Equivalence Definition and 747 Threads

Our math Teacher asked us to find how many equivalence relations are there in a set of 4 elements, the set given is A={a,b,c,d} I found the solution to this problem there are 15 different ways to find an equivalence relation, but solving the problem, i looked in Internet that the number of...

Mass energy equivalence question Hi I'm just wondering how energy is equivalence to the mass if E=mc ^2. I don't understand why you must times it by the speed of the light^2. And if the mass is proportional to the gravity, is it right to consider gravity as energy?

Homework Statement ~ is a equivalence relation on integers defined as: a~b if and only if 2a+3b is divisible by 5 What is the equivalence class of 0 Homework Equations The Attempt at a Solution [0] = {0, 5n} n is an integer My reasoning for choosing 0 is that if a=0...

Homework Statement Relation on set of integers. a~b if and only if 2a+3b is divisible by 5 show that ~ is an equivalence relation Homework Equations The Attempt at a Solution I have already proved that the relation is reflexive and symmetric, but I'm unsure of my approach...

Equivalence Principle: A hint on how to start! Hi, I have no idea where to start. 1. Statement Problem Let X be a non empty set with a equivalence relation ~ on it. Prove that for all x,y\inX, [x]=[y] if and only if x~y. Homework Equations For the Equivalence Relation to exist, it...

imagine we rotate a vector centered at the origin with euler angles alpha,beta,gamma. now the question is, can we do this rotation by the means of defining a vector N(which its length is 1).and rotating the vector zeta radians counter clockwise around N? I think it must be possible and I want...

Since all the observables in QM is on the form \langle \alpha |A| \beta \rangle where A is an observable, one can transform the observables and states like A \to A' = UAU^{-1} \ \ \ |\beta \rangle \to |\beta '\rangle = U |\beta \rangle where U is a unitary transformatioin. These...

Hi I like to play basketball and am curious as to how much force is required to jump a certain height I have tried to calculate it but I get stuck at energy and am not sure how to find the force as it is difficult to measure without equipment and being very accurate and precise (unless there is...

I am studying set theory and I came across various definitions like equivalence relations, partial order relations, antisymmetric and many more. I am aware mathematicians don't care about real life applications but still - why are we defining so many relations? What is the use of defining...

Homework Statement Prove that (\mathbb{R}, <) and (\mathbb{R} \backslash \{0\}, <) are elementary equivalent using the fact that there exist countable models (A, <_0) and (B, <_1) which are elementary equivalent with (\mathbb{R}, <) and (\mathbb{R} \backslash \{0\}, <) respectively...

Hi, I have stumbled upon PF many times through Google, but this is my first time posting. Hopefully, someone will be able to help me out. My question is about the concept of elementary equivalence in logic. According to my book, two structures A and B are elementary equivalent if: for every...

Homework Statement Show that ||A||_1 \le \sqrt{n} ||A||_2 , ||A||_2 \le \sqrt{n} ||A||_1 , where ||A||_1 = \max_{1\le j\le n}\sum_{i=1}^n |a_{ij}| \\ ||A||_2 = (p(A^TA))^\frac{1}{2} \\ p(B) = \max|\lambda_B| with A,B\in \mathbb{R}^{n,n}, i,j\in[1...n] , \lambda_Athe eigenvalues of matrix A...

In a lot of practical situations it is simply assumed the canonical and microcanonical ensemble give the same predictions, and that's fine, but I'm interested in a more exact statement of when they are indeed equivalent (in the thermodynamic limit). First of all, a thermodynamic limit must...

Hi, I am stuck on two problems from Allen Hatcher's book, Algebraic Topology. Homework Statement 4. A deformation retraction in the weak sense of a space X to a subspace A is a homotopy f_t: X→X such that f_0=1_X (the identity map on X), f_1(X) \subset A, and f_t (A) \subset A for all t. Show...

I have been trying to work this out for the last couple of weeks, but I just keep getting the Newtonian deviation in angle for a path of a photon traveling from x=-∞ to x=∞. At first I tried putting the actual path into a computer simulation, transforming back and forth between the hovering...

I have been reading my books section on the weak equivalence principle over and over again, I think I understand it now, this is my understanding. Since all particles are accelerated by gravity at the same rate, no matter what they're composition or mass are, only a frame free falling with...

Let G be a group and let H be a subgroup of G. Define ~ as a~b iff ab-1εH. Define ~~ as a~~b iff a-1bεH. The book I am using wanted us to prove that each was an equivalence relation, which was easy. Then, it asked if these equivalence relations were the same, if so, prove it. My initial...

Homework Statement (b) Show that (p → q) ∨ (p→ r) is equivalent to p → (q ∨ r). Homework Equations the ~ means negate The Attempt at a Solution Im not sure if i did this correctly (p → q) ∨ (P → r) (~p∨q) ∨ (~p∨r) used the conditional law p→q equivalent to ~p∨q...

Homework Statement Show that: 7x≈3 mod(15) Homework Equations From the given above I think it should be: 7x-3=15n The Attempt at a Solution I tried factoring this in various ways to show that either said was a factor of the other, but I'm struggling here. But I don't know...

Hello Forum, According to general relativity, objects in a gravitational field behave similarly to objects within an accelerating enclosure. For example, an observer will see a ball fall the same way in a rocket as it does on Earth, provided that the acceleration of the rocket provides the...

Homework Statement Which of these relations on {0, 1, 2, 3} are equivalence relations? Determine the properties of an equivalence relation that the others lack a) { (0,0), (0,2), (2,0), (2,2), (2,3), (3,2), (3,3) } This one is not reflexive Homework Equations I understand that...

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...

Given the set S, where aSb if and only if a - b \in Z It is asking for the equivalence class and the answer given is S has only one equivalance class for each real number x such that 0 ≤ x < 1. the class [x] is given by {x + k : k \in Z} i dun get it, since S is a set of relation where a...

I'm having a little trouble understanding the equivalence principle explanation of the twin paradox. I understand that the resolution to the paradox according to the equivalence principle is that the non-traveling twin has a higher gravitational potential energy in the pseudo-gravitational...

I've seen a lot of statements regarding Einstein's equivalence principle. Many formulate it to say that no experiment can distinguish between a reference frame in a gravitational field and an accelerating reference frame. But - isn't is true that in a gravitational field, tidal effects are...

I had a physics test at school recently. One of the questions was based on the equivalence principle, going something like this: Two clocks in a spaceship that is accelerating. One at the bottom and one at the top of the space ship. Now think that the spaceship is so far away from any object in...

An accelerating elevator is locally equivalent to a gravitational field. When this is applied to light, it is seen that a horizontal beam of light in the accelerating frame curves and the effect is same in a gravitational field, but wouldn't this violate the constancy of the speed of light?

Why do we say that Galileo's experiment at Pisa is an illustration of Equivalence Principle? All we know is that G* (mass of earth)*(gravitational mass of object)/(R^2) = (intertial mass of object)*a Therefore, a=G* (mass of earth)*(gravitational mass of object)/(R^2 * (inertial mass of...

1 = √(1) = √(-1)(-1) = (√-1)(√-1) = i.i = i^{2} = -1 Is this a correct equation?? anythings wrong with this? i think theoretically it is correct but it seems like √(1) = √(-1)(-1) √(1) = √(1)(1) also! so how to explain this??

Homework Statement Relation is x^y = y^x...x and y belong to integersHomework Equations The Attempt at a Solution Well i have already proven that they are reflexive and symmetric. I have doubt with transitive I did the follwoing way x^y = y^x...(1) and y^z = z^y...(2) from(1) x^z = y^(zx/y)...

Hello All, Let m be a mass, equivalent to energy E such, that E=mc^{2}. Does it follow that c is the cosmic speed limit? ====================================== To say the above with more words: 1) m is a mass 2) in some process, it is established that through...

Homework Statement I have to show that if there is a mobius transformation p such that m=p°n°p^{-1} forms an equivalence class. Homework Equations need to show that aRa, if aRb then bRa, and if aRb and bRc then aRc The Attempt at a Solution well.. for aRa I somehow need to show...

Hello All, is the following in principle, correct: Scenario A:------------ 1A) A box with mass M contains mass m, their weight is g(m+M) 2A) the mass m is (somehow) converted to energy E=mc^{2} 3A) at this moment, the box still has weight g(m+M) Scenario B: ---------- 1B) A box...

Hello, I'm having some issues with Love's equivalence principle. I'm studying Balanis' "Antenna theory" (1997), here's a (legal) fragment of the section in question: http://www.uniroma2.it/didattica/ap1/deposito/02_2-Balanis-Equivalence_Theorems.pdf I'm trying to understand the following...

What defines what element results when energy is converted into matter? i.e. the protons/electrons/neutrons

I'm wondering if someone could briefly explain how I can determine the equivalence class of relation? I understand that first you must test the relation to see if is true for the properties, reflexive, symmetric, and transitive. But my main problem is once that is done how can I get the...

There seem to be two definitions for a regular representation of a group, with respect to a field k. In particular, one definition is that the regular representation is just left multiplication on the group algebra kG, while the other is defined on the set of all functions f: G \to k . I do not...

I am required to show that (i)in the upper limit of very high energies, the Born and eikonal identities are identical. (ii)that the eikonal amplitude satisfies the optical theorem. Regarding (i) I think it will involve changing from an exponential to a trig(Euler's theorem) but I could be...

(X,\rho) is a pseudometric space Define: x~y if and only if ρ(x,y)=0 (It is shown that x~y is an equivalence relation) Ques: If X^{*} is a set of equivalence classes under this relation, then \rho(x,y) depends only on the equivalence classes of x and y and \rho induces a metric on...

Why in equivalence class of N of even number and odd number, the even number are taken as related to 0 and odd number are related as 1 i.e [0] and [1]. Instead of [0], even number can also be related to [2] or [4]? Or [2] or [4] could also be taken, as it is just an convention. Thanks.

Homework Statement Let H and K be subgroups of the group G. Let a,b \in G and define a relation on G by a ~ b if and only if a = hbk for some h \in H and k \in K. Prove that this is an equivalence relation.Homework Equations a = hbkThe Attempt at a Solution The goal is to prove the reflexive...

Homework Statement R is a relation on the integers, xRy if x^2=y^2. Determine the distinct equivalence classes. Homework Equations [x]={yεZ}|yRx} Where Z is the set of integers The Attempt at a Solution [n]={-n, n} where n is an integer is this correct?

Hi all, first post, please bear with me! I am trying to understand Lagrange's Theorem by working through some exercises relating to the Orbit-Stabilizer Theorem (which I also do not fully understand.) I think essentially I'm needing to learn how to show cosets are equivalent to other things or...

Homework Statement To make it simpler just assume n is a positive even integer though it is also true when this is not the case but then the limits on s will be half an odd integer(s). We also assume L is a non-negative integer and s goes by unit steps in the summation as usual...

Hi all, I was just looking for some assistance in reconciling the equivalence principle and the varying intensity of the gravitational field. (I'm in high school so go easy on me, I'm just studying Einstein's Relativity for the general reader). For convenience let's keep with Einstein's example...

Homework Statement The question is to show that Cos^2(x) + Sin^2(y) = 1 is an equivalence relation. The Attempt at a Solution I know that there are three conditions which the equation must satisfy. (reflexivity, symetry, transitivity) For reflexivity I tried: Cos^2(x) - Cos^2(x) = Sin^2(y)...

Homework Statement This isn't strictly a homework question as I've already graduated and now work as a web developer. However, I'm attempting to recover my ability to do physics (it's been a few months now) by working my way through the problems in Analytical Mechanics (Hand and Finch) in my...

This seemed like the least inappropriate place for this. Feel free to move it if I am wrong. Generally speaking, two computational models are equivalent if they recognize the same class of languages. In the case of models that can run indefinitely, we also have the problem of decidability...

The completeness properties are 1)The least upper bound property, 2)The Nested Intervals Theorem, 3)The Monotone Convergence Theorem, 4)The Bolzano Weierstrass, 5) The convergence of every Cauchy sequence. I can show 1→2 and 1→3→4→5→1 All I need to prove is 2→3 I therefore need the proof...

A sphere has charge density \rho=k\cdot r. Using the integral form of Gauss's Law, one easily finds that the electric field is E=\frac{k\cdot r^2}{4\epsilon} anywhere inside the sphere. However, \nabla\cdot E=\frac{k\cdot r}{2\epsilon}, which is half of what should be expected from the...