Set Definition and 1000 Threads

Hello, I was going through the following paper: http://www.emis.de/journals/HOA/AAA/Volume2011/142128.pdf In page 6, immediately after equation (3.15), its written that "functions of the form v(t) are dense in L^2". I have been looking for proofs online which verifies the above statement but...

Homework Statement Let X be a set. Then the set {Y:Y is a subset of X} prove this is a set.Where do i start? Really unsure, i know that i have to use the power set? I have written down; {0,1}^X

Hello! (Nerd) I am looking at the proof of the following sentence: For any natural numbers $m,n$ it holds that: $$n \in m \rightarrow n \subset m$$ Proof: We define the set $X=\{ n \in \omega: \forall m (m \in n \rightarrow m \subset n)$ and it suffices to show that $X$ is an inductive set...

Hi! (Smile) We want to show that the elements of the natural numbers are natural numbers, i.e. $(n \in \omega \wedge x \in n) \rightarrow x \in \omega$ Could you explain me why, in order to show this, it suffices to show that $X=\{ n \in \omega: (\forall y \in n)(y \in \omega)\}$ is an...

Hi! (Nerd) Sentence: There is a unique inductive set that is contained in each inductive set. Proof: Let $A$ be an inductive set (we know that there is such a set from the axiom of infinity) and we define: $$B=\{ X \subset A: X \text{ is an inductive set}\}$$ ($B$ is a set, because if $X...

Hello! (Wave) A set $A$ is called inductive set, if $\varnothing$ is an element of $A$ and for each $x \in A$ its next element, $x'=x \cup \{ x \}$ belongs to $A$. I want to show that if $B$ is a nonempty set of inductive sets, then $\bigcap B$ is an inductive set.That's what I have tried:$B$...

I would like to find the distance that a vehicle travels using a Kalman filter. The vehicle is a car that travels the road between two positions. The vehicle has a GPS/barometer/accelerometer device that collects position data, which I converted from a longitude and latitude to a North, East...

Homework Statement Let ##R## be an ordered relation on a set ##A## that is reflexive and anti-symmetric. If there is a chain of elements in ##R## that begins and ends with the same element, say the element ##x \in A## is it true that all the elements of ##R## sandwiched in between the ones...

Could the mass of a singularity be described or defined as an empty set, or else what is the term to describe it (in at least one sentence).

What I am asking is not a math "problem" but something about presenting the problem in math language. Assume $0<a_{i}<1, i=1,2,3,...,N$ and $a_{i} \neq a_{j}$, I want to have a set $A$ which contains the $M$ indices with the top $M$ values of $a_{i}$. For example, we have $a_{1}=0.2...

Let R be the class of all ordinals. If a subset C of R is unbounded (i.e. for any ordinal \alpha \in R, there is \beta in C with \beta greater than \alpha ), then it seems to me that C cannot be a set, only a class. Is this true, and if so, how does one prove it? My reading on the general...

Hello, I am an electrical engineering student and I was hoping some body here could help me out with a cantilever question. I want to model a vibrating cantilever with a mass at the end. I am doing this for a project where I wanted to model a Wurlitzer 200 Electric Piano. The way they produce...

A mass on the end of a spring which is hanging vertically is raised up and let go. It then oscillates between 2m and 1.5m above the floor and completes 32 cycles in one minute. The height, h metres, of the mass above the floor after t seconds can be modeled by the function h=acos(pi t / 180)...

I am trying to solve the system of simultaneous equations: \frac{\partial \rho_f\phi}{\partial t}+\frac{\partial}{\partial z}(\rho_f\phi v_f)= \frac{\partial F}{\partial t} \frac{\partial \rho_s(1-\phi)}{\partial t}+\frac{\partial}{\partial z}(\rho_s(1-\phi) v_s)=-\frac{\partial F}{\partial t}...

I am reading Paul E. Bland's book, "Rings and Their Modules". I am trying to understand Section 3.1 on Categories. At present I am working on Problem 2 in Problem Set 3.1 and I need some help in understanding the problem and its solution. Problem 2 (Problem Set 3.1) reads as...

I am reading Paul E. Bland's book, "Rings and Their Modules". I am trying to understand Section 3.1 on Categories. At present I am working on Problem 1 in Problem Set 3.1 and I need some help in understanding the problem and its solution. Problem 1 (Problem Set 3.1) reads as follows...

Hello! (Wave)I want to show that the algebraic set of $K^n$, $V$, is irreducible iff $I(V)$ is a prime ideal. That's what I have tried so far: We know that the algebraic set $V$ is irreducible iff $V$ cannot be written as $V=V_1 \cup V_2$, where $V_1, V_2$ are algebraic sets of $K^n$ and $V_1...

Prove that for all x,y\in\omega,\ \ x\subset y\vee y\subset x. If I assume that the conclusion is false then I can prove that for some a\in x,\ b\in y we have a\notin b and b\notin a. Also I am thinking that if assume the contrary then \omega minus \{x\} or minus \{y\} or both is a smaller...

Hey guys, I was reading Kenneth's Discrete Mathematics and I came across this definition in the function chapter: Let f be a function from A to B and let S be a subset of A.The image of S under the function f is the subset of B that consists of the images of the elements of S.We denote...

Let x be a natural number (set). How to prove that there is no bijection between x and x^+, where x^+=x\cup\{x\}? Then I can show that \mathrm{card}\,x<\mathrm{card}\,x^+. I know that x\notin x.

Hi! (Smile) Let $B$ be a nonempty set. Does it stand that $\bigcap \mathcal{P}B=\mathcal{P} \bigcap B$? Is the set $B \times B$ always a function? If not, what condition should $B$ satisfy, so that the relation $B \times B$ is a function? Let $x \in \bigcap \mathcal{P}B$. Then $\forall b \in...

Hey! (Wave) If $x \neq y $, define the sets $\bigcup \langle x,y \rangle , \bigcup \bigcup \langle x,y \rangle$. According to my notes, it is like that: $$\langle x,y \rangle= \{ \{x\}, \{x,y\} \} $$ $$ \bigcup \langle x,y \rangle=\{x,y\} $$ $$ \bigcup \bigcup \langle x,y \rangle=x \cup y$$...

Hello! (Nerd) I am looking at the following exercise: Prove that for all sets $A,B$, there are the sets: $$\{ A \cap x: x \in B \} \text{ and } \{ A \cup x: x \in B \}$$ Show that: $A \cap \bigcup B=\bigcup \{ A \cap x: x \in B\} $ for $B \neq \varnothing$, $A \cap \bigcup B=\bigcap \{ A...

Hello! (Wave) Let $R$ be a relation and $A$ a set. The restriction of $R$ to $A$ is the set: $$R\restriction A=\{ <x,y>: x \in A \wedge <x,y> \in R\}=\{ <x,y>: x \in A \wedge xRy\}$$ For a relation $R$ and a set $A$, it stands that: $$dom(R \restriction A)=dom(R) \cap A$$ Could you give...

ℝ or ℝ2? Both are infinite, but is one greater then the other?

Homework Statement Hi there! I have a data set of r (independent variable) and E (electric field strength) (dependent variable). The question asks for a non graphical method to show if there is an inverse square law relationship between the two data sets. -- My attempt: I picked the equation...

Hi guys, I stumbled upon this lovely quote from the philosopher of science Wesley Salmon: "The fool hath said in his heart that there is no null set. But if that were so, then the set of all such sets would be empty, and hence, it would be the null set. Q.E.D." (in Martin Gardner, Mathematical...

Homework Statement Prove if an ordered set A has the least upper bound property, then it has the greatest lower bound property. Homework Equations Definition of the least upper bound property and greatest lower bound property, set theory. The Attempt at a Solution Ok, I think that my main...

Homework Statement Let ##G## be a group. I need to find a set ##X## and an injective function from ##G## into ##Sym(X)## Homework EquationsThe Attempt at a Solution I am having difficulty with this problem, and I want to make sure I understand exactly what it is asking. If I understand the...

If you take the Fourier series of a function $f(x)$ where $0 < x < \pi$, then would $a_{0}$, $a_{n}$, and $b_{n}$ be defined as, $a_{0} = \displaystyle\frac{1}{\pi}\int_{0}^{\pi}f(x)dx$ $a_{n} = \displaystyle\frac{2}{\pi}\int_{0}^{\pi}f(x)\cos(nx)dx$ $b_{n} =...

Homework Statement Determine whether the set S spans R3. If the set does not span R3, then give a geometric description of the subspace that it does span S = [ (2,0,3) , (2,0,-1) , (6,0,5) , (4,0,6) ] Homework EquationsThe Attempt at a Solution I know S does not span R3 because the system of...

Hello everyone, Here is the definition of path connected and domain my textbook provides: Definition 171. An open set S is path connected if each pair of points in S can be connected by a polygonal line (e.g. a finite number of line segments connected end to end). A domain is an open set that...

Could anyone recommend some good books on set theory?

Mod note: Moved from a technical section, so missing the homework template. Here is what I'm trying to prove. Let f:A->B. If there are two functions g:B->A and h:B->A such that g(f(a))=a for every a in A and f(h(b))=b for every b in B, then f is bijective and g=h=f^(-1). I think I have most...

Hello, I have the following problem. I have a system of differential equations, with two parameters that satisfy certain condition. 0 < 1.5(1-a) < b < 1. So when I fix the value of a I can find values of b satisfying this and its associated equilibrium point. When I calculate (with...

Image is a set of 1D potentials which i need more examples and their solutions containing transmitting states, bounded states, scattering states and coefficients. I searched with "1D potential combinations" "1D potential set" keywords but can not find anything yet. Which keyword should i...

My question is on how to answer if two statements are equal in set theory. Like De'Morgans laws for example. I'm currently reading James Munkres' book "Topology" and am working through the set theory chapters now, and this isn't the first time I've seen the material, but every time I see this...

Homework Statement Determine if the given set of functions is linearly independent or linearly dependent.Homework Equations $$S=x~sin~x, ~ x~cos~x$$The Attempt at a Solution My first instinct was to use the Wronskian. $$W[y_1(x), y_2(x)]=\begin{vmatrix} x~sin~x & x~cos~x\\ x~cos~x+sin~x &...

Let B be a non-zero mx1 matrix, and let A be an mxn matrix. Show that the set of solutions to the system AX=B is not a vector space. I am thinking that I need to show that the solution is not consistent. In order to do so would I need to show that B is not in the column space of A?

Homework Statement Let S = { (x,y) in Z | 5x+7y is divisible by 3 } Show that S is symmetrical. Homework Equations None apart from basic algebraic knowledge. The Attempt at a Solution [/B] The only thing I can think of is starting with 3a = 5x+7y and putting x (or y) into the...

Homework Statement Damped driven oscillator: ruler example. Suppose the ruler used in the classroom demonstration has a length of 12 and 13/16ths inches, a width of 1 ½ inches, is 1/16th inch thick with a density of 1.2 g/cm3. It swings from a pivot point ¼ of an inch from the top end. a) Find...

Hi guys, I've got a philosophical question for you concerning the empty set. I know that in axiomatized set theory the empty set is disjoint with itself. Because it has no members, the empty set cannot have any members in common with itself. This is common sense in set theory. But if we say...

Hi everyone. It's been years since I've done any stats, so I need a bit of help, please. I want to include it in a blog post I'm going to do (not here on PF), so I don't want to give away too many details :p I apologise for my terrible understanding of stats, please be patient! Anyway, over...

Homework Statement This is from Baby Rudin Exercise 2.20- Are closures and interiors of connected sets always connected? (Look at subsets of \mathbb{R}^2 ). Homework Equations The interior is the set of all interior points for a set E that is a subset of a metric space X. A subset Y of a...

Hello! Why does the moon has different rise and set timing for different phases? And also, why does it rise from different directions during different phases? Help me out please!

Find the solution set for x^2/(x+3) < 9/(x+3) So I moved the term 9/(x+3) over to the left side and cross-multiplied the two fractions. Then, I simplified to get x^2-9 (because the x+3 cancel out across the fraction bar). I got x^2-9, which factors to (x+3)(x-3). Then, I created an interval...

Hey! (Wave) Knowing that $A,B$ are sets, and: $$\text{ If } a \in A, \text{ then } \{ a \} \subset A \rightarrow \{ a \} \in \mathcal P A \rightarrow \{ a \} \in \mathcal P (A \cup B)$$ from this: $\{ a \} \in \mathcal P (A \cup B)$, can we conclude that: $$\{ a \} \subset \mathcal P (A \cup...

Hi! (Smile) I want to prove that for each set $A$: $$A \subset \mathcal P \cup A$$ According to my notes, we prove it like that: Let $x \in A$. We want to show that $x \in \mathcal P \cup A$, so, that: $\exists y \in \mathcal P \cup A$, such that $x=y$. It suffices to show that if $z \in x$...

There is a Theorem that states that if G = (V, E) is a graph, then S is an independent set $\Leftarrow\Rightarrow$ V - S is a vertex cover. Suppose the vertices have positive integer weights. Does it follow from the theorem that: S is an independent set with maximum weight...

Homework Statement Suppose that R is a partial order on A, B1 ⊆ A, B2 ⊆ A, x1 is the least upper bound of B1, and x2 is the least upper bound of B2. Prove that if B1 ⊆ B2 then x1Rx2. Homework EquationsThe Attempt at a Solution I split the proof into two different cases: case 1: x_1 is an...