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

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2
votes
1answer
40 views

Proving $R(3,4)\le 9$

I am trying to prove $R(3,4)\le 9$. This is my approach: For any $K_9$ we have (WLOG) at least 4 red edges by the pigeonhole principle. Consider all of the edges between these 4 red edges, if ...
1
vote
0answers
17 views

Show that $\{x\in V| \langle x,e \rangle=0 \forall e\in E\} =\{y\in V ~| ~y\perp w_i, 1\leq i \leq k \}$

Let $E$ be subset of a vector space $V$. Let $B =\{w_1,\dots,w_k\}$ be a basis for $E$. Prove: $E^\perp =\{y\in V | y\perp w_i, 1\leq i \leq k \}$ Is my proof correct? Define two sets: (a) ...
1
vote
0answers
17 views

Proving that a linear system is consistent if and only if the rightmost column of the augmented matrix is not a pivot column.

Is this proof sufficient? Theorem: A linear system is consistent if and only if the rightmost column of the augmented matrix is not a pivot column. Proof: (Showing the implication $\to$) Suppose a ...
0
votes
2answers
58 views

$\mathbb{R}$ is uncountable with Cantors Diagonal argument (how to improve binary expansion specificity?)

I know it's spelled out more than usual, but this is an introduction to higher math class. If there's any way I can improve this, please let me know. Thank you in advance. Let ...
2
votes
1answer
25 views

Proving a collection of subsets is a basis

I am given this definition of a basis: Let $a$ be a point in a metric space $X$. A collection, $\mathfrak{B}_a$, of neighborhoods of $a$ is called a basis for the neighborhood system at $a$ if every ...
1
vote
4answers
31 views

Epsilon-Delta Continuity proof (verification/help)

So, I am really bad at these problems, and I don't know why. Edit: The metric over $\Bbb R$ is assumed to be $|f(a,b)-f(x_1,x_2)|$ Problem statement: Define $f: \Bbb R^2 \rightarrow \Bbb R$ by ...
-4
votes
0answers
39 views

Why is this clause in unsatisfiable? [closed]

Why is the following clause unsatisfiable? By unsatisfiable, I mean that we can derive to the empty clause using the Resolution Proof Method. $\{x\}, \{\lnot x\}, \{x, y, z, w\}$
1
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0answers
35 views

Proof verification: Bolzano-Weierstrass (second proof).

I construct an other proof of Bolzano-Weierstrass theorem (i.e. that all bounded sequence has a subsequence that converge). Do you think that my construction is correct ? Let $a:=\inf_{n\geq 1} x_n$ ...
1
vote
1answer
29 views

Proof Verification: Cauchy Sequences are convergent.

Assume $(a_n)$ is a Cauchy sequence, then it is bounded. By Bolzanno-Weierstrass theorem, there is a convergent sub-sequence $(a_{n_j})$; denote its limit as $a$. Thus we have the following: for every ...
1
vote
0answers
20 views

A sequence with two distinct limits

I just wanted to check I was right about this: Consider $X=\{1,2,3\}$ equipped with the topology $T=\{\emptyset,\{1,2\},X\}.$ Then the sequence $(1,2,1,2,1,2,\ldots)$ converges to both $1$ and $2$ ...
0
votes
0answers
46 views

Looking for a function which can serve as an upper bound to a count of the the pairs (x)(x+2) that have a given least prime factor?

Let $p \ge 7$ be a prime. Let $z > p$ also be a prime. Let $f_p(z)$ be the number of elements $x$ such that $z \le x < z^2$ and the least prime factor of $x(x+2) = p$ I am trying to find ...
2
votes
0answers
58 views

Definite Integral of Series

I'm looking to find someone to either verify or correct my computation of the definite integral $$\int_{0}^{1} f(x)~dx$$ where $$f(x) = \sum_{n=1}^{\infty} \frac{1}{(x+n)^{n}}$$ What I came up ...
3
votes
3answers
110 views

Prove $f=1+x+x^2+x^3+\cdots+x^n$ has no multiple roots.

Prove $f=1+x+x^2+x^3+\cdots+x^n$ has no multiple roots. My attempt: Consider the polynomial $g=(x-1)(1+x+x^2+x^3+\cdots+x^n)$ As $f\mid g, g$ all the roots of $f$ are roots of $g$. This means I ...
0
votes
2answers
49 views

Prove/Disprove $f(x)=e^{x}$ is Injective and Surjective

Dr. Pinter's "A Book of Abstract Algebra" presents the exercise: Prove whether each of the functions is or is not (a) injective and (b) surjective. $$f \implies \mathbb{R} \to (0, \infty)$$, ...
1
vote
1answer
45 views

Show that the following map is a bijection

$$g(x,y) = \frac{1}{2}(x-1)x + y$$ where $g:\mathbb{Z}^+ \times \mathbb{Z}^+ \longrightarrow \mathbb{Z}^+$ Attempt: I am only having problems with proving the injectivity part so that is all I'm ...
6
votes
2answers
86 views

$f(x)=x^{3}+1$ - Injective and Surjective?

Dr. Pinter's "A Book of Abstract Algebra" presents this exercise: Prove/disprove whether the following function $f$ (with inputs/outputs of real numbers) is injective and/or surjective: ...
4
votes
4answers
62 views

$f(x)=3x+4$ - Injective and Surjective?

As a follow-up to Understanding why $f(x)=2x$ is injective, I'm working on proving/disproving that $$f(x)=3x+4,$$ where inputs/outputs live on real numbers, is injective and surjective. Supposing ...
2
votes
1answer
103 views
+100

Proof-verification: Bolzano Weierstrass theorem (modified)

I want to show that every bounded sequence has a subsequence that converge. Do you agree with my proof and if not, what's wrong ? Proof Let $(x_n)_{n\in\mathbb N^*}$ a bounded sequence. Since ...
3
votes
2answers
146 views

Showing planarity of graphs

I am trying to show $G_3$ is planar. I have constructed $G'_3$ as shown. Is it correct to say that by the Jordan curve theorem, $G_3$ cannot be planar, as any drawing will cause edges to overlap. ...
1
vote
0answers
23 views

Prove that a point $(a,b)$ in $\mathbb{R^2}$ has the same homotopy type as $\mathbb{R^2}$.

Prove that a point $(a,b)$ in $\mathbb{R^2}$ has the same homotopy type as $\mathbb{R^2}$. If someone could verify my proof that would be great. I just started this learning this material and I ...
1
vote
0answers
61 views

If there's anything I need to add to this proof, I'd greatly appreciate it!

I'm doing a proof for the following statement: Z (defined as the set of real numbers belonging to all inductive sets of real numbers) is the smallest inductive set of reals in the sense that Z is a ...
0
votes
2answers
29 views

Prove $\tilde d: M\to \mathbb{R}; x\mapsto \tilde d(x) = d(x,A)$ is continous

Let $A$ be a non-empty subset of a certain metric space $M$. Prove that $\tilde d: M\to \mathbb{R}; x\mapsto \tilde d(x) = d(x,A)$ is continous. (where $d(x,a) = \inf\{d(x,a): a\in A\}$) ...
0
votes
0answers
46 views

$\sum a_n$ converges $\implies\ \sum \sqrt{a_na_{n+1}}$ converges?

Let $a_n > 0.$ When $\sum a_n$ converges $\sum \sqrt{a_n a_{n+1}}$ converges or not? For, $$\frac{\sqrt{a_n a_{n+1}}}{a_n}=\frac {\sqrt{a_{n+1}}} {\sqrt{a_n}}$$ $\because$ By comparison test ...
0
votes
1answer
19 views

Integrality conditions and proof by double counting.

Theorem $\mathbf{3.4.}$ In a block design of type $2-(v,k,\lambda)$ every element lies in precisely $r$ blocks, where $$r(k-1)=\lambda(v-1)\textit{ and }bk=vr\;.$$ The letter $r$ stands for ...
2
votes
0answers
25 views

convergence in $L^p$ implies convergence in measure

I am trying to show that if $f_n$ converges to $f$ in $L^p(X,\mu)$ then $f_n\to f$ in $L^p$ in measure, where $1\le p \le \infty$. Here is my attempt for $p>1$ - Let $\varepsilon>0$ and define ...
3
votes
3answers
54 views

Baby Rudin Exercise 4.2

Can someone check my proof? If $f$ is a continuous mapping of a metric space $X$ into a metric space $Y$, prove that $$f(\overline{E}) \subset \overline{f(E)} $$ for every set $E\subset X$. ...
0
votes
1answer
12 views

Linear Second order Differential operator proof questions

I have 3 proof questions from my book that I have tried and I would like to see if my solutions are valid and/or there is a simpler way to prove them. Firstly, the notation $ker(L)$ means all $f$ ...
3
votes
0answers
43 views

When is $x^2 - 75 y^2 = 0$ in $\mathbb{Z}_p$ solvable?

Exercise: For which prime numbers does the equation $x^2 - 75 y^2 = 0$ have non-trivial solution in the $p$-adic integers $\mathbb{Z}_p$? For $p\neq 5$, the non-trivial solvability of the ...
1
vote
2answers
37 views

Relevance of prime being divisble by $4k+1$ in proof that 'There are infinitely many primes of the shape $4k+3$'

Show that there are infinitely many primes of the shape $4k+3$ Proof: $1)$ Suppose that there are only finitely many such primes, say $p_1,...p_n$. $2)$ Consider the integer $Q=4p_1...p_n-1$ $3)$ ...
1
vote
1answer
40 views

Find the group of conformal automorphisms of $U=\lbrace z\in \mathbb{C}: \vert z-1\vert>1\rbrace$

Well $\phi$ is an automorphism of $U$ $\iff$ $1/ \phi$ is an automorphism of $U^C=\lbrace z\in \mathbb{C}:\vert z-1\vert<1\rbrace$ $\iff$ $1/\phi -1$ is an automorphism of the unit disc $\iff$ ...
0
votes
0answers
15 views

Show that if f is differentiable at $x_0$, then it is continuous at $x_0$. (Weierstrass-Caratheodory formulation)

this is an argument for a question which I am unsure whether it is sufficient or not. We are asked to try show the continuity at $x_0$ given that $f$ is differentiable at $x_0$. My argument goes as ...
0
votes
0answers
11 views

$\mathcal{Z}$-transform of differential equations $y(n+2)-3y(n+1)-10y(n)=(-2)^n$

Is defined function: $$y(n+2)-3y(n+1)-10y(n)=(-2)^n$$ with conditions: $$y(0)=0, y(1)=0 $$ And my solution is (Z-transform): $$\mathcal{Z}\{y(n+2)\}=z^2Y(z)-0z^2-2z=z^2Y(z)-2z$$ ...
1
vote
1answer
46 views

There is no equivariant map $f:S^2 \to S^1$

To fix some notation, let $n \geq 2$ and let $p:S^n \to P^n$ be the canonical double cover. Let $\gamma:I \to S^n$ be a lift of a representative of a nontrivial element in $\pi_1(P_n) \cong ...
1
vote
0answers
27 views

Series Proof Question [duplicate]

By considering the partial sums for S, that is Sn =1+2+3+···n show that the infinite series S does not converge. However in this video http://www.numberphile.com/videos/analytical_continuation1.html ...
0
votes
1answer
32 views

Show that if $a, b$ and $m$ are integers such that $m \geq 2$ and $a \equiv b \pmod{m}$, then $\gcd(a, m) = \gcd(b, m)$

Problem 1 (#3.5.32). Show that if $a, b$, and $m$ are integers such that $m \geq 2$ and $a \equiv b \pmod {m}$, then $\gcd(a, m) = gcd(b, m)$. Proof. Let $d = \gcd(a, m)$ Then $d \mid a$ and $d ...
1
vote
0answers
79 views

Proof that a limit is true

In my assignment I have to prove the following limit, by definition: $$\lim _{x \to 2}\sqrt{3x-2}=2 $$ I have made some calculations trying to prove it, and I've made some way but I'm afraid I'm ...
2
votes
0answers
48 views

Dimensions of quotient rings of $K[x,y]$

I have tried to solve the following problem and would be very grateful if someone could check my answer. Let $K$ be an algebraically closed field with $\mathrm{char}(K)=0$. I wish to compute ...
0
votes
1answer
57 views

What is wrong with this proof of a number theory competition problem?

Let $a$ and $b$ be positive integers. Suppose $a^n+n| b^n+n$ for any positive integer $n$, prove that $a=b$. My trial: Clearly $b\geq a$, write $b=a+d$, we must show that $d=0$. Now by assumption and ...
1
vote
0answers
9 views

Moments bounds VS Chernoff bounds

I have to prove that, when bounding tail probabilities of a nonnegative random variable, the moments method is always better than the classical Chernoff method. In mathematical language, I have to ...
2
votes
1answer
28 views

Does this reasoning work?

Consider the following system of ODEs. $$ \theta'=r\\ r'=1-r^2 $$ On the unit circle, $\theta'=1$, and $r'=0$ Now consider the system $$ \theta'=1\\ r'=0 $$ The solution curves to this system are ...
1
vote
2answers
24 views

Expected Value and Variance of transformed Random variable

I am trying to find the expected value and variance of $Y_i=\ln(X_i)$ for $X$ is uniformly distributed between $1$ and $3$. I believe that $E(Y_i)=(\ln3)/2$ and $\operatorname{Var}(x)=(\ln3)^2/12$. ...
0
votes
1answer
36 views

Binomial Distribution with probability $P$ such that $P$ is Uniformly distributed

A number $P$ is random chosen from the uniform distribution from [0,1]. Then a coin with probability $P$ of getting a head is flipped $n$ times. Let $X$ be the number of heads showing and compute ...
1
vote
1answer
27 views

trouble in getting triangle inequality

Let $l_{2}$ be the set of all infinite sequences , $ (x_{n})$ such that $\sum_{n=1}^ {\infty} x_{n}$ converges. Define $$d(x,y)= \sqrt{\sum_{n=1}^{\infty} (x_{n}-y_{n})^{2}}$$ for each $x=(x_{n})$ ...
0
votes
1answer
47 views

Show that there exists a unique function with a certain property

I'm trying to prove the following theorem: "Let $~f: \mathbb{N} \times \mathbb{N} \to \mathbb{N}~$ be a function, and let $~c~$ be a natural number. Show that there exists a unique function $~a: ...
3
votes
1answer
46 views

My answer to this combi problem doesn't match the answer in the book (Problem-Solving Strategies)

[Problems 31 and 32 from Arthur Engel's Problem-Solving Strategies.] Let $n$ children be seated in a line. How many ways can they change their places if they may only move by one place at most? ...
1
vote
1answer
32 views

Existence of a Map

I just wanted to check to be sure I was correct as the more I stare at my proof the more I doubt myself. Suppose you have a commutative diagram of $R$-modules with exact rows: $$ ...
2
votes
0answers
40 views

Unifrom Convergence of series of product of two sequences

Suppose {$f_n$}, {$g_n$} defined on $E$ and, (a) $\Sigma f_n$ has uniformly bounded partial sums; (b) $g_n \to 0$ uniformly on $E$ (c) $g_1(x)\geq g_2(x)\geq g_3(x)\geq ...$ for every $x \in E$. ...
0
votes
1answer
61 views

Are these two definitions of prime numbers equal?

In Coq for instance, prime numbers are defined ${n\ is\ prime} \doteq \forall a\in \mathbb{N}: a|n \rightarrow (a=1 \vee a=n)$ ...
0
votes
0answers
31 views

multivaraible chain rule proof

I wanted to prove the multivaraible chain rule; I had to prove that $df \large \frac {({x(t)},{y(t)})}{dt} = \frac{∂ f}{∂ x}\cdot\frac{dx}{dt} + \frac{∂ f}{∂ y}\cdot\frac{dy}{dt}$ So, I took the ...
0
votes
2answers
51 views

Galois Group of an Extension

Question: Determine the isomorphism type of $ \mathrm{Gal}\,(\mathbb{Q}(\sqrt[8]{2},i)/\mathbb{Q}(i)) $. $\\$ This amounts to finding isomorphisms that send $\mathbb{Q}(\sqrt[8]{2},i)$ to ...