quotient Definition and 356 Threads

Homework Statement Let G be the group of real numbers under addition and let N be the subgroup of G consisting of all the integers. Prove that G/N is isomorphic to the group of all complex numbers of absolute value 1 under multiplication. Hint: consider the mapping f: R-->C given by...

Homework Statement X / 1+sinX The Attempt at a Solution Quotient rule (1+sinX)(1)-X(1+cosX) / (1+sinX)2 To: 1+sinX-X-XcosX / (1+sinX)2 But when I look at the answer in the back of the book, it's wrong.

Homework Statement Differentiate f(x) = 6-5x-x2 / x2-1 ------- A) -5-5x / x2-1 B) 5 / (x+1)2 C) -5-5x / (x2-1)2 D) -5 / (x+1)2 E) -5+5x / x2-1 F) None of the above The Attempt at a Solution (x2)(-5-2x)-(6-5x-x2)(2x) / (x2-1)2 Simplified to: 5x2-10x+5 / x4-2x2+1...

I am trying to show that if X is a topological space, ~ an equivalence relation on X and q:X-->X/~ the quotient map (i.e. q(x)=[x]), then the quotient topology on X/~ (U in X/~ open iff q^{-1}(U) open in X) is characterized by the following universal property: "If f:X-->Y is continuous and...

Homework Statement find the difference quotient and simply your answer. f(t)=1/t, [f(t)-f(1)]/t-1, t doesn't equal 1 Homework Equations the book says the answer is -1/t, t doesn't equal 1 The Attempt at a Solution (1/t-1)/t-1 (1/t-t/t)/t-1 (-1t/t)/t-1 -1/t-1...

I have a question... "Is the quotient set of a set S relative to a equivalence relation on S a subset of S?" I suppose "no",since the each member of the quotient set is a subset of S and consequently it is a subset of the power set of S,but I have e book saying that "yes",I am a bit...

Homework Statement a=g.[(M-m)/(M+m)] how do i go about finding the partial derivative wrt m? Homework Equations The Attempt at a Solution i started by rearranging it to the form a=g.(M-m)(M+m)^-1, i used the chain rule to find the derivative of (M+m)^-1 to be -(M+m)^-2 but...

Im reviewing material for the exam and came across this question: Let pi_1:RxR->R be the projection on the first coordinate. Let A be the subspace of RxR consisitng of all points (x,y) s.t either x>=0 or (inclusive or) y=0. let q:A->R be obtained by resticting pi_1. show that q is quotient...

Hello all I have read about quotient spaces of a vector space in several books and have an understanding of what they are. Looking up Quotient Vector Space in Wiki it says :- The quotient of a vector space V by a subspace N is a vector space obtained by "collapsing" N to zero. I...

For two blades: A \in {\bigwedge}^r B \in {\bigwedge}^{k-r} and a corresponding pseudovector for the wedge product of the two: I \in {\bigwedge}^k intuition tells me that the following scalar quotient: \frac{A \wedge B}{I} can be reduced to A \cdot...

I posted this earlier and thought I solved it using a certain definition, which now I think is wrong, so I'm posting this again: Show that the quotient spaces R^2, R^2/D^2, R^2/I, and R^2/A are homeomorphic where D^2 is the closed ball of radius 1, centered at the origin. I is the closed...

Show the following spaces are homeomorphic: \mathbb{R}^2, \mathbb{R}^2/I, \mathbb{R}^2/D^2. Note: D^2 is the closed ball of radius 1 centered at the origin. I is the closed interval [0,1] in \mathbb{R}. THEOREM: It is enough to find a surjective, continuous map f:X\rightarrow Y to show that...

Let G be a group and let N\trianglelefteq G , M\trianglelefteq G be such that N \le M. I would like to know if, in general, we can identify G/M with a subgroup of G/N. Of course the obvious way to proceed is to look for a homomorphism from G to G/N whose kernel is M, but I can't think of...

Homework Statement Let H and K be normal subgroups of a group G. Give an example showing that we may have H isomorphic to K while G/H is not isomorphic to G/K. Homework Equations The Attempt at a Solution I don't want to look in the back of my book just yet. Can someone give me a...

This is not directly a homework problem, so I opted not to place this question there. From what I have read/gathered from the internet/my textbook, a quotient mapping is any surjective, continuous mapping from a space X to a space comprised of the equivalence classes of all x in X from a...

Homework Statement For all integers m, m^{}2=5k, or m^{}2=5k+1, or m^{}2=5k+4 for some integer k. Relevant equations I'm pretty sure we have to use the Quotient Remainder THM, which is: Given any integer n and positive integer d, there exists unique integers q and r such that...

I just wanted to know if someone can explain to me the basic concept of a quotient space and quotient groups.

Homework Statement THe quotient map f is open but is it also closed? The Attempt at a Solution I think it is. Consider f: X->Y FOr every open set V in Y there exists by definition an open set f^-1(V) in X. There is a one to one correspondence between open sets in X and open sets in Y by...

The question is "find and simplify the difference quotient." Given function f(x)=sq root of x So what I did is insert (x+h) under the radical & got sq root of (x+h), then I subtracted the sq root of x (original function) My answer was sq root [(x+h) - sq root (x)] / h The...

Homework Statement In exercises 7-12, use the Quotient Rule to differentiate the function. 9) h(x) = \frac {\sqrt[3]{x}}{x^3+1} Homework Equations Quotient Rule The Attempt at a Solution I'm trying to figure out basic calculus over the summer in preparation for class...

Help needed to eventually understand Quotient spaces.

Just wondering how you take the second derivative when using the quotient rule. After using the quotient rule to get my first derivative, I tried again and the numerator ended up as 0.

what exactly is a quotient set? I know it "partitions" a large group of numbers into discrete subsets but I still don't know what exactly it is in practical terms. Like, does it relate somehow to Euler's phi function?

im having a lot of trouble using the chain rule product rule and quotient rule..i can do them fine seperatly but when they're put together i can't get them like if you have (x^2-1)^4 (2-3x) i would start with 4(x^2-1)^3(2x)(2-3x)+(x^2-1)^4(-3) have i done something wrong here because i never...

Quotient ring is also know as factor ring but what has it got to do with 'division' in any remote sense whatsoever? I know it is not meant to be division per se but why give the name of this ring the quotient ring or factor ring? What is the motivation behind it? R/I={r in R| r+I} Normally...

suppose q:M -> M/R is a quotient map. i've asked my dad what is the quotient map from MxM to (M/R)x(M/R)? he told me it is qxq: MxM -> (M/R)x(M/R) defined by (qxq)(x,y) = (q(x), q(y)), but there are some conditions to be met, but he could not remember what those conditions are. i...

As one can see, the definition of quotient space, group, ring, field, vector space are very similar. It is similarly defined as an algebraic structure with a ~ on it. I am really having trouble vistualise what a quotient space and group are. My professor told me that we can work more easily with...

I'm having a bit of trouble seeing Vector Quotient Spaces. Lets say I have a vector space $V$ and I want to quotient out by a linear subspace $N$. Then $V/N$ is the set of all equivalence classes $[N + v]$ where $v \in V$. For example, let me try to take $\mathbb{R}^{2} /$ x-axis. This...

Hi, I'm struggling to understand how to find the derivative of something like this... [(2x - 1)^2] / [(x - 2)^3] The answer in my book says it is supposed to be [-(2x - 1)(2x + 5)] / [(x - 2)^4] How do I use the chain rule with the quotient rule at the same time? I even multiplied...

I am attempting to find the second derivative of a function: h(x) = [(x^2)-1] / [2x-(x^2)] I proceeded by using the Quotient Rule, and I found the following as the first derivative. (It is correct.) h`(x) = [2(x^2)-2x+2] / [2x-(x^2)]^2 Next, I tried using the Quotient Rule again, and...

Is there a simple method for finding all the units in a polynomial quotient ring over a finite field? For example: {F_2[x] \over x^7-1} I can see the easy ones like 1, and all power of x, but I wanted a general rule or method for finding all of them if it exists (besides testing each...

My test here asks me to: "Use log5 2 =0.4307 and log5 3=0.6826 to approximate the value of log5 12." According to my textbook I would solve this by subtracting (using the quotient property): 0.6826-0.4307. That = 0.2519. But that number isn't right! log5 12=1.544 (about) Which I found...

I can't understand it. No matter how much I try, Can anyone explain it step by step, and give some examples. How can it be applied to contruct different shapes?

Let X,Y be two spaces, A a closed subset of X, f:A--->Y a continuous map. We denote by X\cup_fY the quotient space of the disjoint union X\oplus{Y} by the equivalence relation ~ generated by a ~ f(a) for all a in A. This space is called teh attachment of X with Y along A via f. i) If A is a...

Say for example, to differentiate x/(x²+1) I would use to quotient rule. However, would it be legal to bring up the denominator to: (x)(x²+1)-¹ and use the product/chain rule instead?

First of all, I'm not sure if this is the right forum, but none of the forums mention topology in their description. But anyway, I'm taking a topology class, and the professor mentioned that the projective plane is obtained by identifying antipodal points on the sphere, ie, points diametrically...

Use the Rayleight quotient to find a good approximation for the principal eigenvalue of the Sturm-Liouville problem. u'' + (\lambda - x^2)u = 0 0 < x < 0 u(0) = u'(1) = 0 Any help?

I need to take the limit of this quotient as n goes to infinity: [2(-1)^(n+1) - 3^(n+1)] ----------------------- [2(-1)^(n) - 3^(n)] It seems to go to infinity over infinity in its current form, which, if I recall correctly, is indeterminate. It seems then that I need to simplify...

sec 0 = 1/cos 0 Write the quotient property expressing tan 0 as a quotient of two other trigonometric functions...is there someone that please lead me in the right direction? Bryce

I'm embarassed because I'm surely missing something obvious... the very first exercise in Categories for the Working Mathematician is: Show how each of the following constructions can be regarded as a functor: The field of quotients of an integral domain; the Lie algebra of a Lie group...

how to define R\Q?(under addition) R\Q={a+Q:? <a<?} a€R but if it is not bounded then it will repeat please help me n

Let f(x) and g(x) be functions. Then if limit of f(x)/g(x) = 1. That implies lim f(x) = lim g(x) right? Consider this proof. lim f(x)/g(x) = 1 lim f(x) x lim 1/g(x) = 1 lim f(x) = 1 / (lim 1/g(x)) lim f(x) = lim g(x).

How does the difference quotient undo what the Riemann sum does or vice versa. In terms of the two formulas? I would assume that working a difference quotient backwards would be similar to working a Riemann sum forward, but in reality as the operations go this couldn't be further from the...

i am being asked to express 1.262626... as a quotient of two whole numbers. would the answer to this be a simple 1.262626.../1000000? I know I am doing something wrong, but I am not sure what.

I am trying to match a result in one of my textbooks. To assist with one of their arguments they are approximating a 2nd order PDE by using a difference quotient and they show the approximation as follows: (d^2u[x,t])/(dx^2) =~ (1/h^2)(u[x+h,t]-2u[x,t]+u[x-h,t]) When I actually use...

Hello all, first time to the site and its very helpful! I wish I would have found it sooner. I am stuck on quotient rings. Here is my question.. How do I find elements of a quotient ring? It asks me to list all elements of a quotient ring. Anybody have any ideas how i can find them...

Can someone explain the concept of Quotient topology. I tried to read it from a book on topology by author "James Munkres" . It was okay but I did not get a feel of what he was trying to do.. He talks about cutting and pasting elements. I kind of got lost in that.. If someone could give me a...

The following problem appears in my textbook (before it discusses the quotient or product rule, so those rules cannot be used for the answer): Find the derivative of the function: \frac{x^3-3x^2+4}{x^2} I brought the denominator to the top and multiplied it out to get {x-3+4x^-2}[/ltex]...

Hi I just wanted to know what a qoutient space is . Is there a physical picture to it? How can one imagine what an equivalence class,equivalence relation is?

http://www.guardian.co.uk/life/news/page/0,12983,937443,00.html I wonder if people here tend to have lower EQs and higher SQs than that of the general population. High SQs are almost essential for science and engineering and ones with high EQs tend to gravitate towards other fields. High SQs...