$ Is my proof correct?
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Prove or disprove that $\frac{(x+n)!}{x!}$ is not divisible by $n$ distinct primes where each prime is greater than $\frac{(x+n)e}{n}$

Is it true that for integers $n > e, x \ge n$, it is impossible for $\dfrac{(x+n)!}{x!}$ to be divisible by $n$ distinct primes where each prime is greater than $\dfrac{(x+n)e}{n}$? Example: $x=n=...
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2answers
60 views

Extreme values of $|z|$ when $|z^2+1|=|z-1|$

Problem statement: Find the extreme values of $|z|$ when $$|z^2+1|=|z-1|,\ z\in \mathbb{C}-\{0\}$$ My try: $$|z^2+1|=|z-1|\implies|z^2+1|^2=|z-1|^2\implies(z^2+1)(\overline{z}^2+1)=(z-1)(\overline{z}-...
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1answer
21 views

Unsure of a proof shorcut in a simple chasing the angle problem

I was given a simple geometry problem. You need to find the angle ADB. Line DB touches the circle. The angle can be easily found by making use of properties of circle and interior angles taught in ...
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1answer
20 views

Hrbacek/Jech Set Theory Axioms Proof

Is there a less tedious means of achieving this proof? The next section of the problem asks to generalize this result to four sets, hence why I ask. Given $A$, $B$, and $C$, there is a set $P$ such ...
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1answer
35 views

Calculate $6^{1866}$ in $\mathbb{Z}_{23}$ [duplicate]

Calculate $6^{1866}$ in $\mathbb{Z}_{23}$ $Solution:$ Note that $1866=22\cdot 84 + 18$ then by Fermat's theorem $$[6^{1866}]=[6^{22}]^{84}[6^{18}]=[1]^{84}[6^{18}]=[6^{18}]$$ Then $6^6=46656=2028\...
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25 views

Finding a bijective correspondence between $X^{\omega}$ and $\mathcal{P}(\mathbb{Z}_+)$

Let $X = \{ 0,1 \}$ and let $\mathcal{P} (\mathbb{Z}_+) $. Find a bijective correspondence between $\mathcal{P} (\mathbb{Z}_+) $ and the cartesian product $X^{\omega} $ or ${\bf show}$ there isn't ...
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29 views

Prove that a sequence of complex numbers $z_{n} = (a_{n},b_{n})$ converges to $z = (a,b)$ iff $a_{n}$ converges to $a$ and $b_{n}$ converges to $b$

The complex numbers $\textbf{C}$ with the distance $d(w,z) = |w - z|$ form a metric space. If $(z_{n})_{n=0}^{\infty} = (a_{n},b_{n})_{n=0}^{\infty}$ is a sequence of complex numbers, and $z = (a,b)$ ...
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23 views

Substitution for equation in two variables

Lets say I start with an equation $$3(1+x+x^2)(1+y+y^2)(1+z+z^2)=4x^2y^2z^2-1$$ and I get a solution in positive integers $(4,4^3,4^6)$. To go about showing that there are no solutions possible in odd ...
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14 views

Proof check: Subset of separable space is separable

I want to know if my proof is correct and, if so, if it could be simpler. I think the last two lines aren't fully convincing. Let $X$ be a separable metric space. Prove that every subset of $X$ is ...
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43 views

Given a sequence $x_{n}$ in the metric space $(X,d)$, prove that $L$ is a limit point iff there is a subsequence $x_{f(n)}$ which converges to $L$.

Let $(x_{n})_{n=0}^{\infty}$ be a sequence of points in a metric space $(X,d)$, and let $L\in X$. Then the following are equivalent (a) $L$ is a limit point of $(x_{n})_{n=1}^{\infty}$ (b) There ...
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2answers
56 views

How do I show that if $p_n q_{n-1} - p_{n-1}q_n = 1$, then $p_n/q_n$ converges?

Let $p_{n}$ and $q_{n}$ be strictly increasing, integer valued, sequences. Show that $$p_{n}q_{n-1}-p_{n-1}q_{n}=1,$$ for each integer $n \ge 1$, then the sequence of quotients $\frac{p_{n}}{q_{n}}$ ...
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1answer
63 views

Alternative proof of $a\times0= 0$

I was trying to find a proof of $a\times0 = 0$ by myself (assuming commutativity, associativity, distributivity, etc) and I came up with $$ a+0=a(1) \implies 1 = \frac{a+0}{a} = \frac aa + \frac 0a = ...
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4answers
71 views

${\log}_{a}{x}\neq {\int}^{x}_{1}{\frac{1}{t}}dt$

In most calculus textbooks, $\ln{x}$ is defined to be ${\int}^{x}_{1}{\frac{1}{t}}dt$. Some textbooks validate this definition by demonstrating that this function $\int^{x}_{1}{\frac{1}{t}}dt$ has all ...
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20 views

Alternative proof of the first order convexity condition

A differentiable function $f: X \subseteq \mathbb{R}^{n} \to \mathbb{R}$ is convex if and only if its domain $X$ is a convex set and $$f( \mathbf{y} ) \geq f( \mathbf{x} ) + \left\langle \nabla f( \...
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0answers
14 views

Placing balls of different colours into bins

I'm trying to prove the following statement: Suppose that you have 25 balls to place into five different bins. 11 of the balls are red, while the other 14 are blue. Prove that no matter how the ...

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