What is Set: Definition and 1000 Discussions

Seth, in Judaism, Christianity, Mandaeism, Sethianism, and Islam, was the third son of Adam and Eve and brother of Cain and Abel, their only other child mentioned by name in the Hebrew Bible. According to Genesis 4:25, Seth was born after Abel's murder by Cain, and Eve believed that God had appointed him as a replacement for Abel.

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  1. lekh2003

    How Does Energy Transformation Occur in Quantum Orbital Mechanics?

    Homework Statement Homework Equations I've used: ##mv^2/r## = Centripetal ##q^2/r^2## = Force pulling the electron in A bunch of other ones which I really can't be bothered listing. The Attempt at a Solution I managed to get Part A of the question using pretty simple methods. I set ##mv^2/r...
  2. nomadreid

    I Cardinality of a set of constant symbols (model theory)

    First, I want to be pedantic here and underline the distinction between a set (in the model, or interpretation) and a sentence (in the theory) which is fulfilled by that set, and also constant symbols (in the theory) versus constants (in the universe of the model) Given that, I would like to...
  3. A

    I Statistics proof: y = k x holds for a data set

    Simple linear regression statistics: If I have a linear relation (or wish to prove such a relation): y = k x where k = constant. I have a set of n experimental data points ...(y0, x0), (y1, x1)... measured with some error estimates. Is there some way to present how well the n data points shows...
  4. R

    How Can We Prove a Summation Identity by Double Counting?

    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...
  5. R

    Can the summation formula be proven by counting a set in two ways?

    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...
  6. Spinnor

    I Set of possible rotations of a 3D object and the rotation history

    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...
  7. Mr Davis 97

    A Set in the plane is never isometric to a proper subset

    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...
  8. R

    Prove that a set of positive rational numbers is countable

    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}...
  9. V

    B How to set up an integral to integrate over a sine wave?

    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...
  10. C

    MHB Proof the set with the multiplication is a group

    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...
  11. karush

    MHB Set of vectors form a vector space

    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?
  12. G

    MHB Identity Proof: (A-B)-C=A-(B∪C)

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

    I Is the Inverse Image of a Computable Function Recursively Enumerable?

    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...
  14. H

    MHB Set vs Class: Exploring the Differences

    What the differences between set and class?
  15. C

    Auto/Motor What is the best way to set up an alternator/battery system?

    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...
  16. F

    Extract anomalies from two-line element set (TLE) on ISS

    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...
  17. binbagsss

    Complex function open set, sequence, identically zero, proof

    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...
  18. C

    Studying Haven't set foot in a classroom in 20 years -- Planning Advice Please

    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...
  19. Mr Davis 97

    I Trying to understand why a set is both open and closed

    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##?
  20. Mr Davis 97

    Showing that a half-open set is neither open nor closed

    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...
  21. IgnacioPR

    I Some questions about toppling of a set of two cylinders

    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...
  22. F

    I Dimension of a set with vector function

    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##?
  23. W

    Set Theory: Power sets of Unions

    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...
  24. Eclair_de_XII

    What is the procedure for loading a data set from a textbook in R?

    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...
  25. S

    B Is 3-point of H20 used for Kelvin a certain set of isotope

    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...
  26. O

    How to design a motor with a given set of requirements?

    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...
  27. hilbert2

    A Analytical function from numerable point set

    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...
  28. Mr Davis 97

    Interior of the set of "finite" sequences

    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...
  29. A

    MHB Proving Set Equivalence with Set Differences?

    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...
  30. J

    MHB Measure Theory - Existence of Fsigma set contained in measurable set

    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...
  31. danielFiuza

    How to write this in Set Theory notation?

    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...
  32. Shackleford

    A Show that set is an orthonormal set in PC(0,l)

    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...
  33. R

    Set theory: Is my proof valid?

    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...
  34. T

    Need help formalizing "T is an open set"

    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...
  35. evinda

    MHB What is the Property of Infimum for Sets?

    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...
  36. FritoTaco

    Venn Diagram for set operations

    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...
  37. T

    Number of indie vectors ##\leq ## cardinality of spanning set

    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...
  38. Arman777

    Python Mandelbrot Set: Code for High-Resolution Image to Zoom In

    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 ?
  39. S

    I Probability that X is less than a set

    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...
  40. Y

    MHB Show that set of points form right-angled triangle

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

    MHB Every Closed Set in R has a Countable Dense Subset

    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...
  42. Gene Naden

    I Showing that the image of an arbitrary patch is an open set

    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...
  43. evinda

    MHB Is this the desired bounded set of the wave equation?

    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...
  44. Mr Davis 97

    For a set S, there is always a sequence converging to sup(S)

    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...
  45. Onezimo Cardoso

    How to Prove Inequality for Convex Sets in R^n?

    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...
  46. C

    MHB Set of points on the complex plane

    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...
  47. T

    MHB Basis for set of solutions for linear equation

    [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...
  48. Mr Davis 97

    Set of least upper bounds multiplied by a constant

    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...
  49. karush

    MHB What is the Equivalence Class for a Fixed Integer in Hurricane Lane's Aftermath?

    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
  50. Spinnor

    I Random set of N points in a unit disc, what is the average nearest distance

    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...
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