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

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2answers
77 views

Showing that ∀x ∈ S ∀y ∈ E ( Q(x, y) → R(x) ) ≡ ∀x ∈ S (∃y ∈ E Q(x, y)) → R(x)

Q(x, y) := “Student x did exercise y in the book” R(x) := “Student x gets an A in the class” So my goal is to show that the following equivalency holds: ∀x ∈ S ∀y ∈ E ( Q(x, y) → R(x) ) ≡ ∀x ∈ S ...
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4answers
41 views

Proving a increasing function with algebra

I'm attempting to prove a quadratic function is increasing without any calculus, just using algebra facts. My question: Consider the function $g(x) = (x + \dfrac{1}{2})^2 + \dfrac{7}{4}$ Prove that ...
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0answers
28 views

A question involving Partial Steiner Triple Systems

I've been given the following question, which I think I've completed, but I just wanted to check whether what I've said is valid. Suppose that a PSTS(23) with a $K_5$ leave is constructed using ...
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2answers
45 views

Proof by induction for $ 2^{k + 1} - 1 > 2k^2 + 2k + 1$ for $k > 4$

I was given this proof for hw. Prove that $ 2^{k + 1} - 1 > 2k^2 + 2k + 1$ So, far I've gotten this Basis: $k = 5$, $2^{5 + 1} - 1 > 2\cdot5^2 + 2\cdot5 + 1$ => $63 > 61$ (So, the basis ...
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2answers
60 views

Variance of sample mean (problems with proof)

Assuming that I have $\{x_1,\ldots, x_N\}$ - an iid (independent identically distributed) sample size $N$ of observations of random variable $\xi$ with unknown mean $m_1$, variance (second central ...
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1answer
101 views

Prove that all subsequential limits are contained within a closed interval

Let $a, b$ be two real numbers such that $a < b$, and suppose that $(s_n)_{n=1}^\infty$ is a sequence such that $\forall\,\, n\,\, a \leq s_n \leq b$. Prove that all subsequential limits are ...
5
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1answer
115 views

Proving a strange identity

Numerically, it would seem the following identity holds true: $$\frac{6}{7}=\lim_{n\to\infty}\sqrt[n]{\sum_{k=3}^\infty{\left(k-\sum_{j=1}^{k}\frac{1}{j}\right)^{-n}}}$$ Down below I have proven ...
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0answers
39 views

Let $a$ be an element of order $n$ in a group $G$. If $a^m$ has order $n$, then $m$ and $n$ are relatively prime.

Let $a$ be an element of order $n$ in a group $G$. If $a^m$ has order $n$, then $m$ and $n$ are relatively prime. Assume $a^m$ has order $n$ and, $m$ and $n$ are not relatively prime. Then ...
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3answers
61 views

Uniform convergence of $f_n(x) = n \sin(\frac{x}{n}) , x \in [-r,r]$

It is asked to prove that $$f_n(x) = n \sin(\frac{x}{n}) , x \in [-r,r]$$ Converges uniformly on the given interval for $r>0.$ The resolution of this suggested considered the fact that the ...
3
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1answer
63 views

Show that $f$ is continuous if it follows the intermediate value property

If $f: [a,b] \to \mathbb{R}$ is $1-1$ and has the intermediate-value property -- that is, if $y$ is between $f(u)$ and $f(v)$, there is at least one $x$ between $u$ and $v$ such that $f(x)=y$ -- show ...
0
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1answer
67 views

Spotting mistake: unnecessary given condition

I have solved the following problem without using a given premise. Could someone please spot whether I have done something wrong? Suppose we have a relation $\geq$ that is transitive, but not ...
0
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1answer
28 views

Show that $\mathbb{N}=\{ 0\} \sqcup S(n)$ and also that that union is disjoint?

Show that $\mathbb{N}=\{ 0\} \sqcup S(n)$ and also that that union is disjoint? I've been assigned this exercise in my lectures of elements of mathematics 2. Three axioms have been given for a Peano ...
0
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1answer
428 views

If $A$ is a subset of $C$ and $B$ is a subset of $C$, then the union of $A$ and $B$ is a subset of $C$

If $A$ is a subset of $C$ and $B$ is a subset of $C$, then $A\cup B$ is a subset of $C$. I was considering letting $x$ be an element of $A$ and $B$ and going from there, but I'm not sure that that is ...
2
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4answers
153 views

Is $x^y$ always irrational if $x$ is rational and $y$ is irrational?

Prove or disprove: "If $x$ is a rational number, and $y$ is an irrational number then $x^y$ is irrational" I am stuck with this, these are my steps. let $x=2$ and $y=\sqrt{2}$ ...
2
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3answers
58 views

Question about the Characteristic of $\mathbb{F}_{p^n}$

We can prove that any finite field of prime characteristic $p$ must have $p^n$ elements. Conversely, let $F$ be a finite field with $p^n$ elements, where $p$ is a prime number. Is the following ...
3
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1answer
89 views

Isomorphism of ring localized twice - Atiyah Macdonald Exercise 3.3

I studied AM before studying universal properties. When I solved the following exercise, I had a tedious solution that involved dealing with elements. Let $ A $ be a ring with multiplicatively ...
4
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1answer
54 views

Show that $\Gamma \cup \{\neg \phi\}$ is satisfiable if and only if $\Gamma\not \models \phi$

Let $\Gamma$ be a set of formulas and $\phi$ be a formula. Show that $\Gamma \cup \{\neg \phi\}$ is satisfiable if and only if $\Gamma\not \models \phi$. This seemed pretty obvious but I wanted ...
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1answer
56 views

what is the negation of ∀x∀y(xy ∈ nN) =⇒ (x ∈ nN ∨ y ∈ nN).

what is the negation of ∀x∀y(xy ∈ nN) =⇒ (x ∈ nN ∨ y ∈ nN). Is this correct? if the negation of p=>q is p∧~q then the answer is ∀x∀y(xy ∈ nN) ∧ ~(x ∈ nN ∨ y ∈ nN) = ∀x∀y(xy ∈ nN) ∧ ~(x ∈ nN ∨ y ∈ ...
2
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1answer
46 views

Short clarification on induction prove with Gamma defnition

Suppose we are asked to prove this one using induction: $$k! = \int_0^\infty e^{-x}x^{k} dx \,\,\, (*)$$ For $k=0$, it is clear after evaluating the appropriate improper integral that, $$0! = ...
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0answers
42 views

Enumerating the rationals in $[-1,1]$ so that the average converges to a prescribed limit $t\in [-1,1]$

Suppose $(q_n)$ is an enumeration of the rationals in $[-1,1]$ (meaning $q:\mathbb{N}\rightarrow \mathbb{Q}\cap [-1,1]$ is a surjection) and let $t\in [-1,1]$. Show that there is a reordering ...
0
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1answer
62 views

Implicit declaration of function 'exp'

Hello I'm trying to inpute answer for 3 hrs, but lon kapa said I'm wrong... $$8x^5 e^{3y} + 11 y^4 e^{2x} = 17$$ so we use chain rule $$40x^4 e^{3y} + 8x^5 e^{3y} 3y(dy/dx) + 44y^3 y(dy/dx)e^{2x} + ...
0
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1answer
239 views

Set of open intervals in R with rational endpoints is a basis for standard topology on R

Show that the set $\mathcal{B} = \{(a,b) \subset \mathbb{R}: a,b \in \mathbb{Q}\}$ is a basis for the standard topology on $\mathbb{R}$ First I'll show that $\mathcal{B}$ is a basis on ...
3
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4answers
56 views

Proving that $\limsup_{n\to\infty}\frac{1}{n}\sum_{m=1}^n s_m\leq \limsup_{n\to\infty}s_n.$

I am reviewing for my first year analysis exam and am stuck on a problem. Let $\sigma_n=\frac{1}{n}\sum_{m=1}^n s_m$. I am trying to show that, if $(s_n)$ is a bounded sequence of real numbers, ...
5
votes
1answer
240 views

Let U, W be subspaces of a vector space V. Suppose U ⊆ W. Prove or disprove: U + W = W

So, I know that W + W = W. And it makes sense that there is no counterargument that the claim isn't true. So, here is my attempt: Claim: $U + W = W$ Proof: $$W \subset U + W$$ Let $w \in W$. Then ...
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3answers
153 views

Truth Table problems

The problem: You are walking in a labyrinth, which contains at its center a vast treasure. Suddenly, you find yourself in front of three possible paths: a gold path to your left, a marble ...
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0answers
54 views

Branch of the nth root of a complex number

Let $n\geq2$ and $f(z)=z^n,z\in \mathbb{C}$. I need to show that if $L$ is a branch of the logarithm function in a domain $D$ then $h_{1/n}(z)=e^{L(z)\over n}$ is a branch of $f^{-1}$ in $D$. (If $L$ ...
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0answers
234 views

Proof of law of reflection using Fermat's principle : are we really proving the law of reflection?

Before you skip reading this, let me tell you that this isn't a "how to derive the law of reflection using Fermat's principle" question. Also, I asked it on MSE instead of the physics site because ...
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0answers
129 views

Finitely additive function on an infinite set, s.t., $m(A)=0$ for any finite set and $m(X)=1$ (constructive approach)

Other exercise which I found in Dudley's Analysis book: Show that there is a measure on a infinite set $X$, defined on $2^X$ s.t. is finitely additive, $m(A)=0$ for any finite set and $m(X)=1$. ...
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2answers
38 views

Find integers $r$, $s$, and $t$ such that $12r + 30s + 18t = 2$

Could someone please explain if such integers exist and how to find them? If not, could someone please explain how to prove that they don't exist? Thank you!
5
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1answer
39 views

Set of all subsets of X that contain a set Q is a topology

Let $X$ be a set such that $Q \subset X$. Show that $\tau = \{\emptyset\} \cup \{U \in \mathcal{P}(X): Q \subset U\}$ is a topology on X. $\emptyset \in \tau$ by definition and $X \in \tau$ ...
3
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1answer
136 views

Pigeonhole principle (I think): colored points in the plane

Suppose that each points in $\Bbb R^2$ is colored red, green or blue. Prove that either there are two points of the same color a distance $1$ unit apart, or there is an equilateral triangle of side ...
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1answer
79 views

Predicting the outcomes of a subset of chess games correctly

Suppose $n$ games of chess are played. In how many ways can I predict the outcomes of $m$ of the games ($A$ wins, $B$ wins, there is a draw) correctly? Here's my solution. I can choose the $m$ ...
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3answers
189 views

Proof verification: if $f,g: [a,b] \to \mathbb{R}$ are continuous and$f=g$ a.e. then $f=g$.

Suppose $f$ and $g$ are continuous functions on $[a,b]$. Show that if $f=g$ a.e. on $[a,b]$, then, in fact, $f=g$ on $[a,b]$. Is a similar assertion true if $[a,b]$ is replaced by a general measurable ...
0
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1answer
45 views

Matrices and bases

Can you please verify my argument: Let $M = \begin{pmatrix} a & b\\ c& d\end{pmatrix}$, where $a,b,c,d$ are all real. $$AM=\begin{pmatrix} c & d\\ a& b\end{pmatrix}$$ Let $B$ be ...
1
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2answers
95 views

Prove the Inequality on $\pi$-function

Prove that for each $y \geq 2$ , we have $\pi(x)+\pi(y)>\pi(x+y)$ for all sufficiently large $x$. I tried searching in the Internet for quite a while. The best result that I have found is L. ...
2
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1answer
53 views

Verification of identity $2\sum_{m,n\geq 1\,;\,m>n}\frac{1}{m^k n^k}=\left(\sum_{n\geq 1}\frac{1}{n^k}\right)^2 -\sum_{n\geq 1}\frac{1}{n^{2k}}$ [closed]

Is this identity true? $$2\sum_{m,n\geq 1\,;\,m>n}\frac{1}{m^k n^k}=\left(\sum_{n\geq 1}\frac{1}{n^k}\right)^2 -\sum_{n\geq 1}\frac{1}{n^{2k}}$$ If so, how to prove it? Could you provide me a ...
0
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1answer
37 views

Proving that a solution involving the Laplacian is unique.

I've been asked the following question; If $u$ is a solution of $\nabla^2u = p(x)u$, for $x \in D$, and $\nabla u \cdot n = g(x)$, for $x \in \partial D$, show that $u$ is unique. So, to begin, ...
2
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2answers
55 views

Show that $\Gamma_f:\mathbb{R}\to\mathbb{R}^2$ by $\Gamma_f(x)=(x,f(x))$ is continuous, with $f$ continuous.

The entire problem statement is, Let $f:\mathbb{R}\to\mathbb{R}$ be continuous. Define $\Gamma_f:\mathbb{R}\to\mathbb{R}^2$ by $\Gamma_f(x)=(x,f(x))$. Show that $\Gamma_f$ is continuous. My attempt ...
1
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1answer
56 views

Asymptotic Normality of MLE when data is modelled with covariates

Say I have data vector $X_1,\ldots,X_n$ which I want to model with some parametric distribution function $f(X_i;\theta,Z_i)$ and covariates $Z_i$. In this case, how can I prove the asymptotic ...
1
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1answer
46 views

Is the set $E=\{0.a_1a_2… \in \mathbb{R}\mid a_i= 4 \text{ or } a_i=7\}$ dense, compact or perfect?

I want to check my reasoning, I found that it's not dense but it's compact and perfect. $1$- It's not dense for 1 is neither in the set of a limit point of it. $2$- It's compact because it's both ...
0
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1answer
39 views

Is this proof with logarithmic exponentials correct?

I was unsure of this proof and some of the log rules I applied, could you check my proof and tell me if this proof is correct and if not, then what specifically is incorrect about the proof? ...
4
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2answers
63 views

How to prove the following bounds expression

Let n be a positive integer. Prove that there are 2^(n−1) ways to write n as a sum of positive integers, where the order of the sum matters. For example, there are 8 ways to write 4 as the sum of ...
0
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1answer
59 views

Double check $G\sim H$ iff $G≈H$

Let $S$ be the collection of all groups. Define a relation on $S$ by $G \sim H$ iff $G ≈ H$. Prove that this is an equivalence relation. So $S$ is partitioned into isomorphism classes. Proof: Let ...
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2answers
42 views

Let $G$ be a group, $x∈G$, $ a,b∈\Bbb Z$ and $a⊥b$. If $x^a=x^b$, then $x=1$.

There is a missing step in this proof: http://math.stackexchange.com/a/106292/135812 Lemma Let $G$ be a group, $x\in G$, $a,b\in \mathbb Z$ and $a\perp b$. If $x^a = x^b$, then $x=1$. ...
0
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2answers
36 views

Choosing a committee with a constraint - where is my reasoning wrong?

Okay, this is an example from Challenge and Trill of Pre-college Mathematics by Krishnamurthy et al. In how many ways can we form a committee of three from a group of 10 men and 8 women, so that ...
1
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3answers
58 views

Show that if $A$ is any square matrix such that $A^n = 0$ for some positive intiger $n$, then $A$ is not invertible. (answer check)

Show that if $A$ is any square matrix such that $A^n = 0$ for some positive integer $n$, then $A$ is not invertible. I'm not sure if my proof is good enough, or enough "work" as my teacher put it ...
2
votes
0answers
57 views

Evaluating a limit…

I was solving a physics problem and this expression came about: $E =\lim_{N \to \infty} \left( \dfrac{k_0Q}{NR²}\displaystyle\sum_{i=0}^{(N/2-1)}\left[ \left( ...
2
votes
2answers
68 views

If $f \in L^2(\mathbb T)$ then $S_n(f) \to f$ in $L^2$ sense.

Theorem: If $f \in L^2(\mathbb T)$, then $S_n(f) \to f$ in $L^2(\mathbb T)$ sense. Proof: Let $f \in L^2(\mathbb T)$, then by definition $\|f\|_2^2 = \frac{1}{2\pi} \int_0^{2\pi} \vert f(x) \vert^2 ...
0
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0answers
57 views

Set of semi infinite intervals on the real line is a topological basis

Show that the set $\{(r,\infty): r \in \mathbb{R}\}$ is a basis for a topology on the set of real numbers but not a topology itself. Any feedback on my proof would be appreciated. A collection of ...
1
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0answers
49 views

Proof Verification/Alternative to Induction- Well ordering proof

This question grew out of Induction and Maximum Principle, which yours truly asked Sep 23, at 11:51. Due to Mauro Allegranza's suggest, I changed focus so as to first prove the equivalence of $(a)$ ...