quotient Definition and 356 Threads

Given positive real numbers $a,\,b,\,c,$ and $d$ that satisfy the system below: $a^2+ d^2-ad = b^2+ c^2+ bc$ and $a^2+ b^2= c^2+ d^2$. Find all possible values of the expression $\dfrac{ab + cd}{ad + bc}$.

Simply and find the difference quotient for f(x)= 1/(x-3) I know the difference quotient formula is f(x+h)-f(x)/h but when I try solving it I keep getting it wrong.

I've highlighted the part in yellow I don't understand. He apparently 'drops' Δx in the last line, but doesn't display how. What I do know is that he is taking the limit as Δx→0, or as Delta x approaches zero. I'm simpy missing what he did though to drop the Δx. I mean, I understand he...

Hey! :o I want to check if the following statement are true. Let $R$ be a ring, $S$ a subring and $I$ an ideal. If $R$ is Noetherian then $S$ is also. If $R$ is Noetherian then $R/I$ is also. If $R$ is Artinian then $S$ is also. If $R$ is Artinian then $R/I$ is also. If $R$ is...

Hey! :o I want to show that if $H\subseteq Z(G)$ and $G/H$ is nilpotent then $G$ is also nilpotent. I have done the following: Since $G/H$ is nilpotent there is a series of normal subgroups $$1\leq N_1\leq N_2\leq \cdots \leq N_k=G/H$$ with $N_{i+1}/N_i\subseteq Z((G/H)/N_i)$. From the...

Hello! As far as I know any subgroup can, in principle, be used to divide group into bundle of cosets. Then any group element belongs to one of the cosets (or to the subgroup itself). And such division still is not qualified as a quotient. Yes, the bundle of cosets in this case will be...

Hello, if we consider a group G and two subgroups H,K such that HK \cong H \times K, then it is possible to prove that: G/(H\times K) \cong (G/H)/K Can we generalize the above equation to the case where HK \cong H \rtimes K is the semidirect product of H and K? Clearly, if HK is a semidirect...

Hello, I am having some trouble truly interpreting what certain notation means when defining quotient groups, etc. (My deepest apologies in advance, with my college workload I simply have not had the time to really sit down and master latex.) Here are a few random examples I've seen in...

Given any integer A, and a positive integer B, there exist unique integers Q and R such that $$A= B * Q + R$$ where $$ 0 ≤ R < B$$. When is says that $$Q$$ and $$R$$ are unique, what does that mean? That they are different from each other?

For any int $$n $$ , prove that $$ 4 | n (n^2 - 1) (n + 2)$$. I know I have to use the quotient remainder theorem, but I'm wondering how to go about this problem. I'm not sure how to phrase this problem in English.

Homework Statement The displacement of a particle can be modeled by the function x(t)=\frac{2x-5}{4x^2+2x}, where t is in seconds, x is in meters, and t ∈ [1,10] a) Determine the derivative of the function without using the quotient rule. b) Hence, find exactly when the particle is...

Hi everyone. So it's apparent that G/N cyclic --> G cyclic. But the converse does not seem to hold; in fact, from what I can discern, given N cyclic, all we need for G/N cyclic is that G is finitely generated. That is, if G=<g1,...,gn>, we can construct: G/N=<(g1 * ... *gn)*k> Where k is the...

Homework Statement Let A be the algebra \mathbb{Z}_5[x]/I where I is the principle ideal generated by x^2+4 and \mathbb{Z}_5[x] is the ring of polynomials modulo 5. Find all the ideals of A Let G be the group of invertible elements in A. Find the subgroups of the prime decomposition.Homework...

I have just received some help from Euge regarding the proof of part of the Correspondence Theorem (Lattice Isomorphism Theorem) for groups ... But Euge has made me realize that I do not understand quotient groups well enough ... here is the issue coming from Euge's post ... We are to consider...

I have just finished a post entitled: http://mathhelpboards.com/linear-abstract-algebra-14/irreducible-polynomials-quotient-rings-rotman-proposition-3-116-a-16163.htmlon the Linear and Abstract Algebra Forum ... I want to have the following code recognised: k(z) \subseteq \text{ I am } \phi...

I am reading Joseph J. Rotman's book: A First Course in Abstract Algebra with Applications (Third Edition) ... I am currently focused on Section 3.8 Quotient Rings and Finite Fields ... I need help with an aspect of the proof of Proposition 3.116 Proposition 3.116 and its proof reads as...

Why is usual respiratory quotient for humans is between 0.7 and1.0

I understand how to find a difference quotient, but afterwards it asks me to then evaluate or approximate each limit, is that just by plugging in the given limit or is there another step?

Homework Statement Let G be any group and a in G, define f: Z → G by f(n) = a^n Apply any isomorphism theorem to show that range of f is isomorphic to a quotient group of Z Homework EquationsThe Attempt at a Solution The range of f is a^n , then quotient group of Z is Z/nZ Apply the first...

The book I am using for my Introduction to Topology course is Principles of Topology by Fred H. Croom. Problem: Prove that if ##X=X_1\times X_2## is a product space, then the first coordinate projection is a quotient map. What I understand: Let ##X## be a finite product space and ##...

Assume that G is some group with two normal subgroups H_1 and H_2. Assuming that the group is additive, we also assume that H_1\cap H_2=\{0\}, H_1=G/H_2 and H_2=G/H_1 hold. The question is that is G=H_1\times H_2 the only possibility (up to an isomorphism) now?

Given a ring $R$ and $R$-modules $A,B,C,D$ such that $$\sigma:A \rightarrow B, \tau: C \rightarrow D, \rho: A \rightarrow C, \kappa: B \rightarrow D, \ \mathrm{and} \ \kappa \circ \sigma = \tau \circ \rho,$$ where $\sigma, \tau, \rho, \kappa$ are homomorphisms and $\rho, \kappa$ are...

HI, Suppose there are two operators A and B , We have to find A /B - Will it equal to AB-1 OR B-1 A , Because i have read that it equals to AB-1 , BUT i could not find reason for that. thanks

This is probably a stupid question. Let R be a domain, K its field of fractions, L a finite (say) extension of K, and S the integral closure of R in L. Is the quotient field of S equal to L ? I believe that not, but I have no counter-example.

I'm tutoring a pupil for a CLEP exam and her book includes the following algebra problem: What is the remainder when 9x^{23} - 7x^{12} - 2x^{5} +1 is divided by x+1 ? I know how to find the answer by computing the quotient of these two expressions, but in this case doing that is so tedious I...

Find the difference quotient f(x+h)-f(x)/h Where h\ne 0, for the function below f(x)=5x^2+4 Simplify your answer as much as possible. How do I do this?

prove that the quotient space obtained by identifying the points on the southern hemisphere, is homeomorphic to the whole sphere.I am trying to define a homeomorphism between the quotient space and the sphere,and i need help doing it. Thank's in advance.

I have a problem: f(x) = 1/x, [f(x) - f(a)] / x- a I am wondering how to approach this problem. I have so far. (1/x - 1/a) / (x-a) ([a-x] / xa) / (x-a) How would I simplify this? By the way, the answer is -1 / ax

I am reading "Introduction to Ring Theory" by P. M. Cohn (Springer Undergraduate Mathematics Series) In Chapter 1: Basics we find Corollary 1.16 on module homomorphisms and quotient modules. I need help with some aspects of the proof. Corollary 1.16 reads as follows: In the above text...

I am reading "Introduction to Ring Theory" by P. M. Cohn (Springer Undergraduate Mathematics Series) In Chapter 1: Basics we find Theorem 1.15 on module homomorphisms and quotient modules. I need help with some aspects of the proof. Theorem 1.15 reads as follows: In the proof of the...

Homework Statement Find f(a), f(a + h),and the difference quotient f(a + h) − f(a), h where h ≠ 0. f(x) = 5x/x-1 f(a) = ? f(a + h) = ? f(a + h) − f(a)/(h) = ? The Attempt at a Solution As I understand it I must substitute the values but I don't understand the question...

1. Simplify the algebraic expression you get for Δy and Δy/Δx for the equation y=2^x 2. Use the difference quotient (f(x+h)-f(x))/h. No use of chain rule or other shortcuts. 3. I've tried a host of things, including raising terms to a natural log power (I.e. e^(ln2)*x*h), using...

Let R be a local ring with maximal ideal J. Let M be a finitely generated R-module, and let V=M/JM. Then if \{x_1+JM,...,x_n+JM\} is a basis for V over R/J, then \{x_1, ... , x_n\} is a minimal set of generators for M. Proof Let N=\sum_{i=1}^n Rx_i. Since x_i + JM generate V=M/JM, we have...

Could anyone help me solve this problem? Let A,B be two subspace of V, a \in A, b \in B. Show that the following operation is linear and bijective: (A + B)/B → A/(A \cap B): a + b + B → a + A \cap B I really couldn't understand how the oparation itself works, i.e, what F(v) really is in...

I am revising vector spaces and am looking at their quotient spaces in particular ... I am looking at the theory and examples in Schaum's "Linear Algebra" (Fourth Edition) - pages 331-332. Section 10.10 (pages 331-332) defines the cosets of a subspace as follows: Following the above...

Definition/Summary The quotient rule is a formula for the derivative of the quotient of two functions, for which derivatives exist. Equations f(x) = \frac{g(x)}{h(x)} Then, f'(x) = \frac{h(x)g'(x)-g(x)h'(x)}{(h(x))^2} here, h(x) \: \neq \: 0 Extended explanation Even...

Definition/Summary A quotient group or factor group is a group G/H derived from some group H and normal subgroup H. Its elements are the cosets of H in G, and its group operation is coset multiplication. Its order is the index of H in G, or order(G)/order(H). Equations...

am given that ϕ is a function from F(R) tp RxR defined by ϕ(f)=(f(0),f(1)) i proved that ϕ is a homomorphism from F(R) onto RxR. i showed that 1) ϕ(f) +ϕ(g)=ϕ(f+g) [for all f,g in F(R)] 2)ϕ(f)*ϕ(g)= ϕ(f*g) how do i show that ϕ is onto and define the kernal??(Wasntme)

G is a group and H is a normal subgroup of G. where G=Z6 and H=(0,3) i was told to list the elements of G/H I had: H= H+0={0,3} H+1={14} H+2={2,5} now they are saying H+3 is the same as H+0, how so?

Hey guys, just trying to understand how the quotient rule is derived, so I head over to wikipedia and saw this: But I'm having some difficulty understanding what goes on between these two steps: Could someone shed some light on this?

Hi All: We know that the quotient ## \mathbb Z /2\mathbb Z ## ~ ## \mathbb Z/2 ## . Is there a nice way of computing the quotient : ## [\mathbb Z(+) \mathbb Z ]/[ 2\mathbb Z(+)2\mathbb Z]## I know the long way, but I wonder if there is a nicer, shorter way to do it. Thanks.

I am reading munkres topolgy and I am struggling with understanding the following sentence: "We say that a subset C of X is saturated (with respect to the surjective map p:X→Y) if C contains every set p-1({y}) that it intersects" if you have the second edition its in chapter 2 section 22...

I am reading Martin Crossley's book, Essential Topology. I am at present studying Example 5.55 regarding the Mobius Band as a quotient topology. Example 5.55 Is related to Examples 5.53 and 5.54. So I now present these Examples as follows: I cannot follow the relation (x,y) \sim (x', y')...

Hi everyone, :) I think I need to refresh my memory about annihilators and quotient rings. Hope you can help me with the following example. I want to find the annihilator of $a'$ and $b'$ of the quotient ring $R=\mathbb{Z}/(a'b')$ where $a',\,b'>1$. So if I go by the definition...

Hi, I'm currently reading Shilov's Linear Algebra and he mentions that Hyperplanes are planes that don't pass through the origin. Wouldn't that be a quotient space? Thank you.

I’ve been trying to understand how having indistinguishable particles in a system changes the nature of the state space. The QM texts I have gloss over this. A typical approach is to define the symmetric and anti-symmetric kets that serve as a basis for the eigenspace containing...

This is how the problem appears in my book(Munkres 2nd edition Topology, sect. 22 pg 145) 6. Recall that R_K denotes the real line in the K-topology. Let Y be the quotient space obtained from R_K by collapsing the set K to a point; let p : R_K → Y be the quotient map. (a) Show that Y...

Can someone check my working. I don't understand why i am getting different answers? u(x,t)=\frac{{e}^{-\frac{x^2}{4Dt}}}{\sqrt{4Dt}} Differentiate w.r.t 't' by quotient rule: \frac{\partial u}{\partial t}=\left[ \frac{1}{\sqrt{4Dt}}\cdot \frac{x^2}{4Dt^2}\cdot...

Suppose we have some two-dimensional Riemannian manifold ##M^2## with a metric tensor ##g##. Initially it is always locally possible to transform away the off-diagonal elements of ##g##. Suppose now by choosing the appropriate equivalence relation and with a corresponding surjection we construct...

I'm learning algebra by myself and this concept is confusing me. Please excuse me if I define anything wrong... I've never expressed myself in this language before. Lets say we have a group G and a group G' and there exists a homomorphism R: G → G' and for any element g \in G, the...