quotient Definition and 341 Threads

"Group A is a quotient of Group B"? What does this phrase mean? I see it every now and again and can't figure it out. Are they say group B is the homomorphic image of group A? I'm familiar qith quotient groups, but with only groups A and B named, how would we know which quotient group of A...

I took a placement test and blew it away. (I tested into calculus, best possible placement for this test.) Everything was rather simple except for this problem which I cannot seem to get right. Can someone show me where I'm going wrong here?Homework Statement Use x2+2 in the Difference...

Homework Statement Given y = x^2 + 6 / x find x if dy/dx equals 0 zero Homework Equations dy/dx = v du/dx - u dv/dx / v^2 The Attempt at a Solution I have got as far as dy/dx = x^2 - 6 / x^2 However then zero must equal Sqrt 6, which 2.44... -...

Homework Statement if p(x) = f(x)/g(x) Prove that p'(x) = g(x) f '(x) - f(x) g '(x) / g(x)ˆ2 Homework Equations The Attempt at a Solution The proof goes like this in my book p(x + h) - p(x) / h = [ f(x+h)/ g(x+h) - f(x) / g(x) ] / h = f(x + h) g(x) - f(x) g(x +...

Homework Statement Hi all well basically i have finished off chain rule and right now i am going through product rule and quotient, as i was going through some questions , i understood the basic rule and so on, but why i don't get is, how do i figure which rule i need to apply given equation...

Can someone guide me on how to express (.a1a2a3a4a5) base n as a quotient of base n integers. There is a bar over a3a4a5.

Homework Statement Write the multiplication table of C_{6}/C_{3} and identify it as a familiar group. Homework Equations The Attempt at a Solution C_{6}={1,\omega,\omega^2,\omega^3,\omega^4,\omega^5} C3={1,\omega,\omega^2} The cosets are C3 and \omega^3C3 I just need help...

Homework Statement For a prime p and a polynomial g(x) that is irreducible in Z_{p}[X], prove that for any f(x) in Z_{p}[X] and integer k > 1, [f(x)]^{k} = [f(x)] in Z_{p}[X]/(g(x)). The Attempt at a Solution I realize this is an extension of Fermat's Little Theorem, however I cannot...

Hello everyone, first time poster here. I've been a lurker for about a week, but finally decided to join because I cannot for the life of me figure out this problem. Homework Statement Zach has trouble with the Quotient Rule; he thinks that d/dx (f(x)/g(x)) = f′(x)/g′(x). On his last...

The question is "without a calculator, long divide 425/836. Round to the nearest hundredth. Estimate your answer before starting." My attemp at a solution: Well, my estimate would be approximately one half, since 400 divided by 800 would be 0.5; however, I'm not sure how to long divide a...

Differentiate with respect to x; (using the quotient rule) 3/2x-1 (3 over 2x minus 1) dy/dx = (2x-1)(0) - (3)(2) / (2x-1)^2 dy/dx = -6/(2x-1)^2 but my book gives -2/(2x-1)^2 now, y = u/v and i take u = 3 and v = 2x-1. dy/dx = v(du/dx) -...

and simplify your awnser f(x)= (X+3) / (x+1) don't know how to do it, my attempt f(x+h)-f(x) /h ((x+3)/(x+1)+h)-(x+3/(x+1) /h bah how do i start and finish this?

im working on an equation using the quotient rule. this is the equation. 8(5+x)/16+X^2 and i have to find the f'= i know 16+x^2 = 2x im not sure about the 8(5+x) it can not be arctan(x) could you please help.

Hello everybody I would also like to solve the following problem using either the Chain,Product, or Quotient Rule but am unsure of the working stages to get to the given answers i) Find the equation of the tangent at the point with coordinates (1,1) to the curve with the equation...

Ok so you can't apply the quotient criteria to the harmonic series because: lim_{k\to \infty}|\dfrac{a_{k+1}}{a_k}| applied to the harmonic series: lim_{k\to \infty}|\dfrac{1/(k+1)}{1/k}| = lim_{k\to \infty}|\dfrac{k}{k+1}| < 1 which does fullfill the quotient criteria, yet the...

Notations: V denotes a vector space S denotes a subspace of V V/S denotes a quotient space V\S denotes the complement of S in V Question: If {s1, ... , sk} is a basis for S, how to find a basis for V/S? I realize that the basis of V\S may determine the basis of V/S, but I don't know...

Homework Statement Let f(x) = x2 + 1 in Z3[x]. Find the order of the quotient ring Z3[x]/<f>. Homework Equations The Attempt at a Solution Note Z3 is a field. Then Z3[x] is euclidean domain. Then for any polynomial g(x) can be written as g(x) = p(x).(x2+1) + r(x) where...

Prove that b_{ijkl}=\int_{r<a} dV x_i x_j \frac{\partial^2}{\partial_k \partial_l} (\frac{1}{r}) where r=|x| is a 4th rank tensor. i've had a couple of bashes and got nowhere other than to establish that its quotient theorem. can i just pick a tensor of rank 3 to multiply it with or...

Find, up to isomorphism, all possible quotient groups of D6 and D9, the dihedral group of 12 and 18 elements. First of all, I don't understand the question by what they mean about "up to isomorphism." Does this mean by using the First Isomorphism Theorem? Also does this question imply that...

I don't take a Calculus class(I'm learning on my own), but I'm just curious as to what are the steps to solving the following equations. Homework Statement These are the following problems that I'm having trouble solving. y(x) = cos(x) y(x) = \sqrt{x} y(x) = sin(x) y(x) =...

Homework Statement In Z4 x Z4, find two subgroups H and K of order 4 such that H is not isomorphic to K, but (Z4 x Z4)/H isomorphic (Z4 x Z4)/K Homework Equations The Attempt at a Solution I know (Z4 x Z4) has twelve elements (0,0), (1,0), (2,0), (3,0), etc. I can generate subgroups of...

Homework Statement I would like to know...when trying to take the derivative of a function with a fraction in it ... should I always turn it into a product and use the product rule, thereby dropping the quotient rule most of the time? Or is the quotient rule needed more so in some cases...

Simplify \frac{sinx + tanx}{cscx + cotx} I start with \frac{sinx + tanx}{cscx + cotx} = \frac{sinx + \frac{sinx}{cosx}}{\frac{1}{sinx} + \frac{cosx}{sinx}} At this point I am stuck, I cannot see how we then get from \frac{sinx + \frac{sinx}{cosx}}{\frac{1}{cosx} + \frac{cosx}{sinx}} =...

Homework Statement Let A be a symmetric n x n - matrix with eigenvalues and orthonormal eigenvectors (\lambda_k, \xi_k) assume ordening: \lambda_1 \leq...\leq \lambda_n We define the rayleigh coefficient as: R(x) = \frac{(Ax)^T x}{x^T x} Show that the following constrained problem...

Homework Statement Prove that is m, n, and d are integers and d divides (m-n) then m mod d = n mod d. Homework Equations Quotient Remainder Theorem: Given any integer n and positive integer d, there exists unique integers q and r such that n=dq + r and 0\leqr<d and n mod d = r. The...

Domain Functions: f(x) = 4sq.root(1-x2), the 4 is on the outside f(s) = sq.root(s -1)/s-4 f(x) = x-4/sq.root x I have absolutely no clue how to go about doing any of these, I take notes but I cannot piece it together; I have never felt so helpless at anything. This isn't homework I...

what is quotient rule for higher order derivatives ? i mean the one analogous to http://en.wikipedia.org/wiki/Leibniz_rule_%28generalized_product_rule%29" .

Homework Statement Find: Lim | x2+x-12 |-8 / (x-4) x --> 4 Homework Equations The Attempt at a Solution My answer is 9. It it right ? or there is not a limit for F(x) when x --> 4

Is a projection a quotient map? I think a quotient map is an onto map p:X-->Y (where X and Y are topological spaces) such that U is open/closed in Y iff (p)-1(U) is open/closed in X. And a projection is a map f:X-->X/~ defined by f(x)=[x] where [x] is the equivalent class (for a...

Hello everybody, I'm a bit stuck here. I have a problem tha goes like this: Let R be a principal ideal domain (PID). Let D a subset of R which is multiplicatively closed. Show that the ring of quotients D^(-1)R is a PID too. I've tried several different ways but I couldn't get to the...

1. Show that every element of the quotient group G = Q/Z has finite order. Does G have finite order? he problem statement, all variables and given/known data [b]2. This is the proof The cosets that make up Q/Z have the form Z + q, where q belongs to Q. For example, there is a...

Can anyone post me a clear example of how to compute the quotient space U/V from a vector space U and subspace V? I've seen many formal definitions but I'm a little stuck on practical use. I'm particularly interested in an example that shows how U and V being over the reals (for example) can...

Find the gradient of F(s,t) = f(x(s,t), y(s,t)) where f(x,y) = y/x x = s^2 + t^2 y = s^2 - t^2. I'm not sure how to even start the problem. Could someone point me in the right direction?

i am confused about how to find the subgroup of a quotient group given a generator. for example, a lot of problems give as the group Z/nZ with n very large. how do you find the subgroup given a generator? thanks!

Homework Statement Using difference quotient I am trying to find f '(0) for 2^x. Basically my question is a questions of algebra but I will show you what I have done thus far. the limit is as x -> 0 \frac{2^x - 2^0}{x - 0} \frac{2^x - 1}{x} So my question is what can I do to get x...

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...