quotient Definition and 341 Threads

I'm just a high school senior who noticed that the derivative has a general formula that we all know is, \frac{f(x+h)-f(x)}{h} but that there is no general formula (at least I haven't heard of it yet) for the integral of a function. I know I cannot simply just take the inverse of the difference...

Homework Statement As in title. Homework Equations Described in my attempt. The Attempt at a Solution Where do I go from here? I need to show that those 2 unioned sets are open in A. I'm not seeing it

I just had a discussion with someone who said he thought about quotient rings of polynomials as simply adjoining an element that is a root of the polynomial defining the ideal. For example, consider a field, F, and a polynomial, x-a, in F[x]. If we let (x-a) denote the ideal generated by x-a...

Homework Statement using the quotient rule, find the derivative of y = (2x-3)/(√(x^2-5)). Do not leave the answer in complex form.Homework Equations the quotient rule g(x)f'(x) - f(x)g'(x) --------------------- (g(x))^2 The Attempt at a...

Homework Statement $$\int \frac{x-1}{(x+1)(x^2+1)} dx$$ Homework Equations N/A The Attempt at a Solution I thought that I would use partial fractions, so: $$\frac{x-1}{(x+1)(x^2+1)} = \frac{A}{x+1} + \frac{B}{x^2+1}$$ $$x-1 = A(x^2+1) + B(x+1)$$ ##x=-1 \Rightarrow (-1)-1 = A((-1)^2+1) +...

How do you integrate quotients? Let's practice with one like... \int\frac{3x + x^{2}}{5x - 1} dx Thanks for the help!

Homework Statement I have attached the problem below. Homework Equations The Attempt at a Solution I have tried to use the natural epimorphism from G x G x ... x G to (G x G x ... x G)/(K1 x K2 x ... x Kn), but I do not believe that this is an injective function. Then I tried...

In abstract algebra (ring theory specifically), when we are dealing with factorization, UFD's, and so on, we are often only interested in the multiplicative structure of the ring, not the additive structure. So here is the basic situation we face: we a start with an integral domain (R,+,*)...

Homework Statement Original function f(x) = -x^2+5x-2 I have calculated the difference quotient as -2x-h+5 Then use difference quotient to calculate the average rate of change: for the following x=2 and x=2+h I need this to go on to the next part of the question and i wanted to ensure...

Suppose S is an Ideal of a Ring R I want to verify the multiplication operation in the Factor Ring R/S which is (S + a)(S + b) = (S + ab) for this i Need to show that : (S + ab) ≤ (S +a)(S + b) please give me some idea about it IT IS GIVEN AS DEFINITION IN THE BOOK I.N Herstein

Does anyone have any good reference to exercises concerning these topics? I would like to understand them better. Thank you.

I have the answer to this problem but I am stumped as how to get there. Here it is h(x)=e^x/5/sqrt2x^2-10x+17, I'm getting stuck moving the square root up. Help

Simple difference quotient question. Function --> simplify answer Homework Statement I just don't understand this question! Here, I will take a picture of it. Can someone please explain how I answer these? Or even show me a website that shows how to do these? Its not in my review book...

This is a basic question but I don't think I've ever seen anything like it before. If Q(x) \ = \ b_0(x \ - \ a_1)(x \ - \ a_2)\cdot \ . \ . \ . \ \cdot (x \ - \ a_s) then \frac{P(x)}{Q(x)} \ = \ R(x) \ + \ \sum_{i=1}^s \frac{P(a_i)}{(x \ - \ a_i)Q'(a_i)}I just don't understand where the P(aᵢ)...

1. Differentiate 2. y = \frac{v^{3} - 2v\sqrt{v}}{v} 3. I am trying to use the quotient rule, but am having trouble understanding how to use the square root. Here's what I have \frac{v(3v^{2}-2*1/2\sqrt{v}) - v^{3}+2v*1/2\sqrt{v}}{v^{2}}

Homework Statement The quotient test can be used to determine whether a series is converging or not. The full description is in the attachment. Homework Equations The Attempt at a Solution ( i ) Why must they both follow the same behaviour? Even if p ≠ 0, it says nothing about...

Hello friends, I would ask if anyone knows the métrqiue on the quotient space? ie, if one has a metric on a vector space, how can we calculate the metric on the quotient space of E?

Suppose that the number ##\pi## is divided by 2. The resulting quotient is divided by 2 again. This process is continued till the current quotient is less than or equal to 0.01. What is the largest quotient that is greater than 0.01? Here is my attempt: r = 1; quo = 1; while (quo > 0.01)...

Prove or disprove that f is a quotient mapping. f:R^3\{(x1,x2,x3):x1=0}--->R^2 defined by (x1,x2,x3)|->(x2/x1,x3/x1)

Hi all, So the equivalence class X/\sim is the set of all equivalences classes [x]. I was wondering if there was a way of writing it in terms of the usual quotient operation: G/N=\{gN\ |\ g\in G\}? From what I've read, it would be something like X/\sim = X/[e]. But, since I'm looking at the de...

Homework Statement Simplify. \sqrt[3]{\frac{5}{4}} The answer according to the textbook is: \frac{\sqrt [3]{10}}{2} Homework Equations -- The Attempt at a Solution Separated numerator and deonominator into individual cube roots and multiplied both by \sqrt[3] {4} ...

It's been awhile since I studied ring theory but here's a question about it: Let R = C[x1, x2, x3, x4, x5, x6, x7, x8] be a complex polynomial ring in 8 variables. Let f1 = x1 x3 +x5 x7 and f2 = x2 x4 +x6 x8. How do \bar{f}1, \bar{f}2 in (f1,f2)R/(x1,x2)R look like? Is...

I have to describle the elements of the quotient ring $$\frac{\mathbb{Z}[x]}{2\mathbb{Z}[x]+x^2\mathbb{Z}[x]}$$ do I use the division algorithm to write any element as $$f(x)=(x^2+2)q(x)+r(x)$$ where $\operatorname{deg}(r(x))<2$ so the elements of the ring are the linear polynomials over...

How do you show that x is a nonzero divisor in C[x,y,z,w]/<yz-xw>? Here's how one can start off on this problem but I would like a nice way to finish it: If x were a zero divisor, then there is a function f not in <yz-xw> so that f*x = g*(yz-xw).Here's another question which is slightly...

Homework Statement M = {(pa,b) | a, b are integers and p is prime} Prove that M is a maximal ideal in Z x Z Homework Equations The Attempt at a Solution I know that there are two ways to prove an ideal is maximal: You can show that, in the ring R, whenever J is an ideal such...

Homework Statement Show that Z18/M isomorphic to Z6 where m is the cyclic subgroup <6> operation is addition The Attempt at a Solution M = <6> , so M = {6, 12, 0} I figured I could show that Z18/M has 6 distinct right cosets if I wanted to do M + 0 = {6, 12, 0} M + 1 = {7, 13, 1}...

hey guys, I just want grasp the whole concept of quotient groups, I understand say, D8/K where K={1,a2} I can see the quotient group pretty clearly without much trouble however I start to get stuck when working with larger groups, say S4 For instance S4/L where L is the...

Hi I am trying to learn about quotient groups to fill the gaps on things I didn't quite understand from undergrad. Anyway I have a question regarding this: Can someone please explain how { 0, 2 }+{ 1, 3 }={ 1, 3 } in Z4/{ 0, 2 }? I would think since 0 + 1 = 1 and 2 + 3 = 1 under mod 4...

How do you solve (analytically or numerically) a differential equation of this form, $$\frac{\mathrm{d}y(x)/\mathrm{d}x}{\mathrm{d}z(x)/\mathrm{d}x} = a[1-y(x)-z(x)] + b$$ where a, b are constants. Also, $$y(0) = z(0) = 0$$

Homework Statement I'm trying to self-study Mary L. Boas' book Mathematical Methods in the Physical Sciences. One of the exercices asks the reader to find the limit of n -> ∞ (n!)2 / (2n)! Homework Equations None The Attempt at a Solution Instinctively I know that (2n)! grows...

I have to prove this problem. For all n integers, if n mod 5 = 3, then n2 mod 5 = 4 Proof: Let n be an integer such that n mod 5 = 3. n = 5k+3 for some integer k by definition of MOD or QRT? Which one would be correct? Am I using the definition of MOD or QRT? I'm thinking its QRT because its...

Homework Statement Let f,g be continuous on a closed bounded interval [a,b] with |g(x)| > 0 for all x in [a,b]. Suppose that f_n \to f and g_n \to g uniformly on [a,b]. Prove that \frac{1}{g_n} is defined for large n and \frac{f_n}{g_n} \to \frac{f}{g} uniformly on [a,b]. Show that this is...

Homework Statement Find y' of y= 1-3ln(7x)/x^4 Homework Equations The Attempt at a Solution I used the quotient rule and got: y'=x^4*d/dx(1-3ln(7x)-(1-3ln(7x)*d/dx(x^4)/(x^4)2 which is: x^4*(0-3*1/7x*7)-(1-3ln(7x))*4x^3/x^8 simplified to: 3x^4/x-1+3ln(7x)*4x^3 3x^3-4x^3+12x^3ln(7x)/x^8 take...

i am trying to re-express the following in terms of a rational function: \frac{(0+x+2x^2+3x^3+...)}{1+x+x^2+x^3+...} . i know that this is supposed to be \frac{1}{x-1} but I can't figure out how to do it. I know the denominator is just \frac{1}{1-x}. so in order for this work out, the...

Homework Statement a) Differentiate y=((x-1)/(x+1))^2 b (non calculus simplification question): How would i simplify 10(5x+3)(5x-1)+5(5x-1)^2 to get 25(5x-1)(3x+1) should i expand all terms then combine and factor? Homework Equations The Attempt at a Solution a) i tried...

Homework Statement Let D4 = { (1)(2)(3)(4) , (13)(24) , (1234) , (1432) , (14)(23) , (12)(34) , (13), (24) } and N=<(13)(24)> which is a normal subgroup of d4 . List the elements of d4/N . Homework Equations The Attempt at a Solution I computed the left and right cosets to...

Homework Statement I want to prove that if y = \frac{u}{v} then \frac{dy}{dx} = \frac{ v \frac{du}{dx} - u \frac{dv}{dx} }{v²} u and v are functions of x. 2. The attempt at a solution y = uv^{-1} y + dy = ( u + du ) ( v + dv )^{-1} then I suppose I could use Newton's Binomial...

Homework Statement \int \frac{r^3-r^5}{\sqrt{1-r^2}}\,.dr The attempt at a solution The presence of \sqrt{1-r^2} suggests that i use the substitution r=sinθ The integrand becomes: \frac{\sin^3\theta-\sin^5\theta}{\cos\theta} \frac{dr}{d\theta}=\cos\theta \int...

Just to make sure that I'm not overlooking anything, is the following an example of a quotient map p: X \to Y with the properties that Y is pathwise connected (i.e. connected by a continuous function from the unit interval), \forall y \in Y: p^{-1}(\{ y \}) \subset X also pathwise connected and...

In my Abstract Algebra course, it was said that if E := \frac{\mathbb{Z}_{3}[X]}{(X^2 + X + 2)}. The basis of E over \mathbb{Z}_{3} is equal to [1,\bar{X}]. But this, honestly, doesn't really make sense to me. Why should \bar{X} be in the basis without it containing any other \bar{X}^n...

Homework Statement Let X be a compact and locally connected topological space. Prove that by identifying a finite number of points of X, one gets a topological space Y that is connected for the quotient topology. Homework Equations The components of a locally connected space are open...

Hi I don't know how to attack the following question, any hints would be appreciated: If G is a simply connected topological group and H is a discrete subgroup, then \pi_1(G/H, 1) \cong H .Thank you James

I'm having some troubles understanding the concepts of quotient algebra. May someone explain me what exactly they are, giving some concrete examples? I know that a quotient set is the set of all equivalence classes, but it sounds very vague for me and i can't make the analogy with quotient...

Homework Statement Show that \mathbb Q[x,y]/(x^2+y^2-1) is not a unique factorization domain. The Attempt at a Solution We have tried a few approaches. Using [] to denote equivalence classes, we note that we can write [x]^2 = [1-y][1+y]. Our goal was to show that this is a non-unique...

So I have an elementary understanding of group theory and the goals of studying sets and operations on them but I noticed that along the way, my understanding of a quotient group is severely flawed. I understand what a so called normal subgroup is, but could someone please give an in depth...

Problem: Let L and M be finite dimensional linear spaces over the field K and let g: L\times M \rightarrow K be a bilinear mapping. Let L_0 be the left kernel of g and let M_0 be the right kernel of g. a) Prove that dim L/L_0 = dim M/M_0. b) Prove that g induces the bilinear mapping g': L/L_0...

So earlier this year I came here to discuss about having fun with groups, rings and isomorphisms and such. I fell upon the idea of finding an isomorphism of the positive rationals to the sequence of the exponents found in their prime factorization. I didn't know what much to do with it since I...

EDIT: I found the mistake, question is answered! Its funny because I spent 40+ minutes trying to get the right answer and looking for the mistake but typing it all out in latex helped me to find it! [/color] Homework Statement \frac{\sqrt{x}}{x^3+1} The Attempt at a Solution...

Homework Statement Let H be a subgroup of K and K be a subgroup of G. Prove that |G:H|=|G:K||K:H|. Do not assume that G is finite Homework Equations |G:H|=|G/H|, the order of the quotient group of H in G. This is the number of left cosets of H in G. The Attempt at a Solution I...

Hey all, We have not covered quotient vector spaces in class, but my homework (due before next lecture) covers a few proofs regarding quotient spaces. I've done some reading on them and some of their aspects, but as it is still a new concept, I am struggling with how to go about this proof...