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

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0
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2answers
13 views

If the sum of the digits of n is equal to the sum of the digits of 5n, then prove that 9|n.

Let $n\in\mathbb{N}$. So far I have: If the sum of the digits of $n$ is $k$, then $n = 9m + k$, where $m$ element of an integer (not sure why). Now consider $5n-n$. Help?
1
vote
0answers
22 views

About a result concerning Mersenne primes

I want to verify the proof of this result Theorem: If $p>2$ is a prime and $$H_{p}=(((√3+2)^{2^{p-1}}+1)/((2^{p}-1)(√3+2)^{2^{p-2}}))$$ is a natural number then $2^{p}-1$ is a prime number. ...
0
votes
2answers
31 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 ...
0
votes
1answer
23 views

Soundness of a simple tree edge count proof by induction

I'm trying practice and get better at proofs. Here is my attempt at a proof of the following simple statement: There are $n-1$ edges in a $n$ vertex tree. We will prove this by induction on $n$ ...
1
vote
1answer
33 views

Use Fundamental Theorem of Arithmetic to prove that if $a >1$, $p$ is prime, and $p|a ^n$ for some $n \in \mathbb{N}$, then $p|a$

So, by the FTOA, since $a >1$, then a can be broken down into a product of a prime factors, so $a = p_1 \times p_2 \times \dotsm \times p_k$. Then, can I say that since $a$ is multiplied by itself ...
4
votes
3answers
29 views

If d is a norm on V, is $\frac{d(x,y)}{1+d(x,y)}$ a norm on V?

Let d be a norm on a vector space V and let $\psi:V \to [0,\infty)$ be a function defined as $\psi(v)=\frac{d(v)}{1+d(v)}$. Is $\psi$ a norm on $V$? It seems that $\psi$ does not satisfy the ...
1
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0answers
28 views

Applying Stone Weierstrass to this isometry of $C^\ast$-algebra

I proved the following theorem but I'd like to confirm the last part of my proof. Statement: Let $A$ be a non-zero commutative $C^\ast$ algebra. Then $\varphi : A \to C_0 (\Omega(A))$ defined by $a ...
0
votes
3answers
31 views

The second derivative of $f^{-1}$ and another question. :)

Suppose both $f$ and $f^{-1}$ are twice differentiable functions. Derive a formula for $(f^{-1})''$. My attempt: We have that by the inverse function theorem that: ...
0
votes
3answers
17 views

In $S_3$, determine the set $T=\{ x\in S_3 | x^2=e\}$. Is $T$ a subgroup of $S_3$?

Here's my solution: Is it right or wrong? $S_3=\{ \begin{cases} 1\mapsto1 \\ 2\mapsto 2 \\ 3\mapsto 3 \end{cases}, \begin{cases} 1\mapsto 2 \\ 2\mapsto 1 \\ 3\mapsto 3\end{cases}, \begin{cases} ...
1
vote
1answer
27 views

Verify proof that ${p \choose r} ≡ 0 \pmod p$

Let $p$ be a prime number. For any $1 ≤ r ≤ p − 1$, prove that $${p \choose r} ≡ 0 \pmod p$$ I'm thinking that it suffices to show $p$ divides ${p \choose r}$. So then: $$\begin{align} p\ |\ {p ...
0
votes
1answer
15 views

Linear functionals and integration verification

Can you please verify my reasoning? (a) Yes as (b) No, as function is squared (c) Yes, same reasoning as (a), squared values of x do not affect linearity. Does the region of integration affect ...
1
vote
2answers
28 views

Is this a valid method of proof?

We are given that $y = a + b$, and we want to prove that $y = a + c$ (using all the usual properties of numbers that we know from grade school). Does it suffice to set $a + b = a + c$, and by ...
1
vote
0answers
18 views

Calculating the mean of geometric distribution: Two methods with two different answers?

I am calculating the mean of geometric distribution with parameter $p$. Method 1: Direct computation $$E[X] = \sum xp(x) = \sum_{i=1}^\infty i(1-p)^{i-1}p = \frac{1}{p}$$ which is the usual value we ...
18
votes
5answers
402 views

Find the value of $\sqrt{10\sqrt{10\sqrt{10…}}}$

I found a question that asked to find the limiting value of $$10\sqrt{10\sqrt{10\sqrt{10\sqrt{10\sqrt{...}}}}}$$If you make the substitution $x=10\sqrt{10\sqrt{10\sqrt{10\sqrt{10\sqrt{...}}}}}$ it ...
0
votes
2answers
15 views

Verifying proof :an Ideal $P$ is prime Ideal if $R/P$ is an integral domain.

I had to write the proof to show that an Ideal $P$ of a commutative ring $R$ is prime Ideal if $R/P$ is an integral domain. let $a,b\in R$ s.t. $ab\in P$ , ...
6
votes
0answers
112 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 $m(X)=1$. ...
0
votes
5answers
64 views

Guessing on the SATs, is it ever better to leave it blank than to guess?

On most SAT questions, there are 5 answers of which exactly one is correct and exactly four are wrong. If one answers correctly you get $1$ point. If you answer incorrectly, you receive $-\frac14$ ...
0
votes
4answers
27 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 ...
1
vote
0answers
52 views

Proof of Definite Integral of Even Function for Improper Integrals

I am trying to prove $\displaystyle \int_{\mathop \to -a}^{\mathop \to a} f \left({x}\right) \ \mathrm d x = 2 \int_0^{\mathop \to a} f \left({x}\right) \ \mathrm d x$ for $f$ which is an even ...
4
votes
1answer
87 views

Prove that $1^2-2^2+3^2-…+(-1)^{n-1} n^2$=$(-1)^{n-1}\frac{ n(n+1)}{2}$ whenever n is a positive integer using mathematical induction.

I am wondering if the third to last equation is correct, where i factored out the $(-1)^k$. The first term is inside the parenthesis is $(-1)^{-1}$. Is this correct? If I multiply it out again,, wont ...
2
votes
2answers
123 views

Prove P∧(Q∨R)→(P∧Q)∨(P∧R) from P∧(Q∨R) using Natural Deduction

I am trying to prove P∧(Q∨R)→(P∧Q)∨(P∧R) from P∧(Q∨R) using Natural Deduction. Here is my attempt using JAPE application. ...
1
vote
2answers
29 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 ...
0
votes
0answers
12 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 ...
0
votes
1answer
28 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, ...
1
vote
2answers
12 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 ...
5
votes
1answer
81 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 ...
0
votes
1answer
30 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 ...
3
votes
1answer
47 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 ...
1
vote
0answers
26 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 ...
0
votes
0answers
11 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
votes
0answers
16 views

Show that $P_A(\cdot):=P(\cdot \mid A)$ is a probability measure

Let $(\Omega,\mathcal{A},P)$ be a probability space. (i) Show that if $P(A)>0$, then $$ P_A(\cdot):=P(\cdot \mid A) $$ is a probability measure on $(\Omega,\mathcal{A})$. (ii) Is ...
0
votes
3answers
31 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 ...
0
votes
1answer
52 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 ...
1
vote
4answers
53 views

Prove or disprove the rationality of $ x^y $

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 $y=\sqrt{2}$ ...
0
votes
1answer
28 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 ...
3
votes
2answers
129 views

Examples that $f(\overline{E})\subsetneq\overline{f(E)}$ for continuous $f: X \to Y$ and subset $E\subset X$ (Baby Rudin Problem 4.2)

I'm working on the second part of problem 2 of chapter 4 in Rudin, and I was wondering if someone could check my example or, even if it's correct, maybe help me find a less trivial one. The problem, ...
-2
votes
0answers
60 views

What do you think of my proof?

I wrote this proof when I was still in high school (I just graduated). Never had the chance to have it checked by anyone. Therefore, I posted it here! :) Please edit if necessary! Our goal is to ...
0
votes
0answers
31 views

Is my proof complete? ($\inf(-A)=-\sup(A)$)

I need to prove the following statement for $A\subseteq \Bbb{R}$, that is not empty and bounded from above: $\inf(-A)=-\sup(A)$ Here is my proof: Let's take $M'=\inf(-A)$. Then it's ...
0
votes
0answers
51 views
+50

Inequality involving a strictly positive sequence

I was asked to prove the following : \begin{array}{l} x_n > 0,\,\forall n \in Z^ + \\ \lim \sup (\frac{{x_{n + 1} + x_1 }}{{x_n }})^n \ge e \\ \end{array} Is my approach correct ? Using ...
2
votes
3answers
42 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 ...
2
votes
1answer
55 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 ...
3
votes
0answers
26 views

Let $\Gamma$ be a set of formulas and $\phi$. Show that $\Gamma \cup \{\neg \phi\}$ is satisfiable if and only if $\Gamma\not \models \phi$

This seemed pretty obvious but I wanted to see if my proof made sense: Proof: $(\Rightarrow)$ To derive for a contradiction, suppose that: $\Gamma \models \phi$. That means for all truth assignments ...
1
vote
2answers
1k views

Proof of The Associative Law and The Commutative Law.

The associative law of multiplication for three positive integers $a,b$ and $c$ can be proved$^1$ from the Commutative Law and the property of "Number of things" easily. We can prove$^2$ the ...
2
votes
1answer
30 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! = ...
1
vote
2answers
66 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. ...
3
votes
4answers
805 views

Prove that the additive groups $\mathbb{Z}$ and $\mathbb{Q}$ are not isomorphic.

Is my proof below correct? What specific property of rationals did I exploit in my proof? It looks like the property I exploited is the following: Given any positive rational, I can always write it as ...
0
votes
3answers
117 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 ...
1
vote
1answer
39 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 ∈ ...
0
votes
0answers
28 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 ...
-1
votes
1answer
37 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} + ...