Tagged Questions

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

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0
votes
3answers
19 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} ...
0
votes
3answers
32 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: ...
1
vote
1answer
45 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 ...
0
votes
1answer
38 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
31 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
16 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
35 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 ...
0
votes
2answers
18 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. $ab\in P$ , ...
1
vote
0answers
31 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
5answers
65 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
2answers
64 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
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 ...
0
votes
0answers
17 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 ...
1
vote
2answers
34 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 ...
1
vote
2answers
15 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 ...
0
votes
1answer
50 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
votes
1answer
87 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 ...
1
vote
0answers
29 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
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
37 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
votes
1answer
49 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
votes
1answer
55 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
votes
0answers
14 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
1answer
29 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 ...
1
vote
4answers
113 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 $y=\sqrt{2}$ ...
-2
votes
0answers
64 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
34 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 ...
2
votes
3answers
49 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
60 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
28 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
1answer
40 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
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! = ...
0
votes
0answers
30 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
42 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
votes
1answer
31 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
votes
4answers
52 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
66 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 ...
0
votes
3answers
120 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
0answers
16 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 $L$ ...
0
votes
0answers
28 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 ...
0
votes
0answers
9 views

Two Column Proof for a System of Linear Equations

Given $8x+10=y$ $\frac{1}{4}y=x$ Prove: The solution to the system of equations is $(\frac{1}{2},6)$ So I have my two columns, and in step 7 I got $x=\frac{1}{2}$, then in 11 I got $y=6$ ...
6
votes
0answers
120 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
2answers
32 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
votes
1answer
35 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
votes
1answer
50 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 ...
0
votes
1answer
31 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 $m$ ...
1
vote
3answers
82 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
votes
1answer
40 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
vote
2answers
72 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
votes
1answer
38 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 ...