Set Definition and 1000 Threads

Homework Statement prove the summation by counting in a set two ways. Mod edit: The summation was later confirmed by the OP to be ##\sum_{k = 1}^n 2^{k - 1}## $$\sum_{n=0}^k 2^{k-1} = 2^{k}-1$$ Homework EquationsThe Attempt at a Solution [/B] ##2^{k-1} = (1+1)^{k-1} = \binom n 0 + \binom n 1...

Mod note: Fixed numerous problems with LaTeX Also, the fractions shown throughout should instead be binomial coefficients 1. Homework Statement prove the summation formula by counting a set in two ways $$\sum_{k=0}^n k\left( \frac n k \right) = n *2^{n-1}$$ Homework Equations LHS = ##k...

I am trying to understand the picture below which is of a contractible and uncontractible loop in what I would call (proper name?) "rotation space", where "rotation space" is a solid ball of radius π with opposite points on the surface of the ball identified, each point of the ball representing...

Homework Statement Is there a bounded set ##H## in the plane which is isometric to a proper subset ##S \subset H##? Homework EquationsThe Attempt at a Solution I'm thinking that the answer is no. Here are my ideas, that by no means constitute a proof. Every isometry of the plane is either a...

Homework Statement Prove that the set of positive rational numbers is is countable by showing that the function K is a 1-1 correspondence between the set of positive rational numbers and the set of positive integers if K(m/n) = p_1^{2a_1}p_2^{2a_2}...p_s^{2a_s}q_1^{2b_1-1}...q_t^{2b_t-1}...

How do I setup an integral to integrate over the following equation: V(t) = 1/(R*C) integral to t Vin(t) dt This is the capacitor voltage over time formula. I want to integrate over a sine wave from 9 to 81 degrees. Frequency of 120Hz, amplitude of 120V. The formula I used in wolframalpha is...

Dear Everyone, $\newcommand{\Z}{\mathbb{Z}}$Suppose the set is defined as: $\begin{equation*} {(\Z/n\Z)}^{\times}=\left\{\bar{a}\in \Z/n\Z|\ \text{there exists a}\ \bar{c}\in \Z/n\Z\ \text{with}\ \bar{a}\cdot\bar{c}=1\right\} \end{equation*}$ for $n>1$ I am having some trouble Proving that...

this is what is given so by addition $$\begin{bmatrix}x_1\\y_1\\5z_1\end{bmatrix} \oplus \begin{bmatrix} x_2\\y_2\\5z_2 \end{bmatrix} = \begin{bmatrix} x_1+x_2\\y_1+y_2\\5z_1+5z_2 \end{bmatrix} = \begin{bmatrix} X\\Y\\10Z \end{bmatrix}$$ uhmmmm really?

Show that(A-B)-C=A-(BUC)

Hello, I am stuck on deciding if given sets are recursive or recursively enumerable and why. Those sets are: set ƒ(A) = {y, ∃ x ∈ A ƒ(x) = y} and the second is set ƒ-1(A) = {x, ƒ(x) ∈ A} where A is a recursive set and ƒ : ℕ → ℕ is a computable function. I am new to computability theory and any...

What the differences between set and class?

Hello, I am a food truck owner and I recently had a situation which made me rethink my power needs for my kitchen on the food truck. I have a 2001 Workhorse forward control Stepvan and I wanted to run an alternator, batteries, and an inverter setup INDEPENDENT of the trucks own alternator and...

Homework Statement I am faced to a problem of interpretation illustrated on figure below : I must precise that I talk about **mean anomaly**. Homework Equations for the 2 questions, I am asked to find : 1) at which anomaly is the ISS at the two lines epoch ? 2) at wchich anomaly is the...

Homework Statement Hi I am looking at this proof that , if on an open connected set, U,there exists a convergent sequence of on this open set, and f(z_n) is zero for any such n, for a holomorphic function, then f(z) is identically zero everywhere. ##f: u \to C##Please see attachment...

Hi all, I'm seriously considering going back to college for a degree in engineering, specifically mechanical. I meet the aptitude tests, all of the interests, want to design and create, etc. I've definitely got the bug. I do DIY carpentry jobs on the side, I've got technical degrees in machine...

Take the subset of ##\mathbb{R}##, ##X = [0,1]\cup [2,3]##. Under the usual metric, the set ##X## is open and closed, according to my text. How is this the case? In particular, how is ##[0,1]## open in ##X##?

Homework Statement Where ##a,b\in \mathbb{R}##, show that ##[a,b)## is not open. Homework EquationsThe Attempt at a Solution I need to show that there exists an ##x\in [a,b)## such that for all ##\epsilon > 0##, ##B_\epsilon (x) \not \subseteq [a,b)##. To this end put ##x=a##, and let...

Dear colleagues of Physics Forum: I am trying to estimate the inclination (angle) at which the figure presented with the enclosed photo, composed by two cylinders of different densities (ρ1=2650 kg/m3 and ρ2=7000 kg/m3), will topple, when progressively inclined from a horizontal position. Let's...

I have a vector ##\textbf{v} \in \mathbb{R}^{3N}## and a function ##\textbf{Ψ} : \mathbb{R}^{3N} \longrightarrow \mathbb{R}^p## such that ##\textbf{Ψ}(\textbf{v})=0##. Why the set ##T=\{ \textbf{x} \in \mathbb{R}^{3N} \ | \ \textbf{Ψ}(\textbf{x})=0 \}## has dimension ##n=3N-p##?

Homework Statement I'm having issues understanding a mistake that I'm making, any assistance is appreciated! I know a counterexample but my attempt at proving the proposition is what's troubling me. Prove or disprove $$P(A \cup B) \subseteq P(A) \cup P(B) $$ Homework EquationsThe Attempt at...

Homework Statement I'm using a textbook called "Introductory Statistics with R" by Dalgaard, and I am having trouble doing one of the problems. It asks me to isolate names from a data set called "juul". But the problem is that I am unable to locate said data set. Homework Equations Functions...

If I had had enough space to enter the proper question, it would have been: Is the triple point of water which is used for the definition of the unit of Kelvin defined as water having a certain isotopes such as 2 Hydrogen-1 & 1 Oxygen-16? Or would the error introduced by using different...

Let's suppose we are given a set of requirements and we are to design a motor that meets them. Requirements are: 3 phase synchronous, 480v, 50Hz and of hysteresis type and alnico is the only available option and it should deliver a 0.6N.m output torque. Dimensions should not exceed 150 mm and...

Sometimes there are functions that are initially defined for only integer values of the argument, but can be extended to functions of real variable by some obvious way. An example of this is the factorial ##n!## which is extended to a gamma function by a convenient integral definition. So, if I...

Homework Statement Identify the boundary ##\partial c_{00}## in ##\ell^p##, for each ##p\in[1,\infty]## Homework Equations The interior of ##S## is ##\operatorname{int}(S) = \{a\in S \mid \exists \delta > 0 \text{ such that } B_\delta (a) \subseteq S\}##. ##\partial S = \bar{S}\setminus...

I am working on a set equivalent (the textbook refers as "equinumerous" denoted by ~) as follows: If $S$ and $T$ are sets, prove that if $(S\backslash T) \sim (T\backslash S)$, then $S \sim T$. Here is my own proof, I am posting it here wanting to know if it is valid. (It may not be as elegant...

Problem: Let $E$ have finite outer measure. Show that $E$ is measurable if and only if there is a $F_\sigma$ set $F \subset E$ with $m^*\left(F\right)=m^*\left(E\right)$. Proof: "$\leftarrow$" To Show: $E=K\cup N$ where $K$ is $F_\sigma$ and $m^*(N)=m(N)=0$. By assumption, $\exists F$, and...

Hello Everyone, I am trying to write the intersection of a physical problem in the most compact way. I am not really familiar with Set Theory notation, but I think it has the answer. It is about the intersection of two circular areas: - Area 1: A - Area 2: B If I want to write this in Set...

Show that set \{ (2/l)^{1/2} sin(n-1/2)(\pi x/l)\}^\infty_1 is an orthonormal set in PC(0,l). Of course, I need to show that <\phi_m, \phi_n> = \delta_{mn} <\phi_m, \phi_n> = (2/l) \int_0^l sin(m-1/2)(\pi x/l) sin(n-1/2)(\pi x/l)\, dx \\ = 1/l \int_0^l cos(m-n)(\pi x/l) cos(m+n+1)(\pi...

Homework Statement Prove the following for a given universe U A⊆B if and only if A∩(B compliment) = ∅ Homework EquationsThe Attempt at a Solution Assume A,B, (B compliment) are not ∅ if A∩(B compliment) = ∅, x∈A ∨ x∈ (B compliment), but not both If x∈A ∧ x∉(B compliment), then x∈B , because...

Homework Statement Let ##S\subseteq \Bbb{R}## and ##T = \{ t\in \Bbb{R} : \exists s\in S, \vert t-s\vert \lt \epsilon\}## where ##\epsilon## is fixed. I need to show T is an open set. Homework Equations n/a The Attempt at a Solution Let ##x \in T##, then ##\exists \sigma \in S## such that ##x...

Hello! (Wave) I want to show that $\ell$ is the infimum of a set $A$ iff $\ell$ is a lower bound of $A$ and for each $\epsilon>0$ there exists an $a \in A$ such that $\ell+\epsilon>a$. I have thought the following so far for the direction "$\Leftarrow$". Let $\ell$ be a lower bound of $A$ such...

Homework Statement Hi, the problem states to draw a Venn Diagram for A\cap(B-C) Homework Equations (B - C) means include all elements in the set B that are not in C. Definition from my book: Let A and B be sets. The difference of A and B, denoted by A - B, is the set containing those...

Homework Statement In a finite-dimensional vector space, the length of every linearly independent list of vectors is less than or equal to the length of every spanning list. It's quite long :nb), hope you guys read through it. Thanks! :smile: Homework Equations N/A The Attempt at a Solution...

Is there a code for the mandelbrot set that creates an image with an high resolution so that we can zoom and see the fractal over and over again ?

Hi everyone, I am currently working through the textbook Statistical Inference by Casella and Berger. My question has to do with transformations. Let ##X## be a random variable with cdf ##F_X(x)##. We want to find the cdf of ##Y=g(X)##. So we define the inverse mapping, ##g^{-1}(\{y\})=\{x\in...

I was thinking of using Pythagoras here but it didn't get me far Any suggestions?

Show that every closed set in R has a countable dense subset. Let's call the set F. I've been thinking about this problem for a little bit, and it just doesn't seem like I have enough initial information! I tried listing some things that I know about closed sets in R: $\cdot$ Countable...

O'Neill's Elementary Differential Geometry, problem 4.3.13 (Kindle edition), asks the student to show that the image of an open set, under a proper patch, is an open set. Here is my attempt at a solution. I do not know if it is complete as I have difficulty explaining the consequence of the...

Hello! (Wave) I want to show for the initial value problem of the wave equation $$u_{tt}=u_{xx}+f(x,t), x \in \mathbb{R}, 0<t<\infty$$ that if the data (i.e. the initial data and the non-homogeneous term $f$) have compact support, then, at each time, the solution has also compact support. I...

Homework Statement [/B] Let ##S\subset \mathbb{R}## be nonempty and bounded above. Show that there must exist a sequence ##\{a_n\}_{n=1}^\infty\subseteq S## such that ##\lim_{n\to\infty}a_n=\sup(S)##. Homework EquationsThe Attempt at a Solution Here is my idea. Let ##\epsilon >0##. Then there...

Homework Statement Let ##C \subset \mathbb{R}^n## a convex set. If ##x \in \mathbb{R}^n## and ##\overline{x} \in C## are points that satisfy ##|x-\overline{x}|=d(x,C)##, proves that ##\langle x-\overline{x},y-\overline{x} \rangle \leq 0## for all ##y \in C##. Homework Equations By definition...

Dear Everybody, I am wanting to check the solution to this question: Sketch the set of points determined by the given conditions: a.) $\left| z-1+i \right|=1$ b.)$\left| z+i \right|\le3$ c.)$\left| z-4i \right|\ge4$ work: I know (a.) is a circle with radius 1 and its center at (-1,1) on the...

[solved] Basis for set of solutions for linear equation Hi, I have this problem I was working through, but I'm not sure that I've approached it from the right way. The problem consists of 3 parts, which build off of each other. I'm pretty confident about the first two parts, but no so much...

Homework Statement Let ##S,T \subseteq \mathbb{F}## be nonempty sets. Assume ##\sup (S)## and ##\sup (T)## both exist in ##\mathbb{F}##. Show that ##\forall a \in \mathbb{F}^+ \cup \{0\}## we have ##\sup(aS) = a \cdot \sup (S)##. Homework EquationsThe Attempt at a Solution First I prove the...

ok need help with these 3 questions I know its fairly easy but its new to me, so we have had hurricane Lane here this week but mahalo much

Pick a random set of N points from the unit disc. Calculate the distance between all pairs of points and call the smallest value r. Do this calculation for many such sets. Please give me a hint how to estimate what the average value of r is. I guess a computer program could quickly come up with...

In the Zermelo-Fraenkel axioms of axiomatic set theory we find: Axiom. Given any set x, there is a set such that, given any set z, this set z is a member of if and only if every element of z is also an element of x. Why is this needed as an axiom? why isn't it merely a definition? Under...

Hi, I'm reading a paper about combustion and I can't understand the composition of the inlet mixture. Here's what the paper says: "A known flow rate of the isobutene (Matheson, TriGas, 99.5%) in nitrogen (General Air, 99.998%) was introduced into a 6mm ID quartz reactor that was heated in an...

I don’t see how the definition of |an> transmorphs into the statement involving the kroneck delta functions.