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

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1answer
84 views

Cyclic group of order $n$

Let $n$ be any integer. We construct a group of order $n$ as follows: $G$ will consist of all symbols $a^i$, $i=0,1,2 \cdots, n-1$ where we insist $a^0 = a^n = e$, $a^{i}\cdot a^{j} = a^{i+j}$ ...
1
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1answer
87 views

Centre of mass of bounded region conformation of numerical answer

Hello I am looking for any help on solving the following; I want to find the centre of mass of the uniform solid that is bounded by the regions $x^2+y^2=2x$, $z=\sqrt{x^2+y^{2}}$ and $z=0$, with the ...
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2answers
17 views

Finding the probability function of the maximal number

$n$ balls in a jug numbered $1,\dots,n$. pulling out a ball $m$ times with return. Let $X$ be the maximal number that we got in the $m$ pullings. Find the probability function of $X$ (HINT:first ...
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1answer
39 views

All elements of a set have a multiplicative inverse

I was reading my groups notes, and was wondering if this is true -- Claim: If all elements in a set $S$ have a multiplicative inverse then the set is closed under multiplication. Proof: Let $x,y\in ...
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3answers
81 views

Exercise about convergent series

DEF. (Formal infinite series) A (formal) infinite series is any expression of the form $\sum_{n=m}^\infty a_n$, where $m$ is an integer, and $a_n$ is a real number for any integer $n \geq m$. DEF. ...
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1answer
27 views

Distribution of the number of students that will fail in the exam

There are $40$ questions in a test, $30$ questions are easy such that everyone can solve, $10$ questions are difficult such that no one can solve. $25$ students are going to do the test and each one ...
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1answer
36 views

Distribution of the number of the correct answers

There are $40$ questions in a test, $30$ questions are easy such that everyone can solve, $10$ questions are difficult such that no one can solve. $25$ students are going to do the test and each one ...
2
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0answers
31 views

Verification of example to show surjective maps of sheaves need not surject onto sections in all open sets

As an exercise in understanding the notion of surjectivity in the category of sheaves, I came up with this example, slightly modifying the standard ones given in my textbooks. I feel like this one is ...
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3answers
22 views

How to show $S$ is a vector space?

Fix vectors $a_1, . . . , a_m ∈ \mathbb{R}^n$, and let $S$ be the set of $x ∈ \mathbb{R}^n$ such that $a_i · x = 0$ for all $i$. I want to show that $S$ is a vector space. A vector space $S$ is a ...
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4answers
61 views

Prove $\lvert x\rvert$ = $\lvert-x\rvert$ for all real numbers $x$ [closed]

Been at this one for a long time. I'm trying to use the fact that $|x|$ = $x$ if $x$ is greater than or equal to 0, and $|x|$ = $-x$ if $x$ is less than 0. Then I want to split the proof into these 2 ...
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0answers
15 views

Prove the generalized solution to the normal equations $X'Xb = X'y$

I am just learning how to write these statistical proofs with matrices. Please check and revise: Prove $(X'X)^-X'y$ is a solution to the normal equations. Let the solution to the normal equations be ...
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0answers
27 views

How to prove col spaces are equal $C(X)=C(Px)$

Here's a rough draft of this proof, can you look over and edit? Given $X$ is an $n$ x $p$ matrix. Prove $C(X)=C(Px)$. (I'm still learning mathjax thanks in advance for any typo / edits as well). ...
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1answer
33 views

Is my basis for $U=\{p \in \mathcal P_4(\mathbb R) \mid p(6)=0\}$ correct?

$\mathcal P_4(\mathbb R)$ is the set of polynomials with degree at most $4$ with real coefficients. $U=\{p \in \mathcal P_4(\mathbb R) \mid p(6)=0\}$. Would $\langle(x-6),(x-6)^2, (x-6)^3, ...
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1answer
28 views

Prove that the series converges to a function which is $N$ times differentiable

Let $a>1$ and $b>0$ such that there's an $N\in\mathbb{N}$ such that $b^N < a$. Let the series $$S(x) = \sum_{n=1}^\infty \frac{\sin b^nx}{a^n}$$ Prove that $S(x)$ converges uniformly ...
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3answers
42 views

$\operatorname{gcd} \, (a,b) = 1$ then $\operatorname{gcd} \, (a^n,b^k) = 1$

Statement: Suppose $(a,b) = 1$ then $(a^n,b^k) = 1$ for $n,k \geq 1$. Attempt at Proof: Let $P$ be the set of all primes. Let $P_a$ be the set of primes $p_i$ such that $$a = \prod_{i=1}^{r_1} ...
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2answers
58 views

Rolling a dice $5$ times

Rolling a dice $5$ times find the probability that we will get the continuum: A. $12345$ B. $1234$ C. $434$ My attempt: A: for each throwing there are $5$ possibilities so ...
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4answers
45 views

Proof verification for $A \subset B$ iff $A - B = \varnothing$

I am trying to write a proof for $$A \subset B\quad\text{if and only if}\quad A - B = \varnothing$$ Starting from the left side: $$x \in A \subset B$$ $$x \in A \land x \in B$$ $$x \in A \cap B$$ ...
0
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1answer
13 views

On a Simple Theorem from Hilbert's *The Foundations of Geometry*

I want my proof writing skills to get better. I am trying to do this through proving theorems from Hilbert's axioms for Euclidean Geometry. I found Hilbert's The Foundations of Geometry here, a ...
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1answer
41 views

$∂(A\cup B)$ is a subset of $∂A \cup ∂B$

Please let me know if you think my proof of the above is correct. ($∂A$ denotes the boundary of $A$). Suppose $\vec{x}\in ∂(A\cup B)$. Then $\vec{x}\in \overline{A \cup B}=\bar{A}\cup\bar{B}$ and ...
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2answers
34 views

What makes for a rigorous proof?

As an undergraduate student, who wants to solidify his mathematical skills, I want to understand what exactly determines if a proof is rigorous.
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1answer
23 views

How to find a formula that is true for the given model in the First Order Logic?

I think I might get lost in the definitions. I am not sure if this is the right way to deal with models and formulas in the First Order Logic. I am not looking for the solution for this particular ...
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3answers
47 views

If $A\subset B$ then $\bar{A}\subset \bar{B}$

Please let me know if you find my proof satisfactory. Let $A\subset B\subset \mathbb{R}^n$, where $\bar{A}$ is the closure of $A$. If $\vec{x}\in \bar{A}$ then $\vec{x}\in A^\circ$ or $\vec{x}\in ...
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3answers
72 views

Prove that $\sqrt{6}-\sqrt{2}$ $> 1$.

I'm trying to prove that $\sqrt{6}-\sqrt{2}$ $> 1$. I need to admit that I'm completely new to proof writing and I have completely no experience in answering that kind of questions. However, I came ...
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0answers
14 views

Is $\lim_{n\to\infty}a\log n-\sum_{k=n^a+1}^{n^{a+1}}\frac{1}{k}=0$ for each integer $a\geq 1$?

I believe that for each integer $a\geq 1$ $$\gamma=\lim_{n\to\infty}\left(\sum_{k=1}^n\frac{1}{k}-\log\left(n+\frac{1}{n^a}\right)\right),$$ where $\gamma$ is the Euler's constant, when I've use an ...
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0answers
37 views

Proving the Commutativity of Set intersection.

Hi this is basically the question: Write down a formula which states that for any two sets $X$ and $Y$ , the set $X \cap Y$ is the same as the set $Y \cap X$. Then, prove this statement. The union ...
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0answers
12 views

Prove that $\sum_{U\in \mathcal{C}}\left(\sum_{s\in U}|f(s)|\right)$ converges if and only if $\sum_{s\in U}|f(s)|$ converges.

Let $f:S\rightarrow\mathbb{R}$, then $\sum_{s\in S} f(s)=\infty$ means For all $M\in\mathbb{R}$, there is a finite set $T\subseteq S$ such that, for all finite sets $T'\subseteq S$ that contain $T$ ...
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2answers
66 views

$L^{2}[-\pi,\pi]$ is unitarily isomorphic to $l^2(\Bbb C)$

So I have countable orthonormal basis of $L^2[-\pi,\pi]$ as $\{e^{inx}\}_{n \in \Bbb Z}$ and countable orthonormal basis of $l^2(\Bbb C)$ as $\{a_n\}_{n \in \Bbb Z}$ such that ...
0
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2answers
62 views

Proof for $A \cup B = B$ if and only if $A \subset B$

I am trying to define a proof for $A \cup B = B$ if and only if $A \subset B$ I started with: $$A \cup B$$ $$x \in A \lor x \in B$$ but I am unsure on how to continue.
0
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1answer
31 views

$P\left(X_1 < X_2 < X_3\right) = P\left(X_1 \le X_2 \le X_3|X_1\ne X_2\ne X_3\right)$?

I have 3 independent random variables, $X_i$, distributed on a continuous uniform distribution between 0 and 1. Does the following hold given the assumptions above? $$ \tag{1} P\left(X_1 < X_2 ...
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0answers
17 views

Is there a closed form of $n$ for :$h(n)=\frac{\sigma(n)}{n}$ for which $n$ is coprime to $\sigma(n)$?

It is well known that $h(n)=\frac{\sigma(n)}{n}$ is quit irrigulrar,I'm very interesting to know more about it's behavior and i would like to know more about coprimality characteristic then it must be ...
37
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12answers
5k views

Why do I get one extra wrong solution?

I'm trying to solve this equation: $$2-x=-\sqrt{x}$$ Multiply by (-1): $$\sqrt{x}=x-2$$ power of 2: $$x=\left(x-2\right)^2$$ then: $$x^2-5x+4=0$$ and that means: $$x=1, x=4$$ But $1$ is not a ...
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0answers
26 views

A stick of unit length is cut into three pieces of lengths $X, Y, Z$ according to its length in two sequence of cuts. Find Cov(X,Y).

A stick of unit length is cut into three pieces of lengths $X, Y, $and $Z$, with $X \le Y \le Z$. First the stick of length $1$ is cut into two pieces at a randomly chosen point uniformly distributed ...
0
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1answer
43 views

Is this a valid proof of “For all integers m and n, if mn is even, then m is even, or n is even”?

Theorem: For all integers $m$ and $n$, if $mn$ is even, then $m$ is even, or $n$ is even. Proof: Assume for all integers $m$ and $n$, if $mn$ is even, then $m$ is odd and $n$ is odd. By the ...
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0answers
39 views

Is this a valid proof of “a, b are rational, b ≠ 0, r is irrational, then a + br is irrational”

Theorem: If a and b are rational numbers, b ≠ 0, and r is an irrational number, then a + br is irrational. Proof: Assume that if a and b are rational numbers, b ≠ 0, and r is an irrational number, ...
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1answer
37 views

Proof that the convergence of a series is unique

Define the following: $\sum_{s\in S} f(s)=a$ means: For all $\epsilon>0$, there is a finite set $T\subseteq S$ such that, for all finite sets $T'\subseteq S$ that contain $T$, one has ...
3
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0answers
31 views

Show that if $f:\mathbb R\longrightarrow \mathbb R$ is measurable, there is $A_\varepsilon$ s.t. $f|_{A_\varepsilon}$ continuous

Show that if $f:\mathbb R\longrightarrow \mathbb R$ is measurable, then for all $\varepsilon>0$, there is $A_\varepsilon\subset \mathbb R$ s.t. $f|_{A_\varepsilon}$ is continuous and ...
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0answers
53 views

Is my proofs correct and am I using the implication sign in my proofs correctly?

Hi I I have written some proofs and would like to know if they are correct. Here are the proofs: Basically I need to prove This: $\forall m \in Z, m \cdot 0 = 0 = 0 \cdot m.$ ...
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0answers
25 views

Use Cauchy-Schwarz inequality to prove that $\langle\,,\rangle : \mathscr H \times \mathscr H \to \Bbb C$ is continuous.

Let $(a,b) \in \mathscr H \times \mathscr H$ be fixed. So we have to prove that for a given $\epsilon \gt 0$, we can find $\delta_1 \gt 0$ and $\delta_2 \gt 0$ such that $\lvert \langle x,y\rangle - ...
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2answers
23 views

How many of these cosets of $\mathbb{Z}^2/H$ are distinct?

My answer: Since all of $(1\; 6), (3\; 5)$ and $(7\; 11)$ are found in $H$, then none of them are distinct. Does this make sense? I'm not sure I understood the question.
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1answer
21 views

Step in a proof in a probability exercise

I'm working through the same book as in this question Limit superior of $\sum_{j=1}^n X_j$ with $\mathbf{P}[X_j = 1] = \mathbf{P}[X_j = -1] = 0.5$, and the top voted answer has a step in the solution ...
3
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1answer
43 views

Correctness of the proof that $ |f(c+h)-f(c)| ≤ w(|h|)$ implies $f$ is continuous at $c$

Question: Let $\omega: [0,\infty)\rightarrow [0,\infty)$ be continuous at $x = 0$ with $\omega(0) = 0$. Suppose for some point c $\in$ $\mathbb{R}$ the function $f$: $\mathbb{R}$ $\rightarrow ...
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1answer
31 views

Find all permutations such that $\sigma=a\tau a^{-1}$

For (b) and (c) we note that $\sigma$ and $\tau$ have different parity so there cannot be any $a\in S_4$ that will fix that parity mismatch. For (a) we have the cycle $a^{-1}=(3 2 4)$ and it is ...
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0answers
17 views

If $H \le G$ commutes with every subgroup of some family, then it commutes with the subgroup generated by that family. Proof Verification.

Let $\mathcal X = \{ X_i : i \in X \}$ be a family of subgroups for some arbitrary index set $I$. Then $\langle \mathcal X\rangle$ denotes the subgroup generated by all $X_i, i \in I$, i.e. the ...
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5answers
53 views

There are infinitely many vectors such that $\|\mathbf u-(1,1,1)\|\le 3$ and $\|\mathbf u+(1,1,1)\|\le 3$

I want to prove that there are infinitely many solutions in $3$-space for $\|\mathbf u-(1,1,1)\|\le 3$ and $\|\mathbf u+(1,1,1)\|\le 3$ (where bold refers to vectors). My proof: What is wrong with ...
3
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4answers
74 views

Calculate: $\lim_{x\to 0} \frac{f(x^2)-f(0)}{\sin^2(x)}$.

Let $f(x)$ be a differentiable function. s.t. $f^\prime(0)=1$. calculate the limit: $$\lim_{x\to 0} \frac{f(x^2)-f(0)}{\sin^2(x)}.$$ SOLUTION ATTEMPT: I thought that because $f$ is differentiable ...
3
votes
2answers
59 views

Convergence of prime zeta function for $\mathfrak R(s)=1$?

By doing some estimates for the partial sums of the Prime zeta function $P(s)=\sum_p p^{-s}$ for $\mathfrak R(s)=1$ I got that $P(1+i\alpha)$ converges for every $\alpha\neq0$... Since I did not ...
2
votes
1answer
41 views

A topological space is compact iff each infinite subset has a complete accumulation point

This is based on the comments and answers provided in this post. However, I have some questions on the proof and the hint given in Kelleys book p.163. I will highlight the hint of the book. My own ...
1
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3answers
53 views

Prove that if an integer $z$ is not divisible by $p$, then it is invertible in the $p$-adic integer ring $\mathbb{Z}_p$.

Let $p$ be a prime number. Define the $p$-adic valuation on $\mathbb{Z}$ as $v_p(p^kx) = p^{-k}$ where $x$ is not divisible by the prime $p$. Let $\mathbb{Z}_p$ (the $p$-adic integers) be a ...
1
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0answers
18 views

Reduction of a quadratic form to a canonical form

I'm supposed to reduce following polynomial to its canonical form. But my result differs from the one given in my book, so I'm not sure if it's correct too. $$ q = u_{xx} - u_{xy} - 2 u_{yy} + u_x + ...
0
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0answers
34 views

Are 2 quadrilaterals similar if they are both inscribed and have congruent angles and have perp diagonals

This is problem 365 from Kiselev's Planimetry book. I have to show that two inscribed quadrilaterals with perpendicular diagonals are similar iff they have respectively congruent angles. Here is my ...