quotient Definition and 356 Threads

Hi everyone, :) This seems like a pretty simple question, but up to now I haven't found a method to solve it. Hope you can provide me a hint. :) Problem: Let \(V\) be a space with basis \(B=\{b_1,\,b_2,\,b_3,\,b_4,\,b_5\},\,U\) the subspace spanned by \(u_1=b_1+b_2+b_3+b_4+b_5\)...

Just for fun, eh...? (Heidy)For z \in \mathbb{R}, and m \in 2\mathbb{N}+1, show that:\frac{\tan mz}{\tan z}=\prod_{j=1}^{ \lfloor m/2 \rfloor } \tan\left(\frac{j\pi}{m}+z\right) \tan\left(\frac{j\pi}{m}-z\right)

In Beachy and Blair: Abstract Algebra, Section 3.8 Cosets, Normal Groups and Factor Groups, Exercise 17 reads as follows: ---------------------------------------------------------------------------------------------------------------------- 17. Compute the factor group ( \mathbb{Z}_6 \times...

I am reading Dummit and Foote Section 3.1: Quotient Groups and Homomorphisms. Exercise 17 in Section 3.1 (page 87) reads as follows: ------------------------------------------------------------------------------------------------------------- Let G be the dihedral group od order 16. G = <...

If it's possible to relate the product rule with the binomial theorem, so: (x+y)^2=1x^2y^0+2x^1y^1+1x^0y^2 D^2(fg)=1f^{(2)}g^{(0)}+2f^{(1)}g^{(1)}+1f^{(0)}g^{(2)} So, is it possible to relate the quotient rule with the binomial theorem too?

Homework Statement Hello, I have an answer to my question but I'm not sure if I've simplified it correctly or if it can boil down further. differentiate; y = \frac{z^{3}-z}{\sqrt{z}} Homework Equations Quotient rule The Attempt at a Solution let u = z3 - z let v = z1/2...

Prove that G is a subspace of V ⊕ V and the quotient space (V ⊕ V) / G is isomorphic to V. Let $V$ be a vector space over $\Bbb{F}$, and let $T : V \rightarrow V$ be a linear operator on $V$. Let $G$ be the subset of $V \oplus V$ consisting of all ordered pairs $(x, T(x))$ for $x$ in $V$. I...

Let A be an m x n matrix with entries in R. Let T_A : R^n -> R^m be the linear map T_A(X) = A_X. Let U be the solution set of the homogeneous linear system A_X = O. Let W be the set of all vectors Y such that Y = A_X for some X in R^n. I don't really know what I'm supposed to do here, any help...

Homework Statement find the following derivative d/dx [g(x + 1)(√(2+ (x + 8)^(1/3))/(cos(tan(sin(tan(sin x))))] at x = 0 Homework Equations The Attempt at a Solution I split the big long derivative into 3 functions: a(x) = g(x + 1) b(x) = √(2+ (x + 8)^(1/3)) c(x) =...

Simplify \frac{\Large 1+\frac{1}{2^a}+\frac{1}{3^a}+\frac{1}{4^a}+\cdots}{\Large1-\frac{1}{2^a}+\frac{1}{3^a}-\frac{1}{4^a}+\cdots} where $a>1$ is a real number.

Homework Statement I'm trying to prove the statement "Show that a subgroup of a quotient of G is also a quotient of a subgroup of G." Homework Equations See below. The Attempt at a Solution Let G be a group and N be a normal subgroup of G. Let H be a subgroup of the quotient...

Hello. I am studying differential calculus and I need help on what these questions are asking and how to solve them. I have attempted some of the question but I need clarification. This week is mainly focused on learning the Quotient Rule. Please help me 1. (a) Differentiate y = (x - A)/(x -...

On the set of Z of integers define a relation by writing m \triangleright n for m, n \in Z. m\trianglerightn if m-n is divisble by k, where k is a fixed integer. Show that the quotient set under this equivalence relation is: Z/\triangleright = {[0], [1], ... [k-1]} I'm a bit new the subject...

Can't figure this one out. Here's the problem. Solid molybdenum (VI) oxide reacts with gaseous xenon difluoride to form liquid molybdenum(VI) fluoride, xenon gas, and oxygen gas. Write the Qc for this reaction. I know Qc is products over reactants and pure liquids and solids are not...

Homework Statement Evaluate the difference quotient of: f(1+h)-f(1) / h if f(x)=x3 and simplify your answer. Homework Equations The Attempt at a Solution I took the derivative of x3 to be 3x2 and solved for when x=1 (given from the difference quotient) and got 3, but when I enter the...

Homework Statement A soccer ball is kicked into the air from a platform. The height of the ball, in metres, t seconds after it is kicked is modeled by h(t) = -4.9t^2 + 13.5t + 1.2. I solved for the expression that represents the average rate of change of height over the interval 2<= t <=...

I am a little confused about when to use the quotient rule. When you have one function over another function, and are taking the derivative, are you required to use this technique? I thought you were, but then I was watching this video on Khan Academy...

Homework Statement X is a compact metric space, X/≈ is the quotient space,where the equivalence classes are the connected components of X.Prove that X/ ≈ is metrizable and zero dimensional. Homework Equations Y is zero dimensional if it has a basis consisting of clopen (closed and open at...

The definition given is... "Let ##\phi: G \rightarrow H## be a homomorphism with kernel ##K##. The quotient group ##G/K## is the group whose elements are the fibers (sets of elements projecting to single elements of H) with group operation defined above: namely if ##X## is the fiber above...

I have been trying to figure out the derivative of (X²-1)/X. When I use the quotient rule, the result I get is 1-1/X². However, when I simplify the expression first, then take the derivative, I get 1+1/X² Why are the results different?

Homework Statement I want to find the orders of the elements in Z_8/(Z_4 \times Z_4), (Z_4 \times Z_2)/(Z_2 \times Z_2), and D_8/(Z_2 \times Z_2). Homework Equations The Attempt at a Solution The elements of Z_2 \times Z_2 are (0,0), (1,0), (0,1), (1,1), and the elements of Z_8 are of course...

Homework Statement I am working on chemical reaction engineering problem and it involves some math, which I am not able to figure out... I have to find the residence time for maximum production, which is in the case when : (dη_p)/dτ=0 I have to find the τ (residence time)...

I am new to group theory, and read about a "universal property of abelianization" as follows: let G be a group and let's denote the abelianization of G as Gab (note, recall the abelianization of G is the quotient G/[G,G] where [G,G] denotes the commutator subgroup). Now, suppose we have a...

Hi, I am trying to prove the following proposition: Let F be a closed subset of the Euclidean space Rn.Then the quotient space Rn/F is first countable if and only if the boundary of F is bounded in Rn. Any ideas?

Homework Statement Let (G,◦) be a group and let N be a normal subgroup of G. Consider the set of all left cosets of N in G and denote it by G/N: G/N = {x ◦ N | x ∈ G}. Find G/N: (G,◦) = (S3,◦) and N = <β> with β(1) = 2, β(2) = 3, β(3) = 1. Homework Equations The Attempt at a Solution I'm...

Homework Statement How can I show that \frac{x}{e^x-1} is real analytic in a neighborhood of zero, excluding zero?Homework Equations The Attempt at a Solution

How can one prove that for homomorphism G \xrightarrow{\rho} G' and H as kernel of homomorphism, quotient group G/H is isomorphic to G'? Thanks.

Homework Statement lim x-> infinity 3n+2/5n

Intuitively, one would assume that the quotient space of a topological space under an equivalence relation would always be smaller than the original space. It turns out this is not remotely true. I'm specifically interested in quotient spaces of ℝ (under the standard topology). We can...

Homework Statement y=\frac{}{}tsint/1+t Homework Equations The Attempt at a Solution What I do (and arrive at the wrong solution) is the fallowing. Can you please tell me where i go wrong. *I am going to leave out the denominator as i go through the problem because it stays the...

Hello PH, This is my first post. I came here while studying partial derivatives and after clicking here and there for over 4hrs for an answer. While practicing the derivatives rules i came across the hideous quotient rule. I've solved around 20 fractional problems trying to find a decision...

Homework Statement Try to apply the First Isomorphism Theorem by starting with a homomorphism from a polynomial ring R[x] to some other ring S. Let I = \mathbb{Z}_2[x]x^2 and J = \mathbb{Z}_2[x](x^2+1). Prove that \mathbb{Z}_2[x]/I is isomorphic to \mathbb{Z}/J by using the homomorphism...

Determine whether (7)1/2/(15)1/3 is either rational or irrational and prove your answer is correct. So I know that (7)1/2 is irrational from previous theorems since it is a prime, I also split up (15)1/3 into (5)1/3 times (3)1/3. I previously had shown that (5)1/3 is also irrational. Doing...

Derivatives with Quotient Law Help! I have a test tomorow, any help is much appreciated! :) Homework Statement Dervive using the quotient rule: [(2-x)^3] / [(x+1)^2] My attempt: = [(x+1)^2 (3(2-x)^2)]-[(2-x)^3 2(x+1)] When I try expanding I get the wrong answer. The...

If I have the unit sphere and I mod out its equator, I get two spheres touching at one point. I have been thinking what the bijection between these could be but can not come up with one.

Homework Statement This is just a small part of a larger question and is quite simple really. It's just that I want to confirm my understanding before moving on. What are some of the elements of Z[i]/I where I is an ideal generated by a non-zero non-unit integer. For the sake of argument...

I have what I hope to be just a simple notation/definition question I can't seem to find an answer to. I'm not going to post my homework question, just a piece of it so I can figure out what the question is actually asking. I have a function i:A --> X I also have a continuous function g: A...

I wasn't

Homework Statement ([2x+1/4x+3]^2) Homework Equations Exponent and quotient rule The Attempt at a Solution Would this become: 2* (2x+1/4x+3) then do the quotient rule?

Homework Statement Let f(x) = x6 + x3 + 1 in Z_2[x]. Show that f(x) is irreducible. Let E = Z_2[x]/(f(x)), and let αdenote the image of x in the quotient field. Show that E* = <α>. Homework Equations I have solved the first part, but what does E* mean? I have seen the asterix in...

Homework Statement Use quotient rule Homework Equations d/dx (cot(x) The Attempt at a Solution d/dx (cot(x)) = cos(x)/sin(x) = (sin(x))(-sin(x)) - (cos(x))(cos(x)) / (sin(x))^2 = -(sin(x))^2 - (cos(x))^2 / (sin(x))^2 = -1 - (cos(x))^2 Does the last step equal 1 +...

Homework Statement Let w_1,...,w_n be a set of n-linearly independent vectors in \mathbb{R}^n. Define an equivalence relation \sim by p\sim q \iff p-q=m_1w_1+...+m_nw_n for some m_i \in \mathbb{Z} Show that \mathbb{R}^n / \sim is Hausdorff and compact and actually homeomorphic to (S^1)^n...

Homework Statement Describe all the subgroups of Z/9Z. How many are there? Describe all the subgroups of Z/3ZxZ/3Z. How many are there? The Attempt at a Solution I don't even know where to start with this question. If someone could just point me in the right direction that would be...

Hello, I have been given a homework problem and I don't want any help on solving the problem, (I'm not even going to post the problem - I want to figure it out myself), I only want to understand what the problem is asking. (That's why I've posted in this section rather than the homework...

Hi, I got the right answer when I used the Quotient Rule but not when I used the Product Rule... I think it might be an algebra mistake... Product Rule Method: f'(x) = (3 - x^2)*(4 + x^2)^-1 = (3 - x^2)[(-1(4 + x^2)^-2)*2x] + [(4 + x^2)^-1](-2x) = [(3 - x^2)(-2x)]/[(4 + x^2)^2] +...

We are just looking for an example of a quotient map that is not open nor closed. Let π: ℝxℝ -> ℝ be a projection onto the first coordinate. Let A be the subspace of ℝxℝ consisting of all points (x,y) such that x≥0 or y=0 or both. Let q:A -> ℝ be a restriction of π. ( Note: assume that q was...

Can someone please explain to me, in as simple words as possible, what a quotient group is? I hate my books explanation, and I would love it if someone can tell me what it is in english?

Hi Guys, I remember back in the days of Calc I learning that there was an easy way to take multiple derivatives of certain functions that needed repeated uses of the quotient rule. I was wondering if anybody remembered that trick.

Good afternoon. Here is the problem: Show that if $R$ is a ring with unity and $N$ is an ideal of $R$ such that $N \neq R$, then $R/N$ is a ring with unity. My answer: Consider the homomorphism $\phi: R \to R/N$. Given $r \in R$ we have that $\phi(r) = r + N = \phi(1 \cdot r) = \phi(r \cdot 1)...

Good afternoon! Along the same lines as the other, here is the question: Show that the quotient ring of a field is either the trivial one or is isomorphic to the field. My answer: Let $N$ be an ideal of the field $F$. Assume that $N \neq \{ 0 \}$. Consider the homomorphism $\phi: F \to F / N$...