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. C

    MHB Can you explain how the law of logic was used to reach this conclusion?

    New to set and graph theory and need help on how to approach these exercise questions: For each of the following relations, state whether the relation is: i) reflexive ii) irreflexive iii) symmetric iv) anti-symmetric v) transitive Also state whether the relation is an equivalence or partial...
  2. e2m2a

    I An exception to the principle of equivalence?

    Not sure of the dynamics of the following situation. Suppose there is a mass that can slide on a straight track with one degree of freedom. Assume no friction in this scenario. On top of this mass is a track that forms a quarter circle. On this track is a sphere that can slide(again assume...
  3. F

    I Inertial & non-inertial frames & the principle of equivalence

    One particular form of the equivalence principle states that The laws of physics for freely falling particles in a gravitational field are locally indistinguishable from those in a uniformly accelerating frame in Minkowski spacetime My question is, does one arrive at this conclusion from a...
  4. F

    I Understanding the Einstein Equivalence Principle

    The Einstein equivalence principle (EEP) states that “The outcome of any local non-gravitational experiment in a freely falling laboratory is independent of the velocity of the laboratory and its position in spacetime.” I’m trying to make sure I’ve understood this correctly. I’m I correct to...
  5. B

    I Clock hypothesis, gravity time dilation and Equivalence Principle

    1. The Clock Hypothesis states that the rate of a clock does not depend on its acceleration but only on its instantaneous velocity. This has been experimentally verified at very high accelerations. 2. A clock in a gravitational field experiences time dilation and runs slower that one not in a...
  6. S

    I Spinning disk, length contraction, & equivalence principle

    Suppose we have a spinning disk with a very fast spin, an observer in the center, and an observer on the edge. Suppose that the observer on the edge measures the circumference of the spinning disk. (1) Now, the observer on the edge at a given instant will be moving at a faster speed than the...
  7. N

    Thermodynamics equivalence between entropy in two ensembles

    So while practicing statistical mechanics problems I was faced with the following problem : calculate the entropy as function of energy for an ensemble of harmonic oscillators ( the Hamiltonian is ##\sum_{i=1}^N \frac {p_i^2} {2m} + \frac {m\cdot\omega\cdot q_i^2} 2##) ). Now the official...
  8. T

    I How to prove gravitational mass & inertial mass equivalence?

    There is a video on YouTube where Sean Carroll says for Newton it was just an accident that inertial mass equals gravitational mass, but with the general theory of relativity it became obvious that it has to be so. How does one see that? My own attempt has been consisting of transforming...
  9. davidge

    B Einstein's Equivalence Principle: What's New?

    The equivalence principle states that the laws of physics are the same in any inertial frame. Translating this into mathematics language, the equivalence principle states that a given equation should retain its form when one transform between the coordinates of two intertial frames, correct...
  10. N

    B Center of momentum and mass energy equivalence

    told so in another thread that two photons traveling in opposite directions "have mass". I understood the concept of centre of momentum, an inertial frame of reference and rest mass. My cynical reply wasn't conveyed and I got well thought out and accurate replies. the COM is an idealized point...
  11. M

    MHB Show Equivalence of AB Parallel to XY | Wondering

    Hey! :o We have the triangle ABC. We have that $X\in AC$, $Y\in BC$ and $Z=AY\cap BX$, where $X,Y\neq A,B,C$. I want to show that AB is parallel to XY iff $\frac{|CX|}{|CA|}=\frac{|ZB|}{|ZX|}=1$. Could you give me a hint what we have to show? When AB is parallel to XY then we have that then...
  12. S

    MHB Formal Proofs in Maths: Establishing Equivalence

    From the book "FORMAL PROOFS IN MATHS "(Amazon.com),page 101 ,exercise19 ,Iread: Establish the equivalence between: 0<1,.........0<A\Longrightarrow 0<\frac{1}{A},............AC<BC\wedge 0<C\Longrightarrow A<B
  13. Y

    MHB Partial Order Relation and Equivalence Relation

    Hello all, If R is a partial order relation, is it true to say that \[R\cup R^{-1}\] \[R^{2}\] \[R\cap R^{-1}\] Are equivalence relations ? Regarding the first one, I think that the answer is yes. If \[xRx\] then it remains after the union. Asymmetry means that \[xRy\] without \[yRx\]...
  14. Daniel Petka

    B Where Did the Energy Go in Light Waves When the Universe Expanded?

    https://www.physicsforums.com/attachments/198775 The expansion of the universe stretches the wavelength of EM-waves. Though the energy equivalence says energy can't be destroyed... so where did the energy go?
  15. K

    B What Does the Notation = in Equivalence Classes Conclude to?

    Please refer to the video at 37:02 from the link above. I am struggling with the notation "=" of the property (a) which concludes to [a]=[m]. shouldn't it be [a]⊆[m], just like [m]⊆M.
  16. Y

    MHB Different Number of Equivalence Relations

    Hello all, I have a few questions related to the different number of equivalence classes under some constraint. I don't know how to approach them, if you could guide me to it, maybe if I do a few I can do the others. Thank you. Given the set A={1,2,3,4,5}, 1) How many different equivalence...
  17. evinda

    MHB Show Equivalence: $a^d \equiv 1 \pmod n$

    Hello! (Wave) Suppose that $p,q$ are two distinct primes , $n=pq$ and $d=gcd(p-1,q-1)$. I want to show that if $a^{n-1} \equiv 1 \pmod{n}$ for some $a$ then $a^d \equiv 1 \pmod{n}$. That's what I have tried: $ a^{pq-1}\equiv 1 \pmod n \Rightarrow a^{pq-p+p-1}\equiv 1 \pmod n \Rightarrow...
  18. A

    Relativity and Equivalence of Mass and Energy

    Homework Statement An electron is accelerated to a speed that is 99 percent the speed of light, and is moving through a 2-km-long tunnel. The rest mass of the electron is 9.11*10^-31 kg. What is the mass of the electron at this speed? c= speed of light Homework Equations t= (tsubscript(o))/...
  19. S

    I Homotopy Equivalence: Definition & Examples

    Hello! I am a bit confused about the definition of homotopy for loops. So it looks like: Let ##\alpha, \beta : I -> X## be loops at ##x_0##. They are said to be homotopic if there is a continuous map ##F : I \times I -> X## such that: ##F(s,0)=\alpha (s), F(s,1)=\beta(s), F(0,t)=F(1,t)=x_0## for...
  20. binbagsss

    Elliptic functions, periodic lattice, equivalence classes

    Homework Statement ##\Omega = {nw_1+mw_2| m,n \in Z} ## ##z_1 ~ z_2 ## is defined by if ##z_1-z_2 \in \Omega ## My notes say ##z + \Omega## are the cosets/ equivalence classes , denoted by ##[z] = {z+mw_1+nw_2} ## Homework Equations above The Attempt at a Solution So equivalance classe...
  21. S

    B Mass-Energy Equivalence: Does E=mc2 Apply in Systems at Rest?

    Let's assume that a system has zero total momentum. The following relationship between mass and energy should apply: E=mc^2. If a system is overall at rest, does that mean that any internal changes to that system, assuming they leave the system with non-negative mass, will not be able to...
  22. M

    Show Regular Homotopy is an Equivalence Relation

    Homework Statement Show that regular homotopy of regular curves γ : I → Rn is an equivalence relation, that is: i) γ ∼ γ (where the symbol ∼ stands for “regularly homotopic”); ii) γ ∼ γ˜ implies ˜γ ∼ γ; iii) γ ∼ γ˜ and ˜γ ∼ γˆ implies γ ∼ γˆ (here you have to use a smoothing function)...
  23. F

    A On the equivalence of operator vs path integral in QFT

    I have read many textbooks and googled google times for a clear explanation, but I could not find one. How does raising and lowering -annihilation/ creation-(is that energy or particle number?) translate to transition probabilities of path integral.
  24. I

    I Proving equivalence between statements about a sequence

    Hello Let ##(x_n)_{n=1}^\infty## be a real sequence and ##L \in \mathbb{R}##. Consider the following conditions on ##(x_n)_{n=1}^\infty## and ##L##. $$\forall \varepsilon > 0,~ \forall n_0 \in \mathbb{N},~\exists n \in \mathbb{N}\mbox{ so that } (n \ge n_0 \mbox{ and } |x_n - L| < \varepsilon)...
  25. Adolfo Scheidt

    I Harmonic Oscillator equivalence

    Hello, I'm studying the section 2.2 of "Introduction to Quantum Mechanics, 2nd edition" (Griffiths), and he shows this equation $$\frac{\partial^2\psi}{\partial x^2} = -k^2\psi , $$ where psi is a function only of x (this equation was derivated from the time-independent Schrödinger equation) and...
  26. FallenApple

    I Einstein's Equivalence Principle.

    So if someone is in an elevator moving up at g in outer space, vs someone in an elevator in a uniform gravitational field of g, there is no way telling whether one is in an elevator in outer space or resting on earth. But what is the big insight about this? Isn't this true classically as well...
  27. A

    B E = mc^2, Wikipedia Mass–energy equivalence Clarification

    Hello, By far, the easiest explanation of E=mc2, Einstein’s famous mass-energy equation, means that mass is a form of energy and that, mass can be turned into energy (heat, light, other particles, etc) and vice versa. It also seems easiest to say that photons are “pure energy” and are...
  28. N

    Static Equivalent force for cyclic load (fatigue related)

    Homework Statement I have a suspension mount (square tube with unknown thickness) with a bolt going through it that undergoes an 800 lb load laterally into the bracket in an on/off fashion (NOT fully reversed) at a rate of 50 Hz. Ultimately we are trying to calculate the dimensions of the bolt...
  29. N

    Elements of an Equivalence Class

    < Mentor Note -- thread moved to HH from the technical physics forums, so no HH Template is shown > Question: Let ~ be the equivalence relation on the set ℤ of integers defined by a~b if a-b is divisible by 5. Let k ∈ Em belong to the equivalence class of m, and l ∈ En belong to the equivalence...
  30. kiwaho

    A Equivalence equation between cross section & half life?

    Is there equivalence equation between cross section & half life? For beta decay, we usually use half life to describe how fast or slow the decay undergo. For nuclear reaction, we use cross section to describe the possibility of reaction. In a sense, they reflect the same root physics spirit. Is...
  31. B

    I How can the Equivalence Principle hold when we consider tidal forces?

    Tidal Forces: "It arises because the gravitational force exerted by one body on another is not constant across it". which implicitly implies that the acceleration is not constant on that body. Equivalence Principle: "weightlessness sensation occurs when one free falls in gravity" - which...
  32. L

    MHB Equivalence Classes of Bracelet Beads: Red, White, Blue

    Each bead on a bracelet with three beads is either red, white, or blue. Define the relation R between bracelets as: (B1, B2), where B1 and B2 are bracelets, belongs to R if and only if B2 can be obtained from B1 by rotating it or rotating it and then reflecting it. What are the equivalence...
  33. E

    Difficulty proving a relation is an equivalence relation

    Homework Statement Homework Equations I don't think there are any in this case The Attempt at a Solution I know that in order to prove R is an equivalence relation, I'd have to show that it is Reflexive, Symmetric, and Transitive. I'm not sure why, but I'm finding this a bit difficult in...
  34. M

    MHB Are These Statements Equivalent to the Negation of a Bounded Sequence?

    Let $A$ be the statement 'The sequence $(x_n)$ is bounded'. Use your mathematical intuition to decide which of the following statements are equivalent to $notA$. You don't need to give a justification. (a) There exists $C > 0$ such that $\left| x_n \right| > C$ for some $n \in \Bbb{N}$. (b) For...
  35. T

    I Equivalence of two complex expressions

    I found the above while going through my textbook, where the textbook was trying to explain that the principal value of the product of two complex numbers raised to an exponent is not necessarily equivalent to the product of the two complex number each raised to the same exponent first. Based...
  36. P

    MHB Proving equivalence to Euclid Parallel Postulate

    I'm having a hard time proving that the Euclid Parallel Postulate is equivalent to this theorem. Can anyone please help? Euclid Parallel Postulate states: For every line l and point P not on l, there exists exactly one line m so that P is on m and m||l. the theorem states: (Proclus’s Axiom)...
  37. E

    Thevenin Voltage & Impedance with Dependent Source: Homework & Solution"

    Homework Statement Homework EquationsThe Attempt at a Solution How do I do Thevenin voltage and impedance with a dependent source? Any hints would be greatly appreciated.
  38. M

    Temporary magnetization of Casimir effect plates?

    Have any experiments on the Casimir effect tried what happens if the plates are temporarily magnetized when they are pulled together by the effect? E.g. measurements of what happens to passing laser beams and the like?
  39. FallenApple

    A Gravitational Redshift in Newtonian Equivalence Principle?

    So I was reading that the equivalence principle of Newton doesn't work because of a thought experiment. They said that an experimenter shoots bullets( 1 per second) from the bottom of the elevator to the top. This happens in outerspace where the elevator moves up with accleration g. And this...
  40. Y

    MHB Solving a Complex Logical Equivalence in CNF Form

    Hello all, I am trying to bring this: (p \iff q ) \implies r into a CNF form. I have started with the logical equivalences: (p \implies q) = \lnot p\lor q (p \iff q) = (p \land q)\lor (\lnot p \land \lnot q) and then I have applied De Morgan's rules and the distribution rules, but...
  41. Glenn Rowe

    I Equivalence of SU(2) and O(3) in Ryder's QFT book

    I've got a question about the identification of SU(2) with O(3) in Ryder's QFT book (2nd edition) pages 34 - 35. The other posts on this topic I could find don't seem to address this question, so here goes. He derives the matrix in eqn 2.47: $$H= \left[\begin{array}{cc} -\xi_{1}\xi_{2} &...
  42. J

    I About the Equivalence Principle

    Hey all, I was trying to compare the similarity between time dilation due to gravity and the scenario of an accelerating rocket through the equivalence principle and there's something tha blocks my understanding about their similarity. I found a good distance-time graph of the accelerating...
  43. Math Amateur

    I K-algebra/associative algebra - equivalence of definitions

    I am reading Dummit and Foote's book "Abstract Algebra" (3rd Edition) and am focused on Chapter 15: Commutative Rings and Algebraic Geometry ... ... On page 657 D&F give a definition of a k-algebra ... as follows: I have to say I do not find that this definition gives me a good intuitive idea...
  44. Math Amateur

    MHB Equivalence of Definitions of Algebras in D&F and Cooperstein | Peter

    I am reading Dummit and Foote's book "Abstract Algebra" (3rd Edition) and am focused on Chapter 15: Commutative Rings and Algebraic Geometry ... ... On page 657 D&F give a definition of a k-algebra ... as follows:I have to say I do not find that this definition gives me a good intuitive idea...
  45. S

    I Conditional Proposition Equivalence

    Hello, I am confused with the equivalence: (p → r) ∨(q → r) ≡ (p ∧q) → r. I checked that truth tables supports it but I cannot imagine an example which justifies it. Suppose: p says “It is raining”, q says “It is snowing” and r says: “we will close”. So (p → r) ∨(q → r) becomes “if it is...
  46. E

    I Meaning of the equivalence principle in General relativity

    It is known that equivalence principle is good for understaning of general relativity (GR). This means comparison of the elevator and falling in homogenic gravitational field. But, I here somewhere something like "in the higher lever of GR, let us forget on principle of equivalence". How it is...
  47. L

    I Galilean Equivalence Principle: Extended Objects

    The Galilean equivalence principle (or weak equivalence principle) is the statement of the universality of free-fall under gravity. For example, according to Wikipedia, it can be stated as follows My question regards the limitation of the principle to point masses. Does universality of...
  48. F

    Trouble understanding modulo equivalence notation

    Homework Statement I'm trying to understand RSA and modular math better. I've studied some of the relations and rules, but I keep getting stuck when i try to follow an example all the way through. I'm also getting confused by how I'm supposed to treat ≡ vs = at some times. Here is the...
  49. Android Neox

    I Clarifying Equivalence Principle

    Frames under identical acceleration, regardless of whether due to gravity or accelerated motion, are locally identical but time runs slower in a gravity well. For example, at the center of a large mass (like the one pictured, below) time runs more slowly than for an inertial observer far from...
  50. Tony Stark

    Equivalence point in acid-base titration

    In acid base titration, we try to neutralize the analyte by adding sufficient titrant and observing the color change in the indicator. But if the analyte is neutralized, then it must always be at ph 7. Why is it that we use indicator of different ranges to check equivalence point?
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