What is Equivalence: Definition and 744 Discussions

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. RisingSun361

    Equivalence principle and time dilation

    I'm currently reading https://www.amazon.com/dp/0199236224/?tag=pfamazon01-20. The book claims that, according to the equivalence principle, acceleration and gravity have the same effects. So if gravity slows down time, shouldn't acceleration also slow down time? The book seems to state this as...
  2. moriheru

    Proving Equivalence of Decimals in Number Theory

    I have a question, in the field of number theory (Hardy and Wright chapter 9 representation of numbers by decimals) concerning the prove by contradiction of the statement: If Σ1∞ an/10n Σ1=∞bn/10n then an and bn must be equivalent, for if not then let aN and bN be the first pair that differ then...
  3. L

    The (asserted) equivalence of first partial derivatives

    In the solution to a differential-equation problem -- proof of the existence of an integrating factor -- the following statements are made regarding a general function "u(xy)" [that is, a function of two variable that depends exclusively on the single factor "x*y"]...
  4. S

    Give the equivalence classes of this relation

    Homework Statement Give the equivalence classes of the relation aRb if and only if a^4 ≡ b^4 (mod 30) on the set {1, 2, 3, . . . , 15}. Homework Equations Definition of modular arithmetic. Definition of equivalence class. The Attempt at a Solution I can successfully do this problem using...
  5. Jimster41

    Understanding the equivalence principle

    Mentor Note: This thread has been separated out of a previous one to keep discussion of this particular topic open, since the previous thread has been closed. Sorry, the opportunity to ask questions is just too hard to resist. I'm still searching for the right understanding of equivalence. Or...
  6. AdityaDev

    Calculating volume occupied by 1 equivalent of gas

    Its given in my textbook that equivalence of gas = (volume of gas) / (Volume of 1 eq. gas at NTP) How do i calculate volume occupied by 1 equivalent of gas? Example: What would be the volume occupied by 1 eq. of CO2 (Its given 11.2l in my book) Is there a formula for this?
  7. Suraj M

    Equivalence Relation Homework: Proving Transitivity

    Homework Statement If a relation R on N × N is (a,b)R(c,d) iff ad(b+c) = bc(a+d) Homework Equations -- The Attempt at a Solution I got the reflexive and symmetric parts but not the transitive part... here's what i have ## (a,b)(c,d)∈R and (c,d)(e,f)∈R## To prove ##(a,b)(e,f) ∈ R## .i.e...
  8. W

    Models of the Reals of All Cardinalities and Equivalence

    We know that different models of the Reals are elementary equivalent but not isomorphic. In the hyperreals, we have that the Archimedean principle does not hold but it does hold in the standard Reals ( i.e., the hyperreals allow for the existence of indefinitely-small and indefinitely -large...
  9. T

    What is the equivalence point on this graph?

    I am attempting to do a lab that requires the determination of an unknown diprotic acid using a titration curve. I have graphed my data but I am unsure what point to use as my eq. point. In my lab manual it says I may use the second or first eq. point but the second is often clearer. I am not...
  10. C

    MHB Proof of Unique Equivalence Relation on Set w/ Partition Classes

    Hello, I've to construct a proof of the following statement: Prove that if S is a set and S_1... S_k is a partition of S, then there is a unique equivalence relation on S that has the S_i as its equivalence classes. I'm really not sure how to go about this proof at all, so any help would be...
  11. N

    Equivalence Relations on Z: Proving m~n and Describing the Partition

    Prove that the following is an equivalence relation on the indicated set. Then describe the partition associated with the equivalence relation. 1. In Z, let m~n iff m-n is a multiple of 10.2. The attempt at a solution Reflexive: m-n = 0 0 ∈ Z, and 0 is a multiple of every number...
  12. M

    MHB Equivalence Between NFA and DFA: Q, Q` & $\Sigma$

    Heloo! :o I am looking at the equivalence between a NFA and a DFA. NFA: Q={q1,q2} DFA: Q`=P(Q} When $a\in \Sigma, Q_I, Q_j \in Q`$ which is sufficient and necessary condition so that $ Q_I \overset{a}{\rightarrow}Q_j$?
  13. 1

    What is the equivalent capacitance of this formation?

    http://<a href="http://imgur.com/UwulXqg"><img src="http://i.imgur.com/UwulXqg.png" title="source: imgur.com" /></a>[/PLAIN] 1. Homework Statement Find equivalent capacitance of this formation from a to b, C1=4.1 uF, C2=6.86 uF, C3=5.03 uF, C4=5.92 uF http://imgur.com/UwulXqg Homework...
  14. M

    Is Mass Equivalence Explained by the Relationship Between Energy and Gravity?

    In formulating GR, Einstein was motivated by the otherwise strange coincidence of the equivalence of gravitational and inertial mass. He removed this by discovering that trajectories determined by gravitation are actually inertial. However, gravity's source is still the tensor T, which is...
  15. Calpalned

    Principle of Equivalence and Acceleration in Deep Space

    Homework Statement I know that the principle of equivalence states that a person in a window-less box will not be able to tell if s(he) is in a stationary box on Earth or in a box that is accelerating upward at a constant rate. I am assuming that the acceleration of the rocket is 9.8 m/s^2...
  16. J

    Electric dipole equivalence principle paradox?

    Imagine an electric dipole with charges ##+q##, ##-q##, mass ##m## and size ##d##. Assume this dipole is oriented horizontally and is sitting at rest on top of some weighing scales on earth. As the charges are at rest the weight of the dipole is just ##-mg##. But by the principle of...
  17. D

    Difference between equivalence and equality

    Apologies if this is in the wrong forum, but I chose to post here as the question pertains to equivalence relations and classes. Sorry if it's such a trivial question, but what is the mathematical difference between equivalence and equality? My understanding is the following, but I'm a little...
  18. E

    Equivalence Principle Test Kits

    I'd like to see for myself the Equivalence Principle. Would there be a commercially available test kit like perhaps a small iron and cotton enclosed in vacuum tubes and they are to fall at same time and hit the bottom at same time with indicator lights to tell if they hit at same time or at bit...
  19. PsychonautQQ

    Equivalence mapping from integers to rationals

    Homework Statement Let * and = be defined by a*b means a - b is an element of the integers and a = b means that a - b is an element of the rationals. Suppose there is a mapping P: (* equivalence classes over the real numbers) --> (= equivalence classes over the real numbers). show that this...
  20. D

    A question on defining vectors as equivalence classes

    I understand that a tangent vector, tangent to some point p on some n-dimensional manifold \mathcal{M} can defined in terms of an equivalence class of curves [\gamma] (where the curves are defined as \gamma: (a,b)\rightarrow U\subset\mathcal{M}, passing through said point, such that \gamma (0)=...
  21. D

    Better Equivalence Principle Explained

    I am new to relativity, so bare with me. I do not want to argue against it; however, one must admit it is a rather awesomely unwieldy theory. In my self education on the subject, I am bothered by the Equivalence Principle, mainly this: An elevator accelerating through space is no different from...
  22. H

    Equivalence Principle: Feel Difference Between Situation A and B?

    Here are the 2 situations: Situation A: a person floating in space far away from any mass (i.e. the person is in zero-gravity environment) Situation B: a person undergoes a free-fall from the top of a roller coaster (assuming that there's no air resistance) My question is, would the person in...
  23. RUber

    2D Green's Function - Bessel function equivalence

    Homework Statement This is not a homework problem per se, but I have been working on it for a few days, and cannot make the logical connection, so here it is: -- The problem is to show that ##\frac{1}{4\pi} \int_{-\infty}^{\infty} \frac{ e^{-\sqrt{\xi ^2 + \alpha^2 } |y-y'| + i \xi (x-x')...
  24. WCOLtd

    Equivalence of Matter Energy and Space-Time?

    This is probably a stupid question, but is there an equivalence between matter, energy and space time? Like for instance if you had a ball of 1 kilogram, could you convert it into volumetric space and have everything around it be further away? When two or more objects are going further away from...
  25. PsychonautQQ

    What's the difference between an equality and an equivalence

    Might sound like a stupid question, but if somebody could give me a mathematical description of the difference between equality and equivalence that might be really interesting.
  26. DiracPool

    GR Journal: Equivalence Principle & Tidal Forces

    Say we have an elevator moving through deep space at a constant velocity. We have a light emitter at the "top" of the elevator pointing towards a detector at the bottom of the elevator. We shine the light and notice that the frequency at which the light was emitted at the top is the same as...
  27. Ahmed Abdullah

    Equivalence of waveparticle duality to uncertainty principle

    http://www.sciencedaily.com/releases/2014/12/141219085153.htm It says "Here's a nice surprise: quantum physics is less complicated than we thought. An international team of researchers has demonstrated that two peculiar features of the quantum world previously considered distinct are different...
  28. M

    Writing Elements in $L^p(X,Y,\mu)$ Mathematically

    \begin{equation*}Let\text{ } (X,\mathcal{A} ,\mu ) \text{ }be \text{ }a \text{ }complete\text{ } \sigma -finite\text{ } measure\text{ } space \\and \text{ }Y \text{ }be \text{ }a \text{ }separable\text{ } Banach\text{ } space\text{ } supplied \text{ }with \text{ }the \text{ }norm\text{ }...
  29. L

    Light travel time since dark energy - matter equivalence

    So I'm looking to find the distance light has traveled since matter - dark energy equivalence. Assume dark energy dominance from equivalence. Space-time has flat geometry and Ω0m = 0.315 , Ω0 = 1 Thus: Ω0Λ will equal Ω0 - Ω0m= 0.685 ρ0m (1 + zeq)3 = ρ0Λ where: ρ0m = ρ0c * Ω0m and: ρ0Λ =...
  30. H

    Mastering Logical Equivalence Proofs: (P<->Q) ⊣ ⊢ ~(P<->~Q)

    ~(P<->Q) ⊣ ⊢ (P<->~Q) I'm suppose to write the proof for this equivalence but I can't figure it in either direction The closest I got was (P->~Q) from ~(P<->Q) but I can't figure anything else out
  31. evinda

    MHB Equivalence relation-equivalence classes

    Hello! (Smile) We are given the set $E=\{d,e,f \}$, $d,e,f$ different from each other and the relation $I_{E}=\{ <x,x>: x \in E\}$. Prove that $I_{E}$ is a set. In addition, show that the relation $I_{E}$ is an equivalence relation in $E$ and find all the equivalence classes. That's what I...
  32. evinda

    MHB Equivalence Relations: Explaining $I_A$, $\rho^{-1}$ and $\rho \circ \rho$

    Hi again! (Smile) If $\rho$ is an equivalence relation, could you explain me why the following relations stand? (Thinking) $I_A \subset \rho$ $\rho^{-1}=\rho$ $\rho \circ \rho \subset \rho$
  33. P

    The (non-?) flatness of a uniform gravitational field and the equivalence principle

    Desloge (1989) published the article: ## '##Non-equivalence of a uniformly accelerating reference frame and a frame at rest in a uniform gravitational field## '##. His result, briefly resumed: a uniform gravitational field is not flat, is quite interesting. But the way he proves the result is...
  34. T

    Proof of equivalence between nabla form and integral form of Divergence

    Does anybody knows how you can reach one form of the divergence formula from the other? Or in general, why is the equivalence true?
  35. C

    Energy and matter equivalence

    The question was : describe the first law of thermodynamics. I said matter can't be created or destroyed. I got marked wrong, my argument will be that the laws of thermodynamics can be written in different ways. Einstein famous equation relates matter and energy. There is also the matter-energy...
  36. H

    Circuit resistance equivalence

    Homework Statement For the circuit shown below, find the R equivalent resistance between terminals A and B. http://puu.sh/bX1Sp/33db39d1c0.png Homework Equations parallel and series circuit definition[/B]The Attempt at a Solution I admit that I don't even know where to begin with this...
  37. E

    Electromagnetism equivalence theorem

    Hello! In http://my.ece.ucsb.edu/York/Bobsclass/201C/Handouts/Chap1.pdf, pages 19-20, the Love's Theorem in Electromagnetism is declared. In presence of some electric sources \mathbf{J} and magnetic sources \mathbf{M} enclosed by an arbitrary geometrical surface S, which produce outside S a...
  38. E

    Understand Mass-Energy Equivalence: Einstein's Thought Exp.

    In the Wikipedia entry on mass-energy equivalence*, there is a reference to an elementary derivation of mass-energy equivalence in an article by Rohrlich (AJP 1990), which apparently expands on the original thought experiment by Einstein (ie observer in middle of moving train, light pulses...
  39. E

    Equivalence Principle in muon experiment?

    Someone noted that the famous muon half-life experiment, supporting time dilation in SRT, seems to violate the equivalence principle of GRT. The very large radial acceleration in the experiment does not appear to cause any additional slowing. The acceleration does not seem to have an equivalent...
  40. J

    Equivalence of Lebesgue Measurability

    Homework Statement Hello Physics Forums Community. I am having some difficulties with the following problem dealing with Lebesgue Measure and its equivalent interpretation. I will first include the definitions which I am using and then the problem statement, they are coming straight from...
  41. P

    Proving Logical Equivalence with Algebraic Substitutions

    Hi, I have to prove the following logical equivalence using algebraic substitutions: (p v ~q) v ~q → (r v p) ∧ ~q ≡ ~r v q → (q v p) ∧ (~p v ~q) I've already done the truth table for this problem and proved they are logically equivalent but am not sure how to go about using algebraic...
  42. J

    Plane impact equivalence to force in time

    The problem comes up as part of a final master's project in FEM analysis. The project consists in calculating the oscillation response of the World Trade Center towers after the plane impacts. The particular problem I'm having trouble is that I need to transform the plane impact of which mass...
  43. J

    Equivalence principle to argue mass and weight are the same?

    Hi pf, I have recently watched a YouTube physics video from SixtySymbols channel which is a channel of short physics videos presented by professors of physics from Nottingham University. The video (pasted below) argues that because of inertial mass and gravitational mass being equivalent this...
  44. agnishom

    Does the Higgs mechanism explain the equivalence of gravitiational and

    I am not well versed in the idea of Higgs mechanism. However, I was wondering if it does explain the problem of equivalence of gravitational and inertial mass. Also, what is the modern accepted definition of mass?
  45. Greg Bernhardt

    What is the Mass–energy equivalence

    Short answer: Mass and energy are NOT equivalent in general. Confusion can arise due to multiple definitions of mass (see our https://www.physicsforums.com/threads/what-is-relativistic-mass-and-why-it-is-not-used-much.796527/ ), but using the modern convention of identifying the word "mass"...
  46. Greg Bernhardt

    What is an Equivalence Relation and its Examples?

    Definition/Summary A relation on a set A is a set of ordered pairs (a,b) of elements ofA (a subset of A \times A). An equivalence relation \sim\ \subseteq A \times A is a relation which is reflexive, symmetric and transitive. In other words, the relation \sim on A is an equivalence...
  47. D

    Equivalence classes of proper classes

    Suppose we have a class equivalence relation ~ on a proper class C. Can we form the class of equivalence classes C/~? What axioms do we need, assuming we base ourself on NBG? I can't seem to prove that this class exists with the axioms stated. I.e., the proposition that there exists a class D...
  48. W

    SU_L x SU_R and SU_V x SU_A equivalence

    I want to show that the group SU_L(N)\times SU_R(N) is the same as SU_V(N)\times SU_A(N) - i.e. that it is possible to rewrite the transformation: \begin{cases} \psi_L \to \psi'_L=V_L\,\psi_L\\ \psi_R \to \psi'_R=V_R\,\psi_R \end{cases} , where V_L and V_R are N\times N SU(N)...
  49. E

    Proof involving numerical equivalence of sets

    Homework Statement Show that for a set A\subsetN, which is numerically equivalent to N=Z+, and the set B = A \cup{0}, it holds that A and B are numerically equivalent, i.e., that A \approxB Hint: Recall the definition of A≈B and use the fact that A is numerically equivalent to N. Note...
  50. Math Amateur

    MHB Definition of epimorphism and equivalence to 'surjectivity'

    I am reading Paolo Aluffi's book Algebra: Chapter 0 which takes a (moderately) category theory oriented and infused approach to algebra. I am studying chapter 1: Set Theory and Categories and need help with formulating a definition of an epimorphism and with then proving it to be surjective...
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