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

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1
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0answers
26 views

Linear Alg. Short proof on determinant

Hi can I get a quick check on my proof to see if it is correct. proof
0
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2answers
29 views

Suppose $1>a_n>0$ for $n\in \mathbb{N}$. Prove that $\prod_{n=1}^\infty (1-a_n)$ converges if and only if $\sum_{n=1}^\infty a_n<\infty$.

Suppose $1>a_n>0$ for $n\in \mathbb{N}$. Prove that $$\prod_{n=1}^\infty (1-a_n)$$ converges if and only if $\sum_{n=1}^\infty a_n<\infty$. I know this question is similar to one I just ...
1
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0answers
26 views

Deductive logic counter-intuitive result

I am working on a small proof in deductive logic. Here is what must be proved: $(\exists x \in T \mid A \implies P(x)) \implies A \implies (\forall x \in T \mid P(x))$ To me that looks unprovable ...
0
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1answer
12 views

Uniformly convergent in a set implies uniformly convergent in the set closure, too.

Let $f_n$:$X\rightarrow \mathbb{R}$ be a sequence of functions uniformly convergent in $X\subseteq \mathbb{R}$ . Suppose that each $f_n$ is continuous in the closure of $X$. Then $f_n$ is also ...
2
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0answers
16 views

Differential equation where Picard-Lindelöf can not be applied

My question is the following : Let $f:\mathbb{R}\to\mathbb{R}$ be continuous function and let $u:[a,b]\to\mathbb{R}$ be a $C^1$ function such that $$\forall t\in[a,b],u'(t)=f(u(t))\text{ and ...
0
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1answer
33 views

Hartshorne Exercise II.2.18(d)

The Exercise: Let $\phi: A \rightarrow B$ be a ring homomorphism and let $X = \operatorname{Spec} A, Y = \operatorname{Spec} B$. Let $f: Y \rightarrow X$ be the morphism of schemes induced by $\phi$. ...
0
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1answer
39 views

Finding a closed formula for: $1\cdot2\cdot3+2\cdot3\cdot4+…+(n-2)\cdot(n-1)\cdot(n)$ [duplicate]

As I calculated the sum of the serie above doesn't exist(sum doesn't converge). How can I prove it using the double computing(combinatorical method)?
1
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3answers
60 views

Alternative Proof that $\sqrt{p}$ is Irrational when $p$ is Prime

I have found various proofs that $\sqrt{p}$ is irrational on this site, but I didn't find one similar to the one that I am about to post, so I am wondering if it is free of logical problems. Here is ...
0
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0answers
14 views

How to calculate the length of this plane curve (loxodrome/rhumb line)?

I am trying to calculate the length of a (what I believe is) a loxodrome, using differential geometry. I am given a curve $\gamma(t)=\big(\theta(t),\varphi(t)\big)\subset \mathbb S^2$ that ...
3
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0answers
22 views

Properties of the Discrete Logarithm Problem

I am self-studying Hoffstein's An Introduction to Mathematical Cryptography, and this is problem 2.3 (p. 107-08). Let $p$ be a prime and $g$ an element in $\mathbb{F}_p^*$. (a) Suppose that ...
0
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1answer
18 views

Prove regular expression with induction

I need help proving the following regular expression via induction. I have the base case (easy of course) but I'm having a difficult time determining the inductive case. A regular expression over ...
3
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4answers
65 views

Solve the equation $7t+[2t] =52 $ ,where $[x]$ denotes the floor function for $x$.

Solve the equation $7t+[2t] =52 $ ,where $[x]$ denotes the floor function for $x$. My effort Using the fact that for any number $x$ we have that $x=[x]+\{x\}$ (where $\{x\}$ is the fractional ...
0
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1answer
21 views

Prove $(H \times 1)(1\times K)= H\times K$ where $H,K$ are groups.

Prove $(H \times 1)(1\times K)= H\times K$ where $H,K$ are groups. Suppose $x=ab,a\in H\times 1,b\in 1\times K$ Then $x=(h,1)(1,k)$ where $h\in H,k\in K$ Hence $x=(h,k)\in H\times K$ Let ...
0
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2answers
49 views

Proof Verification: If $x$ is a nonnegative real number, then $\big[\sqrt{[x]}\big] = \big[\sqrt{x}\big]$

Let $x$ be a nonnegative real number and denote $[x]$ as the greatest integer less than or equal to $x$. We will attempt to prove that $\big[\sqrt{x}\big] = \big[\sqrt{[x]}\big]$. First suppose that ...
0
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0answers
37 views

Hypothetical proof of Goldbach's conjecture? [on hold]

Goldbach's conjecture: Every even number greater than 4 is the sum of two prime numbers. An equivalent statement is; For all $n\geq 2$ there exists a number $e(n)$ such that $n-e(n)$ and ...
3
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1answer
34 views

Proof Verification: Show that $\Big[\frac{x+n}{m}\Big] = \bigg[\frac{[x]+n}{m}\bigg]$

Let $m,n \in \mathbb{Z}$ and let $x \in \mathbb{R}$. Let $[x]$ denote the floor function. We will attempt to prove $$\Big[\frac{x+n}{m}\Big] = \bigg[\frac{[x]+n}{m}\bigg]$$ Suppose without loss of ...
3
votes
2answers
83 views

Prove that $\lim_{x\rightarrow \infty} \frac{x^2 - 1}{x^2 + 1} = 1$ using definition of limit.

Ok, so if I have to use definition, then I should prove something like this: $(\forall \epsilon >0)(\exists k>0)(\forall x \in X)$ then if $ x>k$ then $|f(x) - L| <\epsilon$ $L$ is ...
1
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2answers
44 views

If $\frac{ay-bx}{p}=\frac{cx-az}{q}=\frac{bz-cy}{r} $

If $$\frac{ay-bx}{p}=\frac{cx-az}{q}=\frac{bz-cy}{r}$$, prove that : $$\frac{x}{a}=\frac{y}{b}=\frac{z}{c}$$ My solution, $$\frac{c(ay-bx)}{cp}=\frac{b(cx-az)}{bq}=\frac{a(bz-cy)}{ar}$$. Now, I could ...
1
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2answers
22 views

Product of terms involving complex exponents [on hold]

I have worked out the $\prod_{k = 1}^{50}i^k$. I get answer is $-i$. Is it correct?
0
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2answers
50 views

Prove or disprove: For non-negative integers $m$ and $n$, $m!n! = (mn)!$

I have rewritten the question as "If $m$ and $n$ are non-negative integers, then $m!n!$ = $(mn)!$" Here is my current attempt. I am not sure if I am on the right path. Proof. Let $m$ and $n$ be ...
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votes
4answers
71 views

Find $A$ and $B$ such that $A⊈B$ and $B⊈A$? [on hold]

I need to prove that the subset relation “$⊆$” on all subsets of $\mathbb Z$ is not a total order and I'm going to do this by finding $A$ and $B$ such that $A⊈B$ and $B⊈A$? Is there a simple solution ...
0
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1answer
67 views

Does this prove the sequence $5+(-1)^n$ does not have a limit?

The question is "Consider the sequence $s_n=5+(-1)^n$. Prove that this sequence does not have a limit". My professor in class proved this by choosing $n_1$ to be even, $n_2$ to be odd, and ...
15
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7answers
2k views

What is the flaw of this proof (largest integer)?

Let $n$ be the largest positive integer. Since $n ≥ 1$, multiplying both sides by $n$ implies that $n^2 ≥ n$. But since $n$ is the biggest positive integer, it is also true that $n^2 ≤ n$. It follows ...
3
votes
3answers
45 views

Find the area of the region described by $|5x|+|6y| \le 30 $

Find the area of the region described by $|5x|+|6y| \le 30 $ (where $|z|$ denotes the absolute value of $z$). My effort Imagining a number line and interpreting the problem as the request to ...
0
votes
1answer
30 views

Using pumping lemma

I'm trying to prove that the language $L = \{w \in \{0,1\}^* ∣ w \leq w′ \text{ where }w′ \text{ is any rotation of }w\}$ is not a regular language. Note: The inequality is with respect to ...
2
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1answer
66 views

Would you confirm this as a proof to the Pythagorean theorem?

I'm new in mathematics, and trying to build my way up starting by doing simple tasks. My current one is proving the Pythagorean theorem without looking it up. This photo contains my current "proof" ...
1
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1answer
16 views

Image of Upper Half Disc under $w = 1/z$

I need to find the image of the upper half disc $|z|<1$, $Im\, z >0$ under the inverse transformation $w = 1/z$. Now, since $|z|<1$, $|z|^{2}<1$. Rewriting this as $z\overline{z}<1$, ...
0
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1answer
20 views

$A_n$ is generated by 3-cycles given $n\geq 3$. Is this proof correct?

The elements of $A_n$ is either of the form $(a,b,c,...)...$ or of the form $(a,b)(c,d)...$ In both cases, the element is a product of an even number of transpositions, not pairwise disjoint in the ...
0
votes
1answer
36 views

Proving $\mathbb{R}/\sim$ is homeomorphic to unit circle

Let $S$ be the unit circle in $\mathbb{C}$, standard topology. Define the equiv. rel. $\sim$ on $\mathbb{R}$ as $x\sim y\iff x - y\in\mathbb{Z}$. I would like to prove that $\mathbb{R}/\sim$ is ...
0
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0answers
30 views

Quick proof check [on hold]

Intro to analysis by gaughan 1.4 #45. Show that if x is any real number, there is a sequence of rational numbers converging to x.
1
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2answers
15 views

Prove that the space $\Bbb R_K$ is not regular.

Prove that the space $\Bbb R_K$ is not regular. where the basic open sets on $\Bbb R_K$ is given by $\{(a,b):a,b\in \Bbb R\}\cup \{(a,b)-K\}$ where $K=\{\dfrac{1}{n}:n\in \Bbb Z_+\}$. ...
0
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0answers
18 views

How to integrate to solve a PDE with mixed partials in the integrand

Problem Statement: Determine the equlibrium temperature distribution inside a circular annulus $r_1\leq r \leq r_2$. If the outer radius is at temperature $T_2$ and inner radius at temp $T_1$. So ...
2
votes
1answer
52 views

If $x$ is an isolated point of $S \subseteq \mathbb{R}$, then $x$ is a boundary point of $S$. [duplicate]

Is the following proof valid? (Note: I know there is a post discussing this problem, but I am curious to see if my argument works). This problem is different from another post that is similar with ...
1
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1answer
20 views

Transformations and Dependence

Hi, for these problems I generally get the gist of it. If you have some linearly dependent vectors $v_1, \ldots, v_m$ in $\mathbb{R}^n$ then when you transform those vectors $T(v_1), \dots, T(v_m)$ ...
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0answers
18 views

renewal process and probability(exercice) [on hold]

Let $N_t$ a renewal process. Let $A_t=t-S_{N_t-1}$, $S_{N_t}=X_1+...+X_{N_t}$ with $X_i$ the jumps moments. Let $Z_A(t)=P(A_t \leq u)$ 1) How to show $P(A_t \leq u |X_1=x)=P(A_t \leq u |X_1 \geq t)$ ...
5
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0answers
32 views

Find the inverse of the following piecewise defined function

Find the inverse of $f$ if $f(x)=$ $$ \begin{cases} \sqrt{2-x}, &\text{for $x<0$}\\ 1-x^2, &\text{for $x \ge 0$} \\ \end{cases} $$ My effort For $y=\sqrt{2-x}$ ,we find ...
1
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0answers
17 views

A Possible Logical Problem with Showing that $x$ is a Boundary Point Whenever It is an Isolated Point

Prove: If $x$ is an isolated point of a set $S$, then $x \in \mathrm{bd} \, S$. I have two ways to solving this problem, but I believe the first one has a logical issue which I will explain below. ...
1
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0answers
27 views

Verification: A self-conjugate element in an odd-order finite group is the identity

I think I've found a proof of the following, but my proof is horrible, and I feel like I've made a mistake or that I've missed an important principle: Theorem: Let $G$ be a finite group of odd order ...
11
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2answers
58 views

Solution to $\frac{d}{d\frac{1}{x}} x$

If I want to solve $$\frac{d}{d\frac{1}{x}} x$$ is my approach correct? As $$\begin{align*} \frac{d}{d\frac{1}{x}}x&=\\ \text{with }\frac{1}{x}&=y\\ ...
2
votes
2answers
28 views

Show that if $\prod_\alpha X_\alpha$ is normal then so is $X_\alpha$.

Show that if $\prod_\alpha X_\alpha$ is normal then so is $X_\alpha$. This a question of proof-verification.So please suggest the required edits and fault in the logic but please don't give a ...
1
vote
0answers
57 views

satisfaction of a sentence with two quantifiers

I want to be sure that I understand how to show that a structure satisfies a sentence under a variable assignment, and suspect that I'm handling the computation of multiple quantifiers incorrectly. ...
0
votes
2answers
29 views

$U(n)^2$ is a proper subgroup of $U(n)$

I'm trying to show that $U(n)^2$ is a proper subgroup of $U(n)$. Here $$ U(n)^2 = \{x^2 \mid x \in U(n)\}$$ where $U(n)$ is the group of units modulo $n$. My idea was to argue as follows: Consider ...
1
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0answers
18 views

Extension of co-coercivity in strongly convex functions

I am studying strongly convex functions and they mention if $f(x)$ is strongly convex with Lipschitz gradients $L$, which means $\parallel \nabla f(y) - \nabla f(x)\parallel \leq L\parallel x - y ...
1
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1answer
41 views

Intersection of two ideals

Let $A$ be a commutative ring and let $\mathfrak{a}$, $\mathfrak{b}$ be ideals in $A$. I am asked the following question: Show that $\mathfrak{a} \cap \mathfrak{b}$ is the largest ideal of $A$ ...
2
votes
4answers
58 views

Misconception of infinite prime numbers proof by contradiction?

I'm using the proof on this page, except with $q$ instead of $p$ on the left side. The misconception of the proof is that $q$ has to be a prime number. I found this using $n = 6$, which gets me $q = 1 ...
0
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0answers
13 views

$∀ Z^+$ can be written as $c_r * 3^r + c_{r-1} * 3^{r-1} + …. + c_2 * 3^2 + c_1 * 3 + c_0$

I am trying to prove the following statement: $∀ Z^+$ can be written as $c_r * 3^r + c_{r-1} * 3^{r-1} + …. + c_2 * 3^2 + c_1 * 3 + c_0$ where $c_r =$ 1 or 2, and $c_i$ = 0, 1, or 2 for all integers ...
0
votes
1answer
35 views

Any collection of n coins can be obtained using a combination of 3¢ and 5¢ coins where n ≥ 14

I am trying to prove this statement with strong induction, but I'm a little lost on the inductive step. Proposition: Let P(n) be the sentence ‘any collection of n coins can be obtained using a ...
0
votes
0answers
15 views

Degrees of vertices in a circuit must be even

Let $G$ be a graph with a circuit. Let $C$ denote the subgraph of $G$ consisting of vertices and edges of the circuit. Then for every vertex in $C$, $\deg (v)$ considered in $C$ is even. I would ...
0
votes
0answers
25 views

Does this method of finding the range of rational functions always work?

Consider the irreducible rational function in $\mathbb{R}^2$. $$y=\frac{A(x)}{B(x)}$$ where at least one term is quadratic and the other term has degree either 0, 1 or 2. The classic way of ...
0
votes
0answers
16 views

Can you analyze this identity involving the sum of divisors function and $rad(n)=\prod_{p\mid n}p$?

Let $\sigma(n)$ the sum of divisors function, then by Mobius inversion formula $$\sigma(n)=n-\sum_{\substack{d\mid n,d<n}}\sigma(d)\mu\left(\frac{n}{d}\right),$$ and since this function is ...