Set Definition and 1000 Threads

This is a concise question, so the title pretty much says it all. Also, this is not a HW question, but the idea has subtly popped up in two homework problems that I have done in the past. I cannot justify why the entire set would be bounded, because we know nothing of the nature of the...

Homework Statement -∏<arg(z)<∏ (z≠0) Homework Equations arg(z) is the angle from y=0The Attempt at a Solution Arg(z) spans the entire graph since -pi to pi is the full 360 degrees so I put: -∏<arg(z)<∏ --> 0<arg(z)<2∏+k∏, (k ε Z) --> arg(z) \subset R --> arg(z) = R: all real numbers but I...

Find the value(s) of k such that f(x) is continuous everywhere: x^2-7 if x <= 2 and 4x^3-3kx+2 if x>2 Can you set the two equations equal to each other if only one of them has k in it?

hello How to you rigorously express the orthonormality of a complete set of eigenvectors (|q\rangle)_q of the position operator given that these are necessarily generalized eigenvectors (elements of the distribution space of a rigged hilbert space)? The usual unformal condition \langle...

Hey there! I will start of with saying I´m not very good at English when considering mathematical terms, neither an expert in Math. My question goes as this: I have a set of 1000 questions - which will be given in rounds with a set of 10. So every round, you get 10 questions out of...

Homework Statement I am not sure if set theory is precalc or not but here is my question. Find a pair set such that {a} belongs to the set and {a} is not a subset of S. The Attempt at a Solution So I thought that a set like this would work S = {{a}, b} because {a} belongs to the set...

Homework Statement Build the lagrangian of a set of N electric dipoles of mass m, length l and charge q. Find the equations of motion. Find the corresponding difference equations. Homework Equations Lagrange function L=T-V Lagrange's equations \frac{d}{dt}\left(\frac{\partial L}{\partial...

I'm at a bit of a loss how to do this. I suspect it's the set \left\{t_1^3, t_1t_2, t_2^2) \mid t \in {\mathbb C}\right\} . Certainly the polynomial x^2z^3 - y^6 vanishes on these points, but I'm not sure how to show the other inclusion. The only thing I can think of gets me close, but not...

Okay, so I am having a problem, I'm not the greatest at math, here is the problem: I need a set of reduction gears, my plan is to have a gear attached to a shaft (base diameter of this first gear is 33mm) with 50 teeth (maybe?) To turn a reduction gear that will be on another shaft that will...

In set theory a set is defined to be a collection of distinct objects (see http://en.wikipedia.org/wiki/Set_%28mathematics%29), i.e. we must have some way of distinguishing anyone element from a set, from any other element. Now a topological space is defined as a set X together with a...

The circle seems to be of great importance in topology where it forms the basis for many other surfaces (the cylinder ##\mathbb{R}\times S^1##, torus ##S^1 \times S^1## etc.). But how does one define the circle in point set topology? Is it any set homeomorphic to the set ##\left\{(x,y) \in...

If every Dedekind cut is at a rational it seems that these cuts would only produce a countable set and would not produce the whole real line. So how should I think about it.

I've seen open sets ##S## of a bigger set ##X## being defined as 1) for every ##x\in S## one can find an open disk ##D(x,\epsilon)## centered at ##x## of radius ##\epsilon## such that ##D## is entirely contained in ##S##. Where $$D(x,\epsilon)= \left\{y \in X: d(x,y) < \epsilon\right\}$$...

Homework Statement . I am trying to solve two exercises about complex sequences: 1) Let ##\alpha \in \mathbb C##, ##|\alpha|<1##. Which is the limit ##\lim_{n \to \infty} \alpha^n##?, do the same for the case ##|\alpha|>1##. 2) Let ##\mathcal M## be the set of the complex numbers ##c## such...

The set of the real numbers is closed. For me this is nearly trivial (*) but perhaps I miss something; a colleagues insists that there are some deeper considerations why this is far from trivial - but I don't get his point (*) A) A set is closed if its complement is open; the complement...

Homework Statement 0<\left|x+3\right|<1/4 Homework Equations The Attempt at a Solution (-13/4)<x<(-11/4) and x\neq-3 Thanks in advance. This is my first post and I am unfamiliar with formatting this kind of stuff so I will work on getting better at that aspect.

Hey! :o I am looking at an exercise and I got stuck... $n\epsilon \mathbb{N},n>1$ $φ(n)=|\{1 \leq k \leq n :$ the greatest common divisor of $k$ and $n$ is $1\}|$ I am asked to find $φ(n)$,but I don't know how...

Hi everyone, I am working on a problem in Operations Research but I need to prove a property related to compactness of a set. Although I expect it is quite elementary, I have never studied Analysis at an advanced level so am not sure how to do it. I have an optimisation problem in which a...

I need some help finding an appropriate statistics model for some experimental data. Thanks in advance for any suggestions. I am trying to compare simulated results from a code that models nuclide concentrations in spent nuclear fuel to experimental data. These concentrations have...

Homework Statement Show that if a set has 3 elements, then we can find 8 relations on A that all have the same symmetric closure. Homework Equations Symmetric closure ##R^* = R \cup R^{-1} ## The Attempt at a Solution If the symmetric closures of n relations are the same then...

Homework Statement Hey guys! I am new to this forum but saw the helpful posts on set theory proofs and wondered if I could finally get some help with this problem: Suppose A, B, C, and D are sets with A⊆C and B⊆D. If A∩B=Ø then C∩D=Ø. This is a biconditional so I have to prove it...

Hey guys! I am new to this forum but saw the helpful posts on set theory proofs and wondered if I could finally get some help with this problem: Suppose A, B, C, and D are sets with A⊆C and B⊆D. If A∩B=Ø then C∩D=Ø. This is a biconditional so I have to prove it both ways correct...

Homework Statement Use polar coordinates to set up and evaluate the double integral f(x,y) = e-(x2+y2)/2 R: x2+y2≤25, x≥0 The Attempt at a Solution First I just want to make sure I'm understanding this my double integral would be ∫^{\pi/2}_{-\pi/2} because x≥0 ∫^{5}_{0}...

Homework Statement For a reaction, A + H2O --> B + C We're given that d[A]/dt = k[A]n[H3O+]m And also a table of [A] vs time at T1 and pH 1, pH 2; as well as [A] vs time at T2 and the same pH 1 and 2. From this data, we're to find pseudo-n-order rate constants, and then n itself. Next...

Am a bit confused about the meaning of cardinality. If ## A \subseteq B ##, then is it necessarily the case that ## |A| \leq |B| ##? I am thinking that since ## A \subseteq B ##, an injection from A to B exists, hence its cardinality cannot be greater than that of B? But this cannot be...

Homework Statement . Let ##X## be a complete metric space and consider ##C(X)## the space of continuous functions from ##X## to ##\mathbb R## with the metric ##d_{\infty}##. Suppose that for every ##x \in X##, the set ##\{f(x): f \in C(X)\}## is bounded in ##\mathbb R##. Prove that there exist...

##GL(n,\mathbb{R})## is set of invertible matrices with real entries. We know that SO(n,\mathbb{R}) \subset O(n,\mathbb{R}) \subset GL(n,\mathbb{R}) is there any specific subgroups of ##GL(n,\mathbb{R})## that is highly important.

**Let A and B be two subsets of some universal set. Prove that if $(A\cup B)^c$ = $A^c$ U $B^c$, then A = B.**Attempt: Let $x\in A$. Then $x\in A\cup B$, so $x\notin(A\cup B)^c$. By hypothesis $(A\cup B)^c=A^c\cup B^c$, so $x\notin A^c\cup B^c$. In particular, then, $x\notin B^c$, and therefore...

Hi all, first math post here. I was just wondering- after having read from quite a few textbooks that intuitively, a set is a collection of objects- if there's a rigorous definition of the concept of set. It's just out of curiosity- I mean, is a rigorous definition even necessary? I guess I'm...

Let A be is a set of some p-dimensional points x \in \mathbb{R}^p. Let d_x^A denote the mean Euclidean distance from the point x to its k nearest points in A (others than x). Let C \subset A be a subset of points chosen randomly from A. We have \Phi(A) = \sum_{x \in A} d_x^C. Now suppose that...

I am currently reading Rautenberg's book on mathematical logic, in it he defines a propositional language ##\mathcal{F}##, set theoretically, as the smallest (i.e. the intersection) of all sets of strings ##S## built from propositional variables (##\ p_1,p_2,\ldots##) as well as any binary...

Homework Statement Prove that(power set) P(E) U P(F) is a subset of P(E U F) Homework Equations P(E) U P(F) is a subset of P(E U F) The Attempt at a Solution P(E)U P(F)={x:xεP(E) or xεP(F)} but P(E)={X:X is a subset of E} or P(E)={x:xεX→xεE} so we get P(E)U P(F)={x:xεX→xεE or...

I was thinking of finding a new hobby, and thought that playing around with the double slit experiment might be interesting. I was wondering how feasible it would be to set up the double slit experiment, not simply the laser and slit version shown here...

Hello, I am trying to evaluate the following condition: Let x_1 ≥ x_2 ≥ ... ≥ x_L and x_L ≥ y, where y is a fixed deterministic value. What is the conditional PDF of x_j (1 ≤ j ≤ L), given that x_L ≥ y ? (recall that i < L)Note that the conditional PDF of x_j, for the case when x_1 ≥ x_2 ≥...

Homework Statement In a group of 30 people each person twice read a book from books A, B, C. 23 people read book A, 12 read book B and 23 read book C. (a) How many people read books A and B? (b) How many people read books A and C? (c) How many people read books B and C? Homework...

Homework Statement A set of functions, F, is given below. Determine the size of the largest subset of F which is mutually orthogonal on the interval [-1, 1], and find all such subsets of this size. Show all of your work. F = { 1, x, x2 , sin(x), cos(x), cosh(x), sinh(x)}Homework Equations Not...

Homework Statement . Let ##X## be a compact metric space. Prove that if ##\mathcal F \subset X## is a family of equicontinuous functions ##f:X \to Y \implies \mathcal F## is uniformly equicontinuous. The attempt at a solution. What I want to prove is that given ##\epsilon>0## there...

The topology ## T ## on a set ## X ## generated by a basis ## B ## is defined as: T=\{U\subset X:\forall\ x\in U\ there\ is\ a\ \beta\in B:x\in \beta \ and\ \beta\subset U \}. But if ##U## is the empty set, and there has to be a ## \beta ## in ##B## that is contained in ##U##, the empty set...

Prove that a set $A\subset\mathbb{R}^n$ is (Lebesgue) measurable $\iff$ there exist a set $B$ which is an $F_{\sigma}$ and a set $C$ which is a $G_{\delta}$ such that $B\subset A\subset C$ and $C$~$B$ (C without B) is a null set. $F_{\sigma}$ is a countable union of closed sets, and...

Prove that the interval A = [0 , 2) has the same cardinality as the set B = [5 , 6) U [7 , 8) by constructing a bijection between the two sets Attempt: x ↦ x + 5 for x ∈ [0 ; 1) x ↦ x + 6 for x ∈ [1 ; 2) What to do next?

Algebraically, how is this done? I can do it no problem if there is no fraction, but have problems when there is. (1/n) = (n/100)

Hi, Here I have a question, apparently easy, but that I think it is a bit tricky. Homework Statement Indicate how can a hydrogen atom be prepared in the pure states corresponding to the state vectors ψ2p-1 and ψ2px and ψ2s. It is assumed that spin-related observables are not...

1. Prove that if A is symmetric and B is skew-symmetric, then {A,B} is a linearly independent set. I am going to need some help to solve this. Not sure how to begin. Homework Equations The Attempt at a Solution

How do you show that a set of linear transformations from one vector space to another spans L(V,W)? This isn't a homework question, or even a question that's in the text I'm reading (Friedberg).

I have been looking at the idea of 1:1 correspondence as a method of determining set size/cardinality, and have noticed that the principle allows for inductive proofs, which I think are properly constructed, that can come to conclusions which are clearly wrong under traditional set theory if...

How to show that a set ##C=\{(x,y,z)\in\mathbb{R}^3:x\geq0,z\geq0,xz\geq y^2\}## is convex? I tried a proof by contradiction: Assume that there exist ##c_1=(x_1,y_1,z_1),c_2=(x_2,y_2,z_2)\in C## and ##t\in(0,1)## such that ##tc_1+(1-t)c_2\notin C##. For this to hold, one would have to have...

Homework Statement Explain why ##f(x,y,z) = x + y - z## must attain both a maximum and a minimum on the sphere ##x^2 + y^2 + z^2 = 81)##. Homework Equations None The Attempt at a Solution I know that any continuous function attains both a maximum and a minimum on a compact set. I defined...

Hi Please see table below GDP growth rates of a group 5 countries. I am trying to derive some conclusions from this small sample. 1. can I conclude that countries in the group have very similar growth rates and there is no significant difference between their growth rates? 2. As...

Homework Statement Give f:A→A and g:A→A where f is surjective, g is injective, but f*g is neither surjective nor injecive The Attempt at a Solution I don't know why I can't really think of two... I assume it's easiest to do one in ℝ, but when it comes to producing...

how do I go about doing 3(a) and 3(b)? I'm guessing that for 3(a), it is true, since we have for LHS: P((A or B) and C) we can consider the case P(A and C) by excluding B, and this is a subset of the RHS when we also exclude B: (P(A) and P(C)). We can consider excluding B because...