Tagged Questions

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

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5 views

Let M and N be normal subgroups of a group G such that $G = MN.$ Prove that $G/(M \cap N) \simeq G/M \times G/N$.

Let M and N be normal subgroups of a group G such that $G = MN.$ Prove that $G/(M \cap N) \simeq G/M \times G/N$. My attempt: Claim 1: $M\cap N$ is a normal subgroup of G: Proof: $1_G \in M$ and ...
0
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0answers
18 views

Proving a function constant

The problem is to prove f is a constant function given that: $$f:R \to Q $$ is a continuous function. I was hoping to show two cases given a is rational and b is irrational and the equate the two. ...
0
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0answers
11 views

Eigenvalue and Eigenvector for the linear transformation in $ \mathbb{Z}_2^4$

I'm trying to find the Eigenvalue and Eigenvector for the Linear transformation: $T:\mathbb{Z}_2^4 \to \mathbb{Z}_2^4: (x_1,x_2,x_3,x_4)=(x_1+x_3,-2x_1-x_3,x_2+x_4,x_2-x_4)$ My problem is with ...
2
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2answers
36 views

Proving and Finding a limit

I need to find the following limit and prove using the definition of limits. $$\lim_{x\to1} {x \over x+1} = \frac 1 2$$. Following the definition: $$\forall \epsilon \exists \delta : \lvert x - c ...
3
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1answer
35 views

Show that a group is abelian.

Let G be a group and m be a positive integer. Suppose that for all $\alpha, \beta \in G$, $$(\alpha \beta)^m = \alpha^m \beta^m,$$ $$(\alpha \beta)^{m+1} = \alpha^{m+1}\beta^{m+1},$$ and $$(\alpha ...
0
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1answer
20 views

Prove that every connected undirected graph with n vertices has at least n-1 edges.

I would appreciate it if anyone can verify my proof. It is a proof by induction, but I attempt to reason things out rather than using a purely mathematical approach, in a similar vein to many other ...
0
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1answer
5 views

Show that the entries of the square of diagonal matrix are equal to the square of the entries of the diagonal matrix.

The question seems trivial which is why I have some trouble coming up with a proof that is mathematically correct. BTW I cannot yet use eigenvalues as we have not yet covered them in class. If ...
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0answers
9 views

Let p be a prime number and let n ≥ 1. Show that $\mathbb{F}_p$ contains an element of order n iff p ≡ 1(mod n).

Let p be a prime number and let n ≥ 1. Show that $\mathbb{F}_p$ contains an element of order n iff p ≡ 1(mod n). For the reverse direction, assume $p \equiv 1(mod n). \text{Let } g \in \mathbb{F}_p ...
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1answer
19 views

Prove that if lim sup$_{n} n^{2}a_{n} = 0 \Rightarrow \sum_{k = 1}^{\infty} a_{k}$ converges

Prove that if lim sup$_{n} n^{2}a_{n} = 0 \Rightarrow \sum_{k = 1}^{\infty} a_{k}$ converges. Also assume that the sequence is positive. lim sup$_{n} n^{2}a_{n} = 0$ means that for every $\epsilon$, ...
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1answer
22 views

Prove that lim inf$_{n} na_{n} > 1 \Rightarrow \sum_{k = 1}^{\infty} a_{k}$ diverges

Prove that lim inf$_{n} na_{n} > 1 \Rightarrow \sum_{k = 1}^{\infty} a_{k}$ diverges The solution proof goes like: lim inf$_{n} na_{n} > 1 \Rightarrow$ there exists an $N \in \mathbb{N}$ such ...
1
vote
1answer
36 views

Induction proof for Fibonacci numbers

I am trying to get a hang on the induction method for proof, but I'm still dubious of many aspect of this proof regarding its application to sequences of integers, such as the Fibonacci sequence. ...
1
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0answers
12 views

Question from Stewart's Calculus regarding proof of independence of path and conservative vector fields.

Please look over this proof. In the proof, it says: "Notice that the first of these integrals does not depend on $x$, so..." How is that so? $C_1$ does depend on $x$. How/Why does it not ...
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2answers
19 views

Transitivity relation in the set of Integers

Prove or disprove that R is transitive, where $R=\{ (a,a^2)| a \in \Bbb Z \}$ is a relation on $\Bbb Z$ By definition: $R$ is transitive $iff$ $$ (a,b)\in R \wedge (b,c) \in R\implies (a,c) ...
0
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1answer
14 views

For $f(\theta)= e^{\theta}$. Is it true that $\hat{f}(n)(1-in)=0$ for all $n\in \mathbb Z.$?

(This is motivated from the following question) Fact: If $f \in C^1(\mathbb{T})$, then the Fourier coefficients $\widehat{f'}(n)$ of the derivative $f′$ can be expressed in terms of the Fourier ...
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3answers
33 views

For $A,B,C\subset X$where $X$ is a metric space under some $d$, check if the triangle inequality holds for $d_m(A,B)=\min_{x\in A,y\in B}\{d(x,y)\} $

$$d_m(A,B)=\min_{x\in A,y\in B}\{d(x,y)\} $$ Is it the case that $$d_m(A,C)\leq d_m(A,B)+d_m(B,C)$$ based on the definition of $d_m$ and the fact that $d$ is already some arbitrary metric on $X$? I ...
0
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0answers
42 views

Proving symmetry of metric (single linkage between clusters using arbitrary dissimilarity measure)

I am told to assume that our dissimilarity measure $d$ satisfies the properties required of it, what seems to be the definition of a metric: $d(x,y) \geq0 $ and $d(x,y)=0 \Longleftrightarrow x=y$ ...
0
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0answers
34 views

For what values $q,r$ does the improper integral $\int_0^1 x^q (1-x^2)^r dx$ converge?

Question: For what values $q,r$ does the improper integral $\int_0^1 x^q (1-x^2)^r dx$ converge? I had some ideas from a previous thread but now I have a different attempt that I would like to ...
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1answer
19 views

Define $\phi:G/H \rightarrow G/K$ by $\phi(Ha)=Ka$. Prove: $\phi$ is a well defined function.

Let $H$ and $K$ be normal subgroups of a group $G$, with $H \subseteq K$. Define $\phi:G/H \rightarrow G/K$ by $\phi(Ha)=Ka$. Prove: $\phi$ is a well defined function. [That is, if $Ha=Hb$, ...
0
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2answers
42 views

If $\sum_{k = 1}^{\infty} a_{k}$ diverges, then one of its rearrangement converges to 1

If $\sum_{k = 1}^{\infty} a_{k}$ diverges, then there exists a rearrangment of $\sum_{k = 1}^{\infty} a_{k}$ that converges to 1. This is not true. A counter example I came up with is $\sum_{k = ...
0
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1answer
39 views

Verify f(x) is not integrable

Let $f(x) = \begin{cases} x &\mbox{if } x\in [0,1]\bigcap\mathbb{Q} \\ -x & \mbox{if } x\in [0,1]\bigcap\mathbb{Q}^c. \end{cases}$ I want to show that $f:[0,1]$ is not integrable. My ...
0
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1answer
18 views

prove characterstic polynomial of $2\times 2$ matrix is $C_{A}(x)=x^2-(\lambda_{1}+ \lambda_{2})x+\lambda_{1} \lambda_{2}$

Let $$ A=\begin{bmatrix} a_{11} & a_{12}\\ a_{21} & a_{22}\\ \end{bmatrix}$$ Let $\lambda_{1}, \lambda_{2}$ not necessarily distinct, be the eigenvalues of A. Show that $$ ...
1
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1answer
6 views

Prove that a partial equivalence relation in set Dom(A) is an equivalence relation

We know that $r$ is a partial equivalence relation in set $$Dom(r) = \{x|\exists y.(x,y)\in r\}$$ The problem is to prove that this is an equivalence relation. Here is my proof. Did I do it right? ...
2
votes
1answer
69 views

Problem with proving formally tautology using given rules

Using the rules below prove that the following assumeptions leads to the following conclusion by tautology. $A\vee B \vee C, A\to C, B\to C \Rightarrow C$ What I did: $A\vee B \vee ...
2
votes
2answers
39 views

Proof for any real number $c>0$ there exists $x>0$ such that $\frac{1}{x}+sinx=c$

I have this problem: Prove for any real number $c>0$ there exists $x>0$ so that $f(x)=\frac{1}{x}+\sin(x)=c$ I tried to prove it, but I feel that my proof isn't entirely true. My proof ...
1
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1answer
29 views

Proof Verification for presentation

Can someone verify or discredit my proof? Help.
0
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1answer
13 views

Understanding how to prove a bijection into three sets

I understand how to prove if there is a bijection from A onto B. However, say that there is a bijection from A onto B and a bijection from B onto C. How would I prove that that there is a bijection ...
1
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1answer
36 views

Prove $f''(x)>0$ for some x.

Question: Let $f$ be a twice differentiable function on $R$. Suppose $f(0) = 0$ and $f(x)/x$ is increasing for $x > 0$, prove $f′′(x) ≥ 0$ for some $x > 0$, but not necessarily for all $x > ...
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0answers
9 views

Prove the A x B lexicographical ordering is partially ordered

Is this proof? I think I may have the right ideas, but I'm not sure.
2
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0answers
24 views

Confusion on particular step in this weak induction proof

I am studying for a midterm, and I came across this proof. Use mathematical induction(weak) to show that for all integers n $\geq$ 2 , if x$_{1}$, x$_{2}$, ... x$_{n}$ are strictly between 0 and 1 ...
3
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1answer
29 views

Finding all ideals in $\mathbb{Q}[x]/I$, where $I$ is the ideal generated by $p(x)=x^2(x^2+x+1)$

I want to find all ideals in $\mathbb{Q}[x]/I$, where $I$ is the ideal generated by $p(x)=x^2(x^2+x+1)$. I know that $$\mathbb{Q}[x]/I \cong \mathbb{Q}[x]/(x^2) \times \mathbb{Q}[x]/(x^2+x+1)$$ I ...
0
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0answers
20 views

Analytic continuation past the disk

So I just proved that if $f$ is analytic on $\overline{\mathbb D}$, and $f(\partial\mathbb D) \subset \partial \mathbb D$, then it extends to a rational function on $\mathbb C$. What if we weaken this ...
0
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1answer
31 views

Let H be a subgroup of a group G. Prove that the following statements are equivalent.

Let H be a subgroup of a group G. Prove that the following statements are equivalent. (a) For all $a,b \in G, (aH)(bH)$ is a left coset of $H$ in $G$. (b) For all $a,b \in G, (aH)(bH) = (ab)H$. (c) ...
2
votes
1answer
53 views

Proof of the First Isomorphism Theorem for Rings [duplicate]

The statement is the First Isomorphism Theorem for Rings from Abstract Algebra by Dummit and Foote. I'd like to check if all is ok. In particular I'm a bit worried about the (*) line. It looks a bit ...
3
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1answer
55 views

Proof of the First Isomorphism Theorem for Groups

The statement is the First Isomorphism Theorem for Groups from Abstract Algebra by Dummit and Foote. This proof was left as a exercise, so I'd like to check if all is ok. In particular, I'm a bit ...
0
votes
1answer
42 views

$\mu$ measure on $\cal B_{\mathbb R}$ that is“linear” for some $c\in\mathbb R$ , $\mu (E)=c\cdot m(E)$

i need to show that for all bounded $E\in\cal B_{\mathbb R}$ (while $\cal B_{\mathbb R}$ is the borel $\sigma$-algebra) and for $\mu$ a measure on $\cal B_{\mathbb R}$ and $\mu (E)<\infty$ and ...
0
votes
1answer
29 views

How to prove that lim sup $a_{n} \leq b$

Assume that $(a_{n})$ is a bounded sequence, prove that lim sup $a_{n} \leq b$ iff, for every $\epsilon > 0$, there exists an $N \in \mathbb{N}$ so that $n \geq N$ implies $a_{n} \leq b + \epsilon$ ...
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0answers
38 views

If $\lim_{x \to \infty}f(x) = l$, then $\lim_{x \to 0} f(\frac1{x^2}) = l$ [duplicate]

Suppose that $\lim_{x \to \infty}f(x) = l$, $l \in \mathbb R$. Prove that $\lim_{x \to 0} f(\frac1{x^2}) = l$. Is it enough for me to say: Let $g(x) = \frac1{x^2}$. So, $\lim_{x \to 0} ...
1
vote
1answer
19 views

Writing skills: Proof of the relation between $\epsilon - \delta$ and open sets continuity

In order to check my math writing skills, I worked on writing the following basic proof. Theorem: If a function $f: X \to Y$ is continuous, then $G \subseteq Y$ is open implies that $f^{-1} (G)$ is ...
1
vote
1answer
11 views

Given two pairs of homotopic functions $h,h'$ and $k,k'$, are the respective compositions $k \circ h$ and $k' \circ h'$ homotopic?

I mostly just want to check my work. Claim: If $h,h': x \rightarrow Y$ are homotopic and $k,k':Y \rightarrow Z$ are homotopic, then $k \circ h$ and $k' \circ h'$ are homotopic. Proof: Let $H(t,s): I ...
1
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0answers
30 views

Zeros of nontrivial solution to 2nd order ODE.

Claim: Any non trivial solution to:$\ y''+\dfrac{(2-x^2)y}{4}=0$ has at most $1$ zero in $\mathbb{R} \ $ May I verify if my proof is correct? $y_1=e^{-\frac{x^2}{4}}$ is a solution to the ...
1
vote
0answers
22 views

$L^{2}(\mathbb{R})$ is a separable Hilbert space.

I want to show $L^{2}(\mathbb{R})$ is separable. My idea is $C_{c}(\mathbb{R})$ is dense in $L^{2}(\mathbb{R})$ in $L^2$ norm and polynomials with rational coefficients are dense in $C[a,b]$ in $\sup$ ...
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0answers
18 views

Winding numbers are continuous: The proof was too easy

There's a question in my complex analysis book: Let $G$ be a region and let $\gamma_0$ and $\gamma_1$ be two closed smooth curves in $G$. Suppose $\gamma_0\sim\gamma_1$ and $\Gamma$ is a homotopy ...
6
votes
3answers
91 views

$\sqrt{ab} = \sqrt{a} \sqrt{b}$

I'm trying to show that $\sqrt{ab} = \sqrt{a} \sqrt{b}$ where $a, b > 0$ For contradiction assume $$\sqrt{ab} \not= \sqrt{a} \sqrt{b}$$ $$ \sqrt{ab} - \sqrt{a} \sqrt{b} \not=0 $$ $$ ...
2
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0answers
13 views

Proof check: Continuity of an integral

$\gamma$ is a rectifiable curve in $\mathbb{C}$; For some open $G$, we have a continuous function $\phi:\{\gamma\}\times G\rightarrow \mathbb{C}$. $g(z):=\int_\gamma \phi(w,z) dw.$ We wish to show ...
3
votes
2answers
43 views

Proof that the characteristic polynomial of a $2 \times 2$ matrix is $x^2 - \text{tr}(A) x + \det (A)$

Let $$ A=\begin{bmatrix} a_{11} & a_{12}\\ a_{21} & a_{22}\\ \end{bmatrix}$$ Let $C_{A}(x) := \det(xI-A)$ be the characteristic polynomial of A. Show that ...
0
votes
3answers
16 views

$\varepsilon$ neighbourhood theorem clarification

I am little confused about visualizing $\varepsilon$ neighborhood theorem. Here is the statement of the theorem: Let $X$ be a compact subspace of $\Bbb R^n$; let $U$ be an open set of $\Bbb R^n$ ...
-1
votes
0answers
4 views

batch processing proof of the number of jobs' relationship with service time and waiting time [on hold]

The classical batch processing system ignores the cost of increased waiting time for users. Consider a single batch characterized by the following parameters: M average mounting time T average ...
1
vote
1answer
34 views

Invertible, Positive and Isometry Operator.

Let $T ∈ L(V )$ and $T = SP$, where $S$ is an isometry and $P$ is a positive operator. Prove that $T$ is invertible if and only if $P$ is invertible. Here is my approach: $\implies:$ $T = SP$ by ...
0
votes
2answers
23 views

Prove following argument is valid

Prove the following argument is valid. If Ralph doesn't do his homework or he doesn't feel sick, then he will go to the party and he will stay up late. If he goes to the party, he will eat too much. ...
0
votes
1answer
31 views

Uniform convergence on compact sets.

I think I missed something in this question because it asks me to use compactness in my proof, but I did none of that… Let $K \subset M$ be compact in $(M,d)$. Assume $f_n,g_n \in C(K,\Bbb R)$ for ...