Set Definition and 1000 Threads

Problem: Show that the set of differentiable real-valued functions ##f## on the interval ##(-4,4)## such that ##f'(-1) = 3f(2)## is a subspace of ##\mathbb{R}^{(-4,4)}## This is my first bouts with rigorous mathematics and my brain is not at all wired for attacking problems like this (yet). I...

https://www.theverge.com/2020/11/19/21575025/arecibo-observatory-puerto-rico-decommission-structural-collapse-cable-breakThe world-famous Arecibo Observatory in Puerto Rico, known for helping scientists peer into deep space and listen for distant radio waves, is set to be decommissioned and...

Summary:: Problem interpreting a vector space of functions f such that f: S={1} -> R Hello, Another question related to Jim Hefferon' Linear Algebra free book. Before explaining what I don't understand, here is the problem : I have trouble understanding how the dimension of resulting space...

I'm trying to evaluate the following integral in cylindrical coordinates. $$\int_0^6 \int_0^{\frac{\sqrt{2}}{2}}\int_x^{\sqrt{1-x^2}}e^{-x^2-y^2} \, dy \, dx \, dz$$ After attempting to set the bounds in cylindrical coordinates, I got $$\int_0^6 \int_0^{\frac{\sqrt{2}}{2}}\int_{\rho \cos\varphi...

Hey! I am looking at the following: translate the following statements into set inclusion. (i) Those who drown are not a fish or a swimmer. (ii) Scientists are human. (iii) A person who is not a swimmer is a non-swimmer. (iv) Fish are not human. (v) There was a case of a drowned...

Is there a theorem that states that a set of binary swaps can result in any permutation? For example, the original set (1,2,3,4,5) can have the swap (24) and result in (1,4,3,2,5). is there a set of specific swaps for each net result permutation?

Has anyone come across Quine's New Foundations? https://plato.stanford.edu/entries/quine-nf/ https://en.wikipedia.org/wiki/New_Foundations I'm not very knowledgeable about set theory, mathematical logic, or the foundations of mathematics, but I found what I read interesting. The basic idea (as...

Greetings, could you commend or correct the following: A dense subset ##X## of a set ##Y## is a set such that in each environment of ##y\in Y##, there is at least one element ##x\in X##. In other words, the elements of ##Y## can be approximated arbitrarily well by elements in ##X##. A set...

https://phys.org/news/2020-10-carbon-creation-astrophysics.html Back in 2016 - Recent results in nuclear astrophysics https://arxiv.org/abs/1605.07810 From the abstract -

If I'm given a set of four vectors, such as A={(0,1,4,2),(1,0,0,1)...} and am given another set B, whose vectors are given as a form such as (x, y, z, x+y-z) all in ℝ, what steps are needed to show A is a basis of B? I have calculated another basis of B, and found I can use linear combinations...

Hi! I want to check if i have understood concepts regarding the quotient U/V correctly or not. I have read definitions that ##V/U = \{v + U : v ∈ V\}## . U is a subspace of V. But v + U is also defined as the set ##\{v + u : u ∈ U\}##. So V/U is a set of sets is this the correct understanding...

So I have attempted to plot the scatter diagram. My first query is does the question intend for you to include both subsets of data on one axis, (which I have plotted on the x-axis) or rather does it demand two separate diagrams to investigate if there is any correlation, or a single diagram? I...

Hi everyone! I'm a mechanical engineering undergraduate and am designing a linear motion system for a school project. I hope to be able to use this system to achieve opposing motion for 2 x 200 kg loads (i.e. move them closer/ further apart simultaneously). Referring to the picture below...

This is about the famous, classic experiment of Magdeburg hemispheres with the Wikipedia link below: https://en.wikipedia.org/wiki/Magdeburg_hemispheres. "The experiment was designed to demonstrate the vacuum pump invented by Otto von Guericke - but also the tremendous 'strength' of the...

The set of real numbers x at the interval [0, 2π ] which satisfy $$2sin^2x\geq3cos2x+3$$ takes the form [a, b] ∪ [c, d]. The result of a + b + c + d is ... a. 4π b. 5π c. 6π d. 7π e. 8π What I've done thus far: $$2sin^2x\geq3cos2x+3$$ $$2sin^2x\geq3(cos2x+1)$$...

I am reading Sheldon Axler's book: Measure, Integration & Real Analysis ... and I am focused on Chapter 2: Measures ... I need help in order to fully understand the set of Borel sets ... ... The relevant text reads as follows: My questions related to the above text are as follows:QUESTION 1...

I am reading Sheldon Axler's book: Measure, Integration & Real Analysis ... and I am focused on Chapter 2: Measures ... I need help in order to fully understand the set of Borel sets ... ... The relevant text reads as follows: My questions related to the above text are as follows:QUESTION...

I have to show that $\forall z\in B(0,0.4)$, there exists an $x\in B(0,1)$ such that $f(x)=z$ but I am not sure how to show this. From the reverse triangle inequality $$-|f(x)-f(y)|+|x-y|\leq 0.1|x-y|\implies |f(x)-f(y)|\geq 0.9|x-y|$$ im not sure if this helps.

AB, AB, AD are Ld, that is, the three vectors lie on the same plane, so, "yes, the points lie on the same plane" However, AB CB and AD are Li, that is, the three vectors span the space R3, and don't lie in the same plane, so, how can four points that lie on the same plane, that can generate only...

I attached a file with some explanations of the variables in the code and the plot that I should get. I don't know what is wrong. Any help will appreciated. from scipy.integrate import quad import numpy as np from scipy.special import gamma as gamma_function from scipy.constants import e...

In the question above it, the author (Apostol) states: $$\int_0^n [t]^{2} dt = \frac{n(n-1)(2n-1)}{6}$$ Why can't I set the two equations = and get the result? 2(x-1) = x(x-1)(2x-1)/6 => 12 = 2x^2 - x => 0 = x^2-(x/2) -6 using quadratic equation I get the wrong answer

Hello everyone, I've been struggling quite a bit with this problem, since I'm not sure how to approach it correctly. The inequality form reminds me of the equation of a circle (x^2 + y^2 = r^2), but I have no idea how to be sure about it. Would it help just to simplify the inequality in terms...

Obviously the parenthetical part of the definition of ##F## means ##B\subset C## but we are not allowed to use ##\subset##. I do not know how to express implication with only union, intersection, and set minus without the side relation ##B\cap C = B\Leftrightarrow B\subset C##. This is using the...

If I give you 9 digits ##u_1, u_2, \dots, u_9##, is there an operation/set of operations you can perform to check whether all the digits from 1 to 9 are represented in that set? Just asking because my solution was a boring brute force check. I don't think anything useful becomes of the product...

Hey! 😊 Let $1\leq n\in \mathbb{N}$ and $\mathbb{R}^n$. A basis $B=(b_1, \ldots, b_n)$ of $V$ is an orthonormal basis, if $b_i\cdot b_j=\delta_{ij}$ for all $1\leq i,j,\leq n$. Let $E=(e_1, \ldots,e_n)$ be the standard basis and let $\phi \in O(V)$. ($O(V)$ is the set of all isometries...

I am solving the wave equation in z,t with separation of variables. As I understand it, Z(z) = acos(kz) + bsin(kz) is a complete solution for the z part. Likewise T(t) = ccos(ω t) + dsin(ωt) forms a complete solution for the t part. So what exactly is ZT = [acos(kz) + bsin(kz)][ccos(ωt) +...

I have tried to solve them. I would like to know if my answers are correct. (a) The total number of functions without any restrictions ##=n^m## The number of functions such that ##f(x)## is never ##1## ##=(n-1)^m## The number of functions such that ##f(x)=1## for at least one ##x\in S_m##...

For instance, I attached two problems in the the thumbnail below. I’m curious why A cannot be the empty set in 18b, but A is not excluded from being the empty set in 17a. In 17a, if A is empty, then all the hypothesis can be satisfied (the composition will be empty too, obviously), but g need...

This exercise is located in the vector space chapter of my book that's why I am posting it here. Recently started with this kind of exercise, proof like exercises and I am a little bit lost Proof that given a, b, c real numbers, the set X = {(x, y) E R^2; ax + by <= c} ´is convex at R^2 the...

I am searching for an easy solution to such questions.I have been playing with it for few hours.I can only make a guess because I don't know how to solve such type of questions.Although I tried assuming first term as 'a',common difference as 'r'.And then the last term that is 'arn-1'should be...

For two different coherent states \langle \alpha|\beta \rangle=e^{-\frac{|\alpha|^2+|\beta|^2}{2}}e^{\alpha^* \beta} In wikipedia is stated https://en.wikipedia.org/wiki/Coherent_state"Thus, if the oscillator is in the quantum state | α ⟩ {\displaystyle |\alpha \rangle } |\alpha \rangle it is...

Summary:: the set of arrays of real numbers (a11, a21, a12, a22), addition and scalar multiplication defined by ; determine whether the set is a vector space; associative law Question: determine whether the set is a vector space. The answer in the solution books I found online says that...

In the past, I have asked in this forum about the concept of set membership, in the context of ZFC. I guess it is a normal reaction to be a bit surprised by the usual statement in books that the set membership relationship is "undefined". But I have had this idea: a typical definition of the...

Hello everyone. =) In honor of Pi Day I'm going to be explaining the very beginning of set theory (which I consider the beginning of university math) live on Twitch in about two hours (1 PM GMT). For those who do not know Twitch, it's a completely free streaming platform - you can come in and...

I'm watching this video to which discusses how to find the domain of the self-adjoint operator for momentum on a closed interval. At moment 46:46 minutes above we consider the constant function 1 $$f:[0,2\pi] \to \mathbb{C}$$ $$f(x)=1$$ The question is that: How can we show that the...

According to https://plato.stanford.edu/entries/zermelo-set-theory/ , Zermelo (translated) said: I don't know if that quote is part of his formal presentation. It does raise the question of whether set theory must formally assume that there exists an equivalence relation on "elements" of...

Here is my attempt. Since we have to prove that ##A## is finite, we need to prove that there exists some ##m \in \mathbb{N}##, such that there is a bijection from ##A## to ##I_m##. And hence we have ##A \thicksim I_m##. Now, since there are ##n## elements in ##I_n##, number of elements in ##A##...

Problem: A vertex set $S$ in a graph $G$ is said to be totally t-dominating, for a positive integer t, if $|N(v) \cap S| \geq t$ for all $v \in V (G)$. Suppose that r, t, n are positive integers such that $r > 2t$ and $t \geq \frac{14}{3}\cdot ln(2n)$, and let $G$ be an r-regular n-vertex graph...

##S_3 = \left\{ \ x∈ℝ : x^2+x+1≥0 \right\}## I am not sure if I have done this correctly. The infimum/supremum and maximum/minimum are confusing me a bit. This is how I started: ##x^2+x+1=0## ##x^2+x+ \frac1 4\ =\frac{-3} {4}\ ## ## \left\{ x^2+\frac 1 2\ \right\} ^2 +\frac 3 4\ = 0##...

Suppose I measure the distance between two objects for three trials. The two objects then get farther away, and I measure the distance between them again for three trials. I repeat this for 3 more different distances, getting a total of 15 measurements (3 trials for 5 distances). I then compute...

Let ##_\Omega \left\{ (x,y,z)\in R^3 : - \sqrt{3-y^2-z^2} \leq x \leq z+2 ,y^2+z^2 \leq 3 \right\} ## and consider the function ##f(x,y,z)=y^2x+z^2x## Represent the domain ##\Omega## compute the vector field ##F=\nabla f## compute the inward flux. So I've found that one is a cylinder of...

Edit, the vector that rotates below might not rotate at all. Please forgive any mistaken statements or sloppiness on my part below. I think that by some measure a helicoid can be considered a smooth curved 2 dimensional surface except for a line of points? Consider not the helicoid above...

Hey! :o Let $\displaystyle{a:=\begin{pmatrix}2 & 1 & 0 & 5 \\ 1 & 0 & 1 & 1 \\ 4 & 1 &2 & 7\end{pmatrix}\in \mathbb{R}^{3\times4}}$ and $\displaystyle{b_1:=\begin{pmatrix}1 \\ 1 \\ 1\end{pmatrix} , \ b_2:=\begin{pmatrix}-2 \\ 1 \\ 0\end{pmatrix} \in \mathbb{R}^3}$. I applied the Gauss...

What's the meaning of the "bar" on the basis set of W at bottom right corner?

This is wild. I was always fascinated with the Mandelbrot set, as well as the bifurcation diagram. I had no idea the Mandelbrot diagram was a different visualization of the bifurcation diagram. Question: is this video accurate? I always question the veracity of YouTube science videos.

"The fact that the above eleven properties are satisfied is often expressed by saying that the real numbers form a field with respect to the usual addition and multiplication operations." -what do these lines mean? in particular the line "form a field with respect to"? is it something like...

Hey! :o I am looking at the following exercise: Make a sketch of a regular tetrahedron and label the corners with the numbers $1, 2, 3, 4$. For $1\leq i\leq 5$ the permutations $\pi_i \in \text{Sym} (4)$ are defined as follows: \begin{align*}&\pi_1:=\text{id} \\ &\pi_2:=(1 \ \ 2) \\...

given the following ##\sum_{n=0}^\infty n^2 x^n## in order to find the radius of convergence i do as follows ##\lim_{n \rightarrow +\infty} \left |\sqrt [n]{n^2}\right|=1## hence the radius of convergence is R=##\frac 1 1=1## |x|<1 Now i have to verify how the series behaves at the...

Now, set of even integers is ## A = \{ \cdots, -4, -2, 0, 2, 4, \cdots \} ##. We need to prove that ## \mathbb{Z}^+ \thicksim A##. Which means that, we need to come up with a bijection from ##\mathbb{Z}^+## to ##A##. We know that ##\mathbb{Z}^+ = \{1,2,3,\cdots \} ##. I define the function ##f ...

I tried to name the shaded area of a Venn diagram using numbers to isolate the regions. And I found that there are several ways to get the same region. Can the set notations simplfy