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

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1answer
66 views

If $\mathbb{N}\sim B$ and $A\subseteq B$, then $\mathbb{N}\sim A$ or $A$ is finite, or $A = \emptyset$

A similar question has been asked, however, it concentrates on using induction to prove this statement. I have tried proving the statement differently, so I am interested if what I did is valid or ...
0
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1answer
41 views

Prove that for two continuous curves that “touch” in a single point, the two tangents in that point are equal

Prove that if two continuous, first order differentiable curves reach a common value in a single point without intersecting, the tangent line of the two curves in that point must be the same. My ...
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4answers
51 views

counterexample to a proof.

Prove the following statement; $$\forall a,b \in R (\forall \epsilon > 0 (a \le b + \epsilon) \rightarrow a \le b)$$ I can't see how this is true This means that I can pick a number for all ...
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0answers
11 views

Problems getting the covariance matrix of the ressiduals

In order to get the variance-covariance matrix of the residuals of a linear regression model, I do the following: Considering that the residual vector $e$ is: $e = Y - \hat{y} = XB+\epsilon - Xb$ ...
0
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1answer
30 views

Help with Proposition $2.3.3$ from Elem. Differential Geometry by Pressley

Why can we have $\mathbf v \cdot \mathbf N =d$? Why is $\mathbf v \cdot \mathbf N =d$ a plane? Where did $\gamma \cdot \mathbf N=d$ come from? Why can we do this?
3
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2answers
32 views

Show that all elements of one sequence are less than all elements of another sequence.

Let $\{a_n\}_1^\infty$ and $\{b_n\}_1^\infty$ be two sequences in $\mathbb{R}$ such that $\forall n \in \mathbb{N}$, it is true that $a_n \leq b_n, a_n \leq a_{n+1}, \text{and} b_{n+1} \leq b_n$. We ...
1
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1answer
30 views

Solving the linear recurrence $ f(n) = f(n - 1) + 12f(n-2)$

Solve the linear recurrence: $$ f(1) = 10, f(2) = -2,\quad f(n) = f(n - 1) + 12f(n-2)$$ My solution is below. Assume: $f(n) = x^n$ $$x^n = x^{n-1} + 12x^{n-2}$$ $$x^2 = x + 12 $$ $$x^2 - x ...
3
votes
2answers
87 views

Show that sup$AB$=(sup$A$)(sup$B$)

Where $AB$ is the product of the sets and $A,B \in \mathbb{R^+}$. Since $A,B$ are bounded above sup $A$ and sup $B$ exist. Let $\alpha = $ sup $A$ and $\beta = $ sup $B$. This implies $\forall a \in ...
1
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1answer
19 views

Normality of subgroup of image of homomorphism

I would appreciate one's insight into this proof, if I'm missing something in it. If there exists homomorphism $\phi: G\to H$ and some subgroup $M$ of $H$, and the pull-back of $M$ is $\phi^{-1}(M) ...
3
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4answers
40 views

Two Proofs on a Simple Set Concerning Suprema and Maxima

Suppose we have the set $$\Gamma = \{\frac{n}{n+1} : n \in \mathbb{N}\}$$ We wish to determine what is $\sup \Gamma$ and $\max \Gamma$. We can clearly see that $\sup \Gamma = 1$ and that $\max ...
0
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1answer
49 views

Pullback of subgroup of homomorphism is subgroup of its domain

Given that $\phi: G\to H$ is a homomorphism, and that $K \leq H$, and also that $\phi^{-1}(K) := \{g \in G: \phi(g)\in K\}$; if one wants to prove that $\phi^{-1}(K)$ is closed under product and ...
2
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1answer
34 views

On Abel summation $\sum_{e<n\leq x}\left(\mu(n)\cdot\int_2^n\frac{ds}{\log s}\right)$, where $\mu(n)$ is the Möbius function

By Abel's identity for $Li (x)=\int_2^x\frac{ds}{\log s}$, $a(n)=\mu(n)$ the Möbius function and $[y=e,x]$ (see Theorem 4.2, page 77 of [1]) and an application of Fundamental Calculus Theorem we ...
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0answers
33 views

Prove the prob. of even when draw marbles from urn

I have a question about Hypergeometric distribution. The original question was solved by Graham Kemp in here. I would like to extend that question in some case. This is my question I has an urn with ...
2
votes
2answers
40 views

Getting two answers to: How many monoalphabetic substitution ciphers of $\{A,B,C,D\}$ are possible in which no letter is fixed?

From Combinatorics by Mazur: I am trying this with $\{A,B,C,D\}$ but I am getting two answers. If I enumerate then I get $9$. $ABCD,\ ABDC,\ ACBD,\ ACDB,\ ADBC,\ ADCB$ $BACD,\ BADC(1),\ BCAD,\ ...
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0answers
38 views

Check proof of fact that $x_n \rightarrow x$, $x \ne 0$ implies $1/x_n \rightarrow 1/x$

I'm studying analysis on my own, and wish to check if my solution to the following problem is correct, and if so, whether it can be made more concise. (The problem is from "General Theory of Functions ...
1
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1answer
102 views

Solve $x^2-2\sqrt{x}+1=0$ in $\mathbb{R}$ without using numerical methods?

How do I solve this equation: $$x^2-2\sqrt{x}+1=0$$ in $\mathbb{R}$ without using numerical methods? Note: I have used variable change by letting $y=\sqrt{x}$ I got this ...
0
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2answers
43 views

Abelian normal subgroups proof

If $A$ and $B$ are normal subgroups such that $G/A$ & $G/B$ are both abelian prove that $G/(A \cap B)$ is abelian. I did it as follows Consider the mapping $\phi : G/A \rightarrow G/B$ defined ...
0
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1answer
82 views

Proof writing involving functions: Injective and Surjective [closed]

Given two sets that have the same cardinal number Example: \begin{align*} A & = \{1, 4\}\\ B & = \{1, 2\} \end{align*} How would you prove that the function from $A$ to $B$ is always ...
5
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1answer
88 views

Verify proof of $ \left( 1 + \frac{x}{n} \right)^n < e^x \,\text{and} \, e^x < \left(1 - \frac{x}{n}\right)^{-n}$

The following is from Tom Apostol's Calculus I, on page 250, exercise 42.: If $\mathit{n}$ is a positive integer and if $\mathit{x} > 0$, show that $$ \left( 1 + \frac{x}{n} \right)^n < e^x ...
3
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2answers
90 views

Is this proof correct?($\mathbb{Q}$ is not locally compact)

I am supposed to prove that $\mathbb{Q}$ is not locally compact. The definition of locally compactness is: A space $X$ is said to be locally compact at x if there is some compact subspace C of X ...
4
votes
1answer
53 views

Limit of a weird sequence

I found the following question in a book:- $Q:$Let $a_1, a_2, ... , a_n$ be a sequence of real numbers with $a_{n+1}=a_n+\sqrt{1+a_n^2}$ and $a_0=0$. Prove that $$\lim_{n \to \infty} \frac ...
2
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2answers
55 views

Let $(x,y)$ be the smallest solution $\in \mathbb N^+$ of $x^2+xy-y^2=0$, then $y-x<x$. Why?

The book shows why $x^2+xy-y^2=0$ doesn't have any solutions in $\mathbb N^+$: Let $(x,y)$ be the solution with smallest $x \in \mathbb N^+$ of $x^2+xy-y^2=0$ (where $y$ must be $> x$). ...
1
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2answers
44 views

Proving that $\frac{\sin \alpha + \sin \beta}{\cos \alpha + \cos \beta} =\tan \left ( \frac{\alpha+\beta}{2} \right )$

Using double angle identities a total of four times, one for each expression in the left hand side, I acquired this. $$\frac{\sin \alpha + \sin \beta}{\cos \alpha + \cos \beta} = \frac{\sin \left ( ...
3
votes
1answer
86 views

Function is continuous iff it's graph is compact

Let $\text{Dom}(f)=E$ be compact and $G_f=\{(x,f(x)): x\in E\}$ the graph of $f$. $G_f$ is compact iff $f$ is continuous on $E$. Proof: $\Rightarrow$ Let $f$ is not continuous at some point $x_0\in ...
0
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1answer
101 views

Assume $X \subset \operatorname{span}(Y)$ and $Y \subset \operatorname{span}(X).$ Prove $\operatorname{span}(X) = \operatorname{span}(Y).$

Let $X,Y$ be sequences or subsets of a vector space $V$ over a field $F.$ Assume $X \subset \operatorname{span}(Y)$ and $Y \subset \operatorname{span}(X).$ Prove $\operatorname{span}(X) = ...
3
votes
1answer
24 views

Proof that compact discrete set is finite with sequences

Let $(M,d)$ be a metric space and $A\subset M$ be a discrete set. I want to show that if $A$ is compact, then $A$ is finite using sequences. I proceeded as follows, but I'm unsure about one step of my ...
0
votes
0answers
34 views

Is my limited understanding of division and gcd on track?

Hello I am trying to make sense of some beginner theorems and propositions in number theory. I am wanting to also know if what I am saying is valid or just completely wrong. I am wanting to show that ...
3
votes
3answers
65 views

Prove that $5^{2n-1} - 3^{2n-1} - 2^{2n-1}$ is divisible by 15 for n $\in$ $\mathbb{N}$

The book I am using for my Combinatorics course is Combinatorics:Topics, Techniques, and Algorithms. Prove that $5^{2n-1} - 3^{2n-1} - 2^{2n-1}$ is divisible by 15 for n $\in$ $\mathbb{N}$ This is ...
3
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2answers
60 views

Prove that 'isomorphism' is an equivalence relation (on any set of sets)

Consider identity function $1_A: A \to A.$ The identity function is bijective, so isomorphism is reflexive. Inverse of a bijection is bijection, so isomorphism is symmetric. Composition of two ...
0
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0answers
35 views

if $0<a \le b \le c$ prove $(x_n) = (a^n+b^n+c^n)^{1/n} \longrightarrow c$ [duplicate]

if $0<a \le b \le c$ prove $(x_n) = (a^n+b^n+c^n)^{1/n} \longrightarrow c$ $\mid \sqrt[n]{a^n+b^n+c^n} - c \mid \, < \,\mid \sqrt[n]{a^n}+\sqrt[n]{b^n}+\sqrt[n]{c^n} - c \mid$ doesn't get me ...
1
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2answers
44 views

Please explain this basic proof

In my freshman math course book there's a proof of associativity of addition on the natural numbers using mathematical induction. The author proves the base case and assumes the hypothesis, $a+(b+c) = ...
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2answers
39 views

Is this a valid way to prove a beginer supremum question

Hello suppose I have a set $$S=\left\{1-\frac{1}{n} : n \in \mathbb{N}\right\}$$ and I want to show that $\sup(S)=1$. Is the following valid? I see that $S$ is bounded from above , in particular by ...
2
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0answers
115 views

(algorithms) Show that in any base b >= 2, the sum of any three single-digit numbers is at most two digits long

So, I'd like someone to review my 'proof' and pick on it for incompleteness, and state how it could be improved. The question (reviewing algorithms) asks, "show that in any base b>=2, the sum of any ...
2
votes
3answers
136 views

On the proof that if a set is open and arc-connected then it's connected by broken lines.

Let $(X,d)$ be a metric space, and $\gamma : [a,b] \rightarrow X$ be a continuous function then $\gamma([a,b])$ is called a continuous arc. I want to prove that if a set $C$ is open and arc-connected ...
2
votes
2answers
55 views

Solutions $\in \mathbb N$ for $x^2+xy-y^2=c$ where $0<c\leq 10$.

I am currently stuck at $c=3$. Here's what I figured out for $c=2$ (hopefully correct): Since $x^2+xy-y^2$ is even only if x and y are even, we can write it as $$(2m)^2+2m2n-(2n)^2=2$$ which means ...
2
votes
1answer
102 views

Starting index of a sequence is irrelevant

"Let $(a_n)_{n=m}^{\infty}$ be a sequence of real numbers, let $c$ be a real number, and let $k \geq 0$ be a non-negative integer. Show that $(a_n)_{n=m}^{\infty}$ converges to $c$ iff ...
4
votes
1answer
118 views

Formally derive $\displaystyle\int_{x=-\infty}^{x=\infty} f(x) \delta(x) \, \mathrm{d}x = f(0)$

I have been searching for a derivation of the defining property for the Dirac-delta function: $\displaystyle\int_{x=-\infty}^{x=\infty} f(x) \delta(x) \, \mathrm{d}x = f(0)$ and found this derivation ...
0
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1answer
25 views

The span of a vector space with elements as linear combinations of no more than $r$ vectors has $dim V \leq r$

If $V=Span \{ \vec{v}_1, \dots, \vec{v}_n \}$ and if every $\{ v_i \}$ is a linear combination of no more than $r$ vectors in $\{ \vec{v}_1, \dots, \vec{v}_r \}$ excluding $\{ v_i \}$, then $dim V ...
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3answers
101 views

Question about an inequality on a proof

I'm stuck on a proof. There's a step that says: $$ \left| \Im\left(\frac{1-e^{i(N+1)x}}{1-e^{ix}}\right)\right| \leq \left| (\frac{1-e^{i(N+1)x}}{1-e^{ix}}) \right|,\quad \text{with } N \in ...
2
votes
1answer
59 views

What do I do next when trying to find the derivative of this fraction?

I'm trying to find the derivative of this equation: $-\frac{3(x-6)}{2\sqrt{9-x}}$ The quotient rule: $\frac{d}{dx}[\frac{f(x)}{g(x)}]=\frac{g(x)f'(x)-f(x)g'(x)}{(g(x))^2}$ where $g(x)$ and $f(x)$ are ...
0
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1answer
31 views

Solution Sets of Homogeneous Systems

I had to prove the following theorem: Suppose that $A\mathbf x=\mathbf b$ is consistent for some given $\mathbf b$, and let $\mathbf p$ be a solution. Then the solution set of $A\mathbf x=\mathbf b$ ...
2
votes
1answer
25 views

Prove composition of two surjections is surjection

Suppose $f: A \to B$ and $g: B \to C$ are both surjections. Since $f$ is surjective, then for every $b \in B$ there exists $a \in A$ such that $f(a)= b$. Since $g$ is surjective, for every $c \in ...
0
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0answers
34 views

Linear algebra - isomorphisms proof confirmation

I asked a question here before about a certain problem which troubled me, and I want to verify if my proof is correct with the new knowledge I obtained. The problem is as follows: Let $\rho : ...
1
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1answer
77 views

Are the fibers of a group homomorphism the cosets of its kernel?

Given a group $G$ and a homomorphism $f : G \to G'$, $f$ induces a relation on $G$, which we will denote $\sim$. The relation is $g_1\sim g_1$ iff $f(g_1) = f(g_2)$. The fiber of an element $y \in G'$ ...
0
votes
1answer
47 views

$K[x]$ is not isomorphic with $K[[x]]$

Let $K$ be a field. Prove that $K[x]$ is not isomorphic with $K[[x]]$ as $K$-vector spaces. My solution: since $K $ is a field, $K[x]$ is pid then is noetherian ring. So every ideal of $K[x]$ is ...
3
votes
1answer
46 views

Intersection of Normal Subgroups is normal, set approach

Assuming that we know that, given two normal subgroups $H,K$ of a group $G$ that their intersection is also a subgroup of $G$, the goal is to show that $H\cap K$ is also normal. I saw a couple of ...
0
votes
1answer
39 views

Work against friction is proportional to length of path

If, given that the frictional force is constant, one wants to show that the work done against friction is proportional to the length of the path, would this line of reasoning be correct? We can use ...
0
votes
1answer
42 views

Linear combination of gradient vector fields

Suppose we have that $\bf{F}=\vec{\nabla}$ ${f}$ and $\bf{G}=$ $\vec{\nabla}g$ are three gradient vector fields in $\mathbb{R}^n$, such that $\bf{H}=$ $c_1\bf{F}+$ $c_2\bf{G}$ for some $c_1, c_2 \in ...
2
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0answers
121 views

Proof that $\inf A = \sup B$

Exercise Suppose $A$ is a nonempty set of reals that is bounded below. Let $B$ be the set of lower bounds for $A$, and assume further that $B$ is not empty and bounded above (I have proven that $B ...
1
vote
1answer
31 views

find the p.d.f. first by determining their d.f.’s, and secondly directly..

If the r.v. $X$ is distributed as Negative Exponential with parameter $λ$, find the p.d.f. of each one of the r.v.’s $Y, Z$, where $Y = e^X,\, Z = \log{X}$, first by determining their d.f.’s, and ...