Set Definition and 1000 Threads

##m##, ##n##, ##a##, and ##b## are in ##\mathbb{Z}##. The set of natural numbers is an abelian group, so ##\{ma+nb|m,n\in\mathbb{Z}\}## is a subset of ##\mathbb{Z}##. ##a ## and ##b## are either relatively prime or not relatively prime. If ##a ## and ##b## are relatively prime, then there are...

How do the roster method and set builder methods when combined give modern meaning to traditional topics? For example: Find the roots of x^2-4x+3=0 Suppose we have no knowledge of the algebraic techniques for solving this equation. Had we wished to write the solution set for this equation using...

Hello! Lately, I've been struggling with this assignment. (angle brackets represent closed interval) I figured out that: a) union = R intersection = {0} b) union = (0, 2) intersection = {1} I asked my prof about this and she explained to me that it should be shown that if a set is an...

I'm not sure if this belongs to astronomy or GR. But as it - once again - proves Einstein right, I posted it here for all who need another paper to conquer all who doubt. And I think it is an interesting paper (53 pages), at least from my layman's point of view...

Suppose that I can generate the result of a function ## c_{x,y}=f(x,y)## by a method not involving the function ##f##. I need to find ##f(x,y)## now. The expression of ##f(x,y)## is expected to contain basic algebra operation (+-*/), power, absolute value and factorial. I have tried to find it...

##S## is a set of all vectors of form ##(x,y,z)## such that ##x=y## or ##x=z##. Can ##S## have a basis? S contains either ##(x,x,z)## type of elements or ##(x,y,x)## type of elements. Case 1: ## (x,x,z)= x(1,1,0)+z(0,0,1)## Hencr, the basis for case 1 is ##A = \{(1,1,0), (0,0,1)##\} And...

Here we go!

Hello, I tried looking for an existing solution for the following problem: "Assume that S is a set of d elements, and R is a total order relation on S. Assume that n elements are randomly extracted from S, and then they are ordered according to R. Which is the probability that in the i-th...

Hey! :giggle: We have the set $M=\{(x,y,z)\in \mathbb{R}^3 : x^2+y^2-z^2\leq 1, \ 0 \leq z\leq 3\}$. Draw $M$ and calculate the volume of $M$. I have done the following : \begin{equation*}\int_M\, dV=\int\int\int\, dx\, dy\, dy\end{equation*} Which are the boundaries of the integrals? Do we...

Let ##L\in E##. By definition, there is a subsequence ##\{x_{n_k}\}_{k\in\mathbb{N}}## that converges to ##L##. There is a natural number ##N## s.t. if ##n_k\geq N##, ##L\in(x_{n_k}-1,x_{n_k}+1)\subset(\inf\{x_n\}-1,\sup\{x_n\}+1)##. Hence, ##E## is a bounded set. If ##E## is a finite set, then...

##A=(-1,1)## ##B=[0,1)## Define ##f:A\longrightarrow B## by ##f(x)=x^2## Set ##X=A##. ##f(X)=\{f(x):x\in X\}=\{x^2:x\in(-1,1)\}=B## ##f^{-1}(f(X))=\{x\in A:f(x)\in f(X)\}=\{x\in (-1,1):x^2\in B\}=X## Now choose a non-zero point ##y\in f(X)##. There are two pre-images of this point: ##x,-x\in...

Homework Statement:: Write each of the following sets in set-builder notation. Relevant Equations:: None. {2, 4, 8, 16, 32, 64...} =2·{1, 2, 4, 8, 16, 32...} What should be the next step in this work?

I've boxed around my answer but I don't know if this is the right answer. I also used a substitution with c=3a+b in my work. I don't know if my work is precise or not.

https://www.cs.bham.ac.uk/~mhe/papers/omniscient-journal-revised.pdf Might be my favorite article I’ve ever came across I would like to see some interpretations on it to broaden my currently very narrow point of view… Have fun! -oliver

Can anyone please check/confirm if the set is correct? I've boxed around my answer.

I would like to know if any of you think there's any sort of connection, analogy, or common features between, sets in set theory and wave functions in QT? Wave functions lack trajectories, so do sets. Wave functions also distribute over areas, as sets can do. To my understanding, wave...

QUESTION: If A is a finite set, its cardinality, o(A), is the number of elements in A. Compute (a) o(A) when A is the set consisting of all five-digit integers, each digit of which is 1, 2, or 3. (b) o(B), where B = {x ∈ A : each of 1,2 and 3 is among the digits of x} and A...

I am confused with some basic definitions in general topology. Topological space is defined as a set ##X## and a collection ##T## of its subsets that satisfies some general properties (which I will not list here, but I will assume that the reader knows them). But such a very general definition...

Hi, I am new to the ANSYS Fluent. I have a question that is how to model a moist object in the Fluent as I am trying to study the drying process with a 2D simulation. I have tried to do it on my own by searching through google but still nothing working out.

In January of the year 2000, I was one more than eleven times as old as my son Will. In January of 2009, I was seven more than three times as old as him. Note: Do NOT solve. Set up the correct system of equations. This one can easily throw someone struggling with age word problems into a loop...

John is twice as old as his friend Peter. Peter is 5 years older than Alice. In 5 years, John will be three times as old as Alice. How old is Peter now? Do not solve. Just set up the correct equation. This one involves more people. I think a table helps out. ......Age Now...Age In 5 Years...

Here is the fuzzy definition of a function as presented by Ron Larson. Definition of Function A function f from a set A to a set B is a relation that assigns to each element x in the set A exactly one element y in the set B. The set A is the domain (or set of inputs) of the function f, and...

Hello everyone. I want to calculate the covariance matrix of a stochastic process using Matlab as cov(listOfUVValues) being the dimensions of listOfUVValues 211302*50. I get the following error: Requested 211302x211302 (332.7GB) array exceeds maximum array size preference. Creation of...

^because in my textbook, the set up doesn't even show the direction of dot/cross at points A/D, so we won't know the direction of current. then how are we supposed to use fleming's left hand rule to determine the direction of force?

[Mentor Note -- Thread moved to the ME forum to get better views] Let's consider an incompressible block of Neo-Hookean material. Let the initial reference geometry be described by ##B=[0,b] \times [0,b] \times [0,h]##. The professor gave me the following task: Of course there can be many...

Hey! :giggle: In $X=\mathbb{R}\setminus \mathbb{Q}$ with the usual metric, we consider $A=\{x\in X: \ 0\leq x\leq1\}$. Which of the following statements are correct? (a) $\partial A$ is singleton (b) $A$ is open and closed (c) $A$ is not open (d) $A$ is not compact (e) $A$ is as a...

Hey! :giggle: The set of $2$-dimensional orthogonal matrices is given by $$O(2, \mathbb{R})=\{a\in \mathbb{R}^{2\times 2}\mid a^ta=u_2\}$$ Show the following: (a) $O(2, \mathbb{R})=D\cup S$ and $D\cap S=\emptyset$. It holds that $D=\{d_{\alpha}\mid \alpha\in \mathbb{R}\}$ and...

I am trying to solve the following set of 5 first order ODEs Where the variables are ##\Theta_0, \Theta_1, \Phi, \delta## and ##v##. The initial conditions (ICs) are (Note that there is a typo in the above ICs; it is ##v## instead of ##u##). I am following this solved sample, following...

These days most measurement instruments in m,y lab are controlled via an ethernet connection. In my lab the PCs that control the experiments therefore have two network cards: one is used of "normal" internet/intranet access the other is set to static IP address (typically 192.168.0.x) and is...

My dad has been asking about some speakers for his boat, something that will move a bit of air, not too big though. I'm thinking 1x12. He also wants it bluetooth and powered externally, so it doesn't drain the boat battery. So I'm thinking 1x12 speaker in a small box that can fit inside the...

For those who forgot the previous chapters: https://www.physicsforums.com/threads/ultra-hyperbolic-pde-and-f-theory.766711/#post-4827614 So I now found in Google the following book which I ran to buy: https://www.amazon.com/dp/3838130510/?tag=pfamazon01-20 unfortunately it's only 200 pages...

I'm not fully aware when ## { n \choose k } ## represents a set of subsets of cardinality ## k ## and when does it represent a set of lists of size ## k ##. For example, look at the following two problems: Problem 1: How many 7-digit binary strings have three 1's? Answer: ## { 7 \choose 3 }...

I've come up with a problem that's important for an algorithm I'm developing, and I don't know how to solve it. Wondering if anyone here can help? I have an initial set S1 of size N. S2 is created by randomly sampling M samples from S1 with replacement (meaning the same item could be selected...

Problem: Find the set of all harmonic functions ##u(x,y,z)## that satisfy the following inequality in all of ##R^3## $$|u(x,y,z)|\leq A+A(x^2+y^2+z^2)$$ where ##A## is a nonzero constant. Work: I removed the absolute value bars by re-writing the expression $$-C-C(x^2+y^2+z^2)\leq u\leq...

Hi, I saw this video by numberphile, and near the end they mention how at the point of a right angle the equation shows infinite velocity for fluids. I'm wondering if this isn't perhaps related to Cantor's solution to Zeno's Paradox of distance (there's always a midpoint). Because I feel like at...

This is a somewhat trivial question, but I never managed to learn much logic back in the day...so: The definition of a subset can be written as: ## A \subset B \Leftrightarrow \forall x (x \in A \Rightarrow x \in B) ## However, over which set is ##\forall## supposed to quantify? It seems to...

$\tiny{311.1.1.26}$ Construct 3 augmented matrices for linear systems whose solution set is $x_1=3, \quad x_2=-2, \quad x_3=-1$ ok the only thing I could think of is just rearrange the rows of an RREF matrix. albeit losing the triangle format hopefully no typos $\left[\begin{array}{rrr|rr} 1&...

120, 60,30 cos 120 = -0.5, cos 60 = 0.5, cos 30 = 0.866

I get 2^5 x 3^80 Am I correct?

Find the probability that this number is in set A but not in set B. (please see attachments) Is the answer to this simple maths secondary school homework 3/14. If not please help?

I have two independant experiments have measured ##\tau_{1},\sigma_{1}## and ##\tau_{2},\sigma_{2}## with ##\sigma_{i}## representing errors on measures. From these two measures, assuming errors are gaussian, we want to get the estimation of Ï and its error (i.e with a combination of two...

ok, just now looking at some examples of how to do this $x_4$ is just a row with all zeros

In the book: SET THEORY AND LOGIC By ROBERT S.STOLL in page 19 the following theorem ,No 5.2 in the book ,is given: If,for all A, AUB=A ,then B=0 IS that true or false If false give a counter example If true give a proof

Hello, I am studying change of basis in linear algebra and I have trouble figuring what my result should look like. From what I understand, I need to express the "coordinates" of matrix ##A## with respect to the basis given in ##S##, and I can easily see that ##A = -A_1 + A_2 - A_3 + 3A_4##...

##A \subset B## means that ##\exists x \in B ## such that ##x \not\in A##. Is this logically equivalent to ##\exists x \in B \land x \not\in A##? Formally, $$A \subset B \iff {(\exists x \in B \land x \not\in A)\land(\forall y \in A \land y\in B)}$$ I have tried consulting...

What is the difference between "Hexagon socket set screw with dog point DIN 915" and "Hexagon socket set screws with cup point DIN 916" about usage field?

Hey! 😊 I am looking the following: Prove that for $m,n\in \mathbb{N}$, $m>n$ it holds that $m-n\in \mathbb{N}$. Hint: Consider $A=\left \{n\in \mathbb{N}\mid \forall m\in \mathbb{N}, m>n:m-n \in \mathbb{N}\right \}$ and show that $A$ is inductive. I have done the following: To show that...

Hey! 😊 We have the matrices \begin{equation*}s:=\begin{pmatrix}1 & 0 \\ 0 & -1\end{pmatrix}, \ d:=\frac{1}{2}\begin{pmatrix}-1 & -\sqrt{3} \\ \sqrt{3} & -1\end{pmatrix}\end{equation*} and the points \begin{equation*}p:=\begin{pmatrix}2 \\ 0 \end{pmatrix}, \ q:=\begin{pmatrix}-1 \\ \sqrt{3}...