Equivalence Definition and 747 Threads

In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric and transitive. The relation "is equal to" is the canonical example of an equivalence relation.
Each equivalence relation provides a partition of the underlying set into disjoint equivalence classes. Two elements of the given set are equivalent to each other, if and only if they belong to the same equivalence class.

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  1. H

    Conformal Equivalence: Help with Penrose's Road to Reality

    Hi fellas I have been reading Road to Reality by Roger Penrose, but can't go beyond chapter 8. I do not understand why topological equivalence does not imply conformal equivalence. In particular I cannot really make sense of his argument as to why a thin torus is not conformally the same as a...
  2. N

    Logical Equivalence Made Easy: Simplify (-p^q)v-(pvq) with Standard Rules

    Use the standard logical equivalences to simlify the expression (-p^q)v-(pvq)... fanx folks!
  3. lemma28

    Ellipse: geometric equivalence of two definitions

    I've been stuck with this problem: An ellipse can be defined as 1) locus of points for which is constant the sum of the distances from two fixed points (foci) 2) locus of points for which is constant the ratio between the distances from a fixed point (focus) and a fixed line (directrix)...
  4. P

    Showing the Equivalence of (1-w)(1-w^2)...(1-w^{n-1}) and n

    Homework Statement if w is the nth root of unity, i.e. w= exp(2pi/n i) show: (1-w)(1-w^2)...(1-w^{n-1})=n Homework Equations The Attempt at a Solution since w^(n-a)= complex congugate of w^a terms on the left hand side are going to pair up to give |1-w|^2 |1-w^2|^2... but I'm...
  5. N

    Describe the partition for the equivalence relation T

    For the set A = {1,2,3,4,5,6,7}, determine whether script A is a partition of A. script A = {{1,3,},{5,6}, {2,4},{7}} Describe the partition for the equivalence relation T defined for x,y \in \mathbbc{R} by X T y iff \left[ \left[x \right] \right] = \left[ \left[y \right] \right] where...
  6. C

    Lagarias’ equivalence to the Riemann hypothesis

    Lagarias’ equivalence to the Riemann hypothesis should be discussed, i.e., if hn := n-th harmonic number := 1/1 + 1/2 + · · · + 1/n, and σn := divisor function of n := sum of positive divisors of n, then if n > 1, hn + ehn ln hn > σn. There is a $1,000,000 prize for the proof of this at...
  7. M

    Equivalence Relations and Quotient Sets - Verifying a Claim

    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...
  8. C

    Equivalence of Time Dilation in Different Gravitational and Accelerating Frames

    Two clocks in a gravitational field separated by an altitude of x, exhibit constant time dilation. Two clocks accelerating in free space separated by the length x of their ship, exhibit non-constant time dilation. How is the principle of equivalence sustained when this fundamental measure of...
  9. T

    Solve Equivalence Relation Homework: Prove ~ is Equivalent

    Homework Statement We define a relation ~ for N^2 by: (n, m) ~(k, l) <=> n + l = m + k Show that ~ is a equivalence relation Homework Equations A relation R on a set A is equivalent if R is: reflexive if x R x for all x that is an element of A symmetric if x R y implies y R x...
  10. B

    Proving Equivalence of f(x) and (1/n) Summation of f(x_k)

    Q1. f is a continuous real valued function on [o,oo) and a is a real number Prove that the following statement are equivalent; (i) f(x)--->a, as x--->oo (ii) for every sequence {x_n} of positive numbers such that x_n --->oo one has that (1/n)\sum f(x_k)--->a, as n--->oo (the sum is taken...
  11. L

    Prove that if and [j] are equivalence classes modulo

    Prove that if [i] and [j] are equivalence classes modulo 1. Prove that if [i] and [j] are equivalence classes modulo n such that [i]=[j], then gcd(i,n)=gcd(j,n) 2. Prove that if gcd(a,b)=1 and if c divides b, then gcd(a,c)=1. please help
  12. E

    Equivalence of prime power decompositions

    Homework Statement Let G be a finitely generated abelian group and let T_p be the subgroup of all elements having order some power of a prime p. Suppose T_p \simeq \mathbb{Z}_{p^{r_1}} \times \mathbb{Z}_{p^{r_2}} \times \cdots \times \mathbb{Z}_{p^{r_m }} \simeq \mathbb{Z}_{p^{s_1}}...
  13. S

    Exploring the Equivalence of r=arctan(tan(x)) and r=x

    Homework Statement if i have a polar equation: r=arctan(tan(x)) is that the same as: r=x ?? Homework Equations The Attempt at a Solution
  14. C

    Is Idempotent Equivalence in Rings Transitive?

    One of my books defines a relation which is "evidently" an equivalence relation. It says that two idempotents in a ring P and Q are said to be equivalent if there exist elements X and Y such that P = XY and Q = YX. The proof that this relation is transitive eludes me. There is so little...
  15. E

    Equivalence Relations on the Set of Integers - Homework Solution

    Homework Statement Let S be the set of integers. If a,b\in S, define aRb if ab\geq0. Is R an equivalence relation on S? Homework Equations The Attempt at a Solution Def: aRb=bRa \rightarrow ab=ba assume that aRb and bRc \Rightarrow aRc a=b and b=c since a=b, the substitute a...
  16. M

    Confused about unit vector equivalence

    x is a unit vector \in \Re^{2}. My textbook states that \frac{x}{||x||}=\frac{1}{||x||}x. What is the point of including \frac{1}{||x||}; why do they divide the vector by its length? Edit: I just looked at a book in Google's database, and from what I understand: e.g. \sqrt{{2^2+2^2+1^2}}=3...
  17. D

    Proving Equivalence of a Relation on Real Numbers

    So, here is the problem: Let x,y\in\mathbb{R}, R=\{{(x,y)\in\mathbb{R}^{2}|x= y r^{2}, for some r \in\mathbb{R}\}. Prove that R is an equivalence relation on \mathbb{R}. Relevant equations: R is an equivalence relation on \mathbb{R} if 1. (x,x)\in R for all x\in\mathbb{R} 2...
  18. D

    Vacuum vs. Centrifuge - equivalence

    I've been using a vacuum system to filter solutions. I would like to move it over to a centrifuge. I have been having trouble equating the pressure in the vacuum to the required amount of force required by a centrifuge. Can someone help me out with the math? I'm looking a the vacuum pressure in...
  19. J

    Are the Metrics \rho^{(p)} and \rho^{(q)} Equivalent on \Re^n?

    Show equivalence of family of metrics on \Re^n: \rho^{(p)}:(x,y)\rightleftharpoons(\sum_{i=1}^{n}|x_i-y_i|^p)^{\frac{1}{p}} for p\geq1 The attempt at a solution Want to show for p,q\geq1, p\neq q, that \rho^{(p)} and \rho^{(q)} generate the same topology. I tried two methods: 1. Show that a...
  20. I

    Exploring Equivalence Classes in Rings: Why a-b Instead of a+a?

    So I'm kind of confused about the definition: a-b\in I Why a - b instead of a + b?
  21. Y

    Problem with equivalence principle.

    Einstien gives the example of an observer riding on the rim of a spinning disc. Using his length contracted rulers he measures the perimeter to be greater by a factor of gamma (y), than the 2\pi R he would have measured when the disc was stationary. When he measures the radius with the same...
  22. Y

    Mach's Principle and Equivalence

    In free space far removed from significant particulate matter, inertial reaction of a test mass will be isotropic. But if Mach's Principle is the root cause of inertia, then a nearby massive object should modify the inertia of such a test mass so that its reactance to acceleration will be...
  23. M

    Proving K is Not EC: Equivalence Classes

    QUESTION: Let L be the language with = and one binary relation symbol E. Let epsilon be the class of all L-structures A such that E interpreted by A is an equivalence relation on |A|. K = { A in epsilon | E interpreted by A has infinitley many equivalence classes }. DEFINITIONS: EC...
  24. O

    Galileo's experiment and Equivalence Principle

    FREE FALL OF BODIES 3-body problem (special case) According to the Newton's Mechanics: Let us consider a mass M of radius Ro. At a distance h from the center of this mass M, we place a spherical shell (e.g a spherical elevator) of mass m1 and radius R. Moreover, at the...
  25. H

    Proving an Equivalence Relation: Tips & Examples

    Iam not clear on how to prove a equivalence relation? I know that is has to have three properties reflexivity, symmetry, and transtivity, but I am unsure how to check. For example Iam given f is a function from z to d. R(z,d) = binary relation How do I prove that R(z,d) is a equivalence...
  26. R

    Difference between equivalence and equality with functions

    I feel aggravatingly close to the answer to this one, but have caved in. Using ~ for "is equivalent to" A given question reads: Given that f(x) ~ 3 - 5x + x^3, show that the equation f(x)=0 has a root x = a, where a lies in the interval 1 < a < 2. Clearly asking for the answer here...
  27. P

    Equivalence Relation and the Unit Circle: Understanding R/Z and S^1

    Homework Statement If Z acts on R by n.x=n+x then R/Z is just S^1. CLaims the book But I think R/Z is (0,1) The Attempt at a Solution Any number greater than or equal to 1 is dealt with by the equivalence relation. How does the unit circle come into it? We are dealing only with one...
  28. radou

    Equivalence of continuity and boundedness

    I need a push with the following theorem, thanks in advance. Let X and Y be normed spaces, and A : X --> Y a linear operator. A is continuous iff A is bounded. So, let A be continuous. Then it is continuous at 0, and hence, for \epsilon = 1 there exists \delta > 0 such that for all x from...
  29. A

    Logical Equivalence of x <=> y and (x-->y) ^ ((~x)-->(~y))

    Prove x <=> y is logically equivalent to (x-->y) ^ ((~x)-->(~y)).
  30. N

    Gravitational Charge - Equivalence between Gravitational and Inertial Mass

    Gravitational "Charge" - Equivalence between Gravitational and Inertial Mass My mind is currently in a mess regarding the equivalence of gravitational mass and inertial mass. Yes, I know which comes in which equation and that they have been experimentally observed to be equal, etc., but I'm...
  31. Y

    Equivalence of Clocks in Gravitational Fields: A Thought Experiment

    In a G field, clocks at a lower potential (closer to the mass producing the field) are known to run slower. When the two clocks are brought togther, the upper clock should be found to have accumulated more time than the lower clock. A rocket accelerating at "a" is equivalent to a G field...
  32. maverick280857

    Equivalence of DAlembert's principle and Action Principle

    Hello everyone. I'm reading Goldstein's Classical Mechanics (2nd Ed) and I have worked out the derivation of Euler-Lagrange equation of motion from DAlembert's Principle as described in Chapter 1 and also the Action Integral approach in Chapter 2. I want to understand the equivalence of the...
  33. S

    Equivalence betwen diferent definitions of charge

    I am trying to fit together a few definitions of charge which are being commonly used. On one side we have the Noether charge associated with any invariance. Sure most of you know how it goes, you have a Lagrangian invariant under a (Lie) symmetry group and for every group generator (Lie...
  34. D

    Proving Equivalence Relations with a Given Condition

    Hi, Here is my question. I need to prove the following an equivalence relation. Let A = {1,2,3,4,5} X {1.2,3,4,5} and define a relation R on A by (x1,y1)R{ x2,y2) if x1+y1=x2+y2. I am bit confused how to use the condition x1+y1=x2+y2 to prove for transitive, symmetric and reflexive properties...
  35. Q

    Unitary equivalence vs. similarity

    Two matrices A and B are defined to be unitarily equivalent if there exists a unitary matrix P such that A = P*BP. If one is given two matrices A and B with no P, how do we know if the matrices are unitarily equivalent? I guess what I'm asking is, is it sufficient to check whether the...
  36. JasonRox

    Trouble understanding the path-homotopy equivalence class

    Ok, I'm having trouble understanding the path-homotopy equivalence class. It's kind of blurry when they apply the operation... [f]*[g] = [f*g] ...where [f] is the path-homotopy equivalence class of f. I can see that an element in [f]*[g] is in [f*g], but not the other way around. For...
  37. Y

    Rotation Matrix: Finding Two Expressions & Verifying Equivalence

    Homework Statement A vector x in R^2 is rotate twice through an angle theta (it is rotated through theta and again through theta). Find two expressions for the matrix representing this rotation. Verify that these two expressions are equivalent Homework Equations rotation matrix R=[cos...
  38. J

    Linear functional equivalence in vsp and subsp

    I am writing a solution for the following problem, I hope someone can correct it, because I am not sure what I am missing. Q. V is a finite dim. vsp over K, and W is a subspace of V. Let f be a linear functional on W. Show that there exists a linear functional g on V s. t. g(w)=f(w). Ans...
  39. T

    Equivalence principle & nonlinearity

    How does nonlinearity of the gravitational field equations follow from the equivalence principle? I remember hearing a handwaving example of this & I'm interested in more details.
  40. G

    A simple observation regarding the equivalence of acceleration and gravity

    This is not an attempt at refuting the validity of relativity, i adore Einstein like most girls love their fave rock star. To me, there is a glaring discrepancy between Einstein and Newton and it behooves me, out of shear respect, to make this right in my mind. Take the very simple...
  41. MathematicalPhysicist

    Question on equivalence relation.

    let R be a reflexive and symmetric relation on a set X. let's define a relation S on X. by (x,y) in S iff there exists a finite sequence x0,x1,...,x_n of terms from X such that x0=x xn=y and (x_i,x_i+1) in R for i=0,1,...n-1. now I've proven that S is an equivalence relation, i need to show...
  42. Q

    How Does Equivalence Class Equality Work in Set Theory?

    Homework Statement Prove the following statement: Let R be an equivalence relation on set A. If b is in the equivalence class of a, denoted [[a]] then [[a]]=[[b]]. Homework Equations [[a]], [[a]]=[[b]]; definition of equivalence: a relation R on a set A that is reflexive, symmetric...
  43. L

    Formal definition of the Equivalence Principle(s)

    Considering how the various forms of equivalence principle can lead to ambiguous discussions, I would like to know if more formal definitions for the various forms of the Equivalence Principles are available. I would consider as "formal" all definitions that could be used (formally) to check...
  44. L

    Tests of equivalence principle based on fluid mechanics

    Were there some tests of this kind? Would that have some meaning? Would the fluid world be very strange if the EP was (somewhat) in default? Some brainstorming on fluid mechanics and the Equivalence principle, to take another point of view? Thanks, Michel
  45. D

    Circuit analysis (thevenin equivalence)

    Homework Statement http://img509.imageshack.us/img509/5658/circuituz5.jpg Find the current through resistor X for V = 7V Homework Equations Thevenin equivalence? The Attempt at a Solution Right I was advised a way to do this by assuming a 1A current flows through the resistor...
  46. Loren Booda

    Non-trivial field limits' equivalence?

    What is the field F(r) with the least symmetry and which obeys lim F(r) as r --> oo = lim F(r) as r --> 0 ?
  47. N

    Equivalence Principle: Understand Constant Acceleration & Warped Spacetime

    Sorry for so many questions. In Einsteins equivalence principle, it states that an observer in a close room at 1G accel would not know the difference between that or whether he was standing on the Earth. This makes fine sense to me. But it also states as well that it is independent of...
  48. G

    Quantifier equivalence in set theory

    Homework Statement I have been asked to show that \exists xAP(x)\vee\exists xBP(x) is equivalent to \exists x(A\cup B)P(x) Homework Equations 1) P\rightarrow Q \equiv \neg P \vee Q 2) \neg(P\vee Q)\equiv \neg P \wedge \neg Q 3) P \vee (Q\vee R) \equiv (P\vee Q) \vee R \equiv P...
  49. L

    How can I show that G=8pT implies the equivalence principle ?

    If I am asking this question, this is maybe a proof that I need a strong "back-to-the-basics". Could you give the way? Thanks, Michel
  50. K

    Does the Equivalence principle holds in QM.

    Does the "Equivalence principle" holds in QM. If we know that in QM under an "small" height so z<< R_{earth} (radius of the earth) so the QM equation is (SE): -D^{2}\Phi (z) +2m^{2}gz\Phi (z) =2mE_{n} \Phi (n) SO the wave function is just a "Airy function" and | \Phi (z) |^{2} is an...
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