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. M

    Equivalence of differential operator terms in action

    Hi guys, I'm sure I'm being very stupid here but I'm reading through notes which contain various actions for fields, most of which are very similar, however there is some discrepancies with the way differential operators are shown acting on the fields and I can't for the life of me work out...
  2. M

    Equivalence Classes: Solve for x in A

    Homework Statement Let A be the set, A = {0, 1, 2, 3}. Consider the relation R on A give by xRy iff there exists k \in \mathbb{Z} such that x - y = 3k. Describe the equivalence classes of R. Homework Equations The Attempt at a Solution E_x = \{y \in A: y - x = 3k, k \in...
  3. J

    Confused about the Equivalence Principle and Inertial Reference Frames

    Hey everyone, I started reading up on GR a couple of days ago, and I'm somewhat stuck on the concept of a free-falling IRF. I understand that an observer on a free-falling small spaceship would experience the laws of physics in a rather simple form, eliminating the need for a force of gravity...
  4. L

    Entery level Mass-energy equivalence questions

    Homework Statement The diagram shows a proton moving with Ek of (0.178)(10-12)J towards a stationary nucleus X. The proton hits X and is absorbed. The resulting nucleus splits into two alpha particles, which move off in the directions shown with equal Ek of (1.481)(10-12)J 1. Calculate the...
  5. Z

    'Is $ℜ$ an equivalence relation?

    Homework Statement The **mean value theorem** says that there exists a $c∈(u,v)$ such that $$f(v)-f(u)=f′(c)(v-u).$$ Here is my question. Assume that $u$ is a root of $f$, hence we obtain $$f(v)=f′(c)(v-u);$$ assume that $f$ is a non-zero analytic function in the whole real line. We...
  6. B

    Equivalence Point: Finding Vtitrant of Titrant Solution

    If we are titrating our analyte with volume Vtitrant of the titrant solution, and we have Vtitrant as a function of various known constants and [H+], would the comprehensive method of finding first the [H+] at the equivalence point and then (by back-substitution) the Vtitrant at which each...
  7. ShayanJ

    What are the different equivalence principles and how are they related?

    In my studies about equivalence principle,I learned that there are three equivalence principles:Weak,Einsteinian and Strong equivalence principles.I'm just confused with them.Also looks like EEP and SEP can be derived from WEP making them seem superfluous. Can someone explain clearly? Thanks
  8. M

    Equivalence Classes: Example of 2 Classes

    Homework Statement Consider the relation on \mathbb{N} given by aRb if there exists k \in \mathbb{Z} such that a/b = 2k. Give an example of two different equivalence classes (that is, find x, y \in \mathbb{N} such that Ex \neq Ey, where Ex and Ey are the equivalence classes of x and...
  9. F

    Summation of Powers of Two Mod N equivalence to # of odd residues

    Homework Statement Prove or disprove that: \frac{{\sum_{i=0}^{ord_N (2) - 1}} (2^i \bmod N)}{N} Is equal to the number of odd residue classes of 2^x \bmod N for all odd numbers N greater than 1. Homework Equations Residue Classes are the residues that are generated by a function...
  10. Z

    Difficulty understanding a logical equivalence

    \exists{x}(P(x)\Rightarrow{Q(x)})\equiv{\forall{x}P(x)\Rightarrow{\exists{x}Q(x)}} I am able to derive this equivalence by using the standard equivalences of symbolic logic, but when I try to verify this semantically, with an example, I just can't see why these two expressions are equivalent...
  11. C

    Testing Equivalence Principle: Prying Gravity From Inertia

    Hi, So I came across this article on the internet, and it was about the Equivalence Principle. So I were interested and read it all, and in the article it said that the EP might be only approximately true. What interested me more about the article was it said that any theory of quantum...
  12. B

    Is E=mc2 About Converting Mass to Energy?

    Hello all . Is the paragraph right ? I think there are three mistakes in the sentences . First mistake is that mass and matter is not same and the text is not right . Second mistake is that energy can not transform to massive particles but it can convert to massive particles , because massive...
  13. B

    Frame of reference question about equivalence principle

    I've been giving some thought to the new scientist article about the equivalence principle. What initially seems like a very simple issue is remarkably tantalising. (please note- I'm not a physicist so this may be quite stupid) This led me on a little to the question of gravity and direction...
  14. S

    Understanding the Principle of Equivalence: A Closer Look

    What’s your understanding of the principle of equivalence? In the literature, I find two meanings: (1) Gravitational mass is numerically equal to inertial “mass.” (This is a postulate.) (2) A mass at rest in a frame is equivalent to being in a “gravitational field” in an accelerated frame...
  15. M

    Equivalence relation demonstration - confirmation needed, please.

    Homework Statement 'Show that row equivalence is an equivalence relation'. Homework Equations The definition for 'row equivalence' given in the text is, 'two augmented matrices corresponding to linear systems that actually have solutions, are said to be (row) equivalent if they have the...
  16. J

    Proving Equivalence of p and (q\Rightarrowr)

    \RightarrowHomework Statement I'm given two statements, p and (q\Rightarrowr) and I need to prove that the two statements are equivalent. So I need to show that p \Leftrightarrow(q\Rightarrowr) I know that p\Rightarrow(q\Rightarrowr) \Leftrightarrow (p\wedgeq)\Rightarrowr but I don't know...
  17. B

    Boolean algebra: Logical equivalence

    Homework Statement I've been trying to prove the equivalence between the two statements for quite a while now, any ideas? (A+C)(B+C') = BC + AC' Homework Equations The Attempt at a Solution I used the distributive property to simplify the LHS to AB + BC + AC' Unsure what...
  18. B

    How to prove that R is equivalence relation

    Homework Statement let X = {1,2,3,..,10} define a relation R on X x X by (a,b)R(c,d) if ad=bc. show that R is an equivalence relation on X x X. Homework Equations The Attempt at a Solution I think that the R have to be reflexive (because ad=bc). Eg. one of the subset is (1,1)...
  19. C

    Equivalence between 2 solutions

    Homework Statement A particle of mass ##m## is projected from ##x(0) = x_o## in the vertical direction, with an initial velocity, ##\dot{x}(0) = v_o##. It is subject to gravity and linear drag, mk|v|, against the motion. 1) Show that the body follows ##v(t)## such that:$$v(t) = -g/k + (v_o...
  20. A

    Are These Logical Equations Equivalent?

    Hello hello, I'm not looking just for an answer per say, but am also wondering the thought process in solving problems such as the following: Hopefully this doesn't take up too much of someones time. Determine whether the following equations are logically equivalent: 1) (∃x)( P(x) →...
  21. M

    Principle of Equivalence: Understanding the Foundations of General Relativity

    I have just began my journey in General Theory of relativity through the book of Weinberg-Gravitation and cosmology principles and applications of the general theory of relativity I don't know if this topic is just a repeat of an older one, if it is please let me know the link to it... I...
  22. S

    What is the pH at halfway to the equivalence point?

    If 25.00 mL of 0.20 mol/L HCO2H(aq) is titrated with 0.26 mol/L NaOH(aq) determine the pH at halfway to the equivalence point. So far I have: Moles of HCO2H = 0.005 mol; at half way = 0.0025 I don't know the next step.
  23. D

    Showing the equivalence of lagrangians

    Homework Statement I have a lagrangian written as: \mathcal{L}_H = \text{Tr}\left[\,(D_\mu \Phi)^\dagger D^\mu \Phi\right] - \mu^2 \text{Tr}\left[\,\Phi^\dagger \Phi\right] - \lambda (\text{Tr}\left[\,\Phi^\dagger \Phi\right])^2 Where the field is: \Phi \equiv \frac{1}{\sqrt{2}}(i...
  24. J

    Equivalence principle implies uniformly accelerated charge doesn't radiate?

    Does a uniformly accelerated charged particle radiate em waves? The equivalence principle says that a particle in uniform acceleration is equivalent to a particle at rest in a gravitational field. A particle at rest in a gravitational field is clearly not going to radiate em waves therefore by...
  25. C

    Help Me Find Mass-Energy Equivalence Formula!

    Simple algebra to find the strict mass–energy equivalence formula, and I can't do it! find ((m^2)*(c^4)=(E^2)-(p^2*c^2)) from (E=m*c^2*y) (p=m*y*v) (y=1/(1-(v^2/c^2))^(1/2)) Yeh i can find the m^2c^4 parts and the E^2 obvioulsy. but then i am stuck with E^2(v^2/c^2) when instead i want...
  26. S

    Equivalence Principle and Gravity

    I'm reading a book wherein the Earth's rotation is supposedly slowing down. The author claims that a ball thrown in the air would fall faster and harder... But if the rotation slows, wouldn't the equivalence principle say that the smaller acceleration could also be interpreted as a smaller force...
  27. R

    Information - Energy equivalence

    I was having a discussion with an individual who was claiming that information is a form of energy. I disagreed with that claim. Over the course of the conversation, he provided me with the following article http://physicsworld.com/cws/article/news/2010/nov/19/information-converted-to-energy...
  28. E

    Mass to energy equivalence question

    Homework Statement An unknown amount of mass lost from a fusion reaction is converted to energy. Determine the amount of mass equal to 17.6 MeV of energy. Homework Equations E=mc2? The Attempt at a Solution 17.6=m(931.5) m=0.01889u Not sure if this is correct
  29. E

    What type of energy is the e in mass–energy equivalence equation?

    We recently started learning about special relativity. Today my teacher said that ET = E0 + EK when talking about masses in motion. So what is the name for the energy stored in mass? We have already learned about kinetic, gravitational, elastic, magnetic and I'm sure I forgot a few.
  30. D

    Group action and equivalence relation

    Given a group G acting on a set X we get an equivalence relation R on X by xRy iff x is in the orbit of y. My question is, does some form of "reciprocal" always work in the following sense: given a set X with an equivalence relation R defined on it, does it always exist some group G with some...
  31. T

    Equivalence of maps on l-infinity (involves limits, suprema and sums)

    Homework Statement Normed space (l^\infty,\|\cdot\|_\infty) with subspace S\subset l^\infty consisting of convergent sequences x=(x_n)_{n\in\mathbb{N}}. Given a sequence of maps A_n:l^\infty\to\mathbb{R} defined as $$A_n(x)=\sup_{i\in\mathbb{N}}\frac{1}{n}\sum_{j=0}^{n-1}x_{i+j}$$need to...
  32. J

    Composition of two equivalence relations

    Homework Statement The question is let E1 and E2 be equivalence relations on set X. A new relation R is defined as the E1 o E2, the composition of the two relations. We must prove or disprove that R is an equivalence relation.Homework Equations The Attempt at a Solution I know that we must...
  33. S

    Proving equivalence for a collection of subsets?

    Homework Statement Let f:X→Y where X and Y are sets. Prove that if {S\alpha}\alpha\inI is a collection of subsets of Y, then f-1(\cup\alpha\inIS\alpha)=\cup\alpha\inIf-1(S\alpha) Would I prove this by showing set inclusion both ways? And any hints on how to begin? Thanks.
  34. B

    Equivalence relations and classes

    Show that if R1 and R2 are equivalence relations on a set X, then R1 is a subset of R2 iff every R2-class is the union of R1 classes. Attempt: I don't understand that if R2 has elements nothing to do with the elements of R1, how can an R2 class be a union of those elements belonging to an R1...
  35. X

    Proving equivalence of two sinusoidal formulas

    Homework Statement While doing a physics assignment today, I came up with an expression in my answer that appears to be equivalent to sin(θ), however, i couldn't find a way to manipulate the expression to prove that it was. Could anyone point me in the right direction as to a way that I could...
  36. D

    Chemistry Equivalence point concentration/number of moles

    Homework Statement Fe[3+]+V[2+]-->Fe[2+]+V[3+] If you start with 0.1000 M solutions and the first-named species is the titrant, what will be the concentration of each reactant and product at the equivalence point of the titrations above? Assume that there is no change in [H+] during the...
  37. A

    Energy, Mass, & Volume: Exploring the Equivalence of E=mc^2

    1. Energy is proportional to mass. 2. Mass (of nucleus) is proportional to volume. 3. Volume can be determined from cross-sectional area. If this is the case, then is E=mc^2 equivalent to kE=(pi)r^2 where r is the radius of the nucleus and k is some constant?
  38. B

    Equivalence between a set and the subset of its subset

    Is it true that: If A is not equivalent to its subset A1. Then A is not equivalent to any subset of A1?
  39. J

    The Full Equation for Mass-energy Equivalence

    Hi. I've seen a video by MinutePhysics that talked about the mass-energy equivalence equation, usually known as E=mc^2. It said that there is an extra part to it, and I didn't really understand what it meant. (E^2)=((mc^2)^2)+((pc)^2) seems to be the full one (p being momentum) So...
  40. B

    Are Equivalent Wave Functions Physically Equivalent in Quantum Mechanics?

    Hello I have a problem in understanding wave functions Let q \mapsto \phi(q) a position wave function for a single particle of mass m The equivalent momentum wave function is said to be computable with using Fourier transform: \psi : p \mapsto \int_q \phi(q) \cdot e^{-i/\hbar \cdot \langle p...
  41. T

    Digital Filter Equivalence in a difference equation form

    Homework Statement Find the digital filter equivalence of the circuit in a difference equation formHomework Equations Is this the difference equation form?The Attempt at a Solution If that is the proper form.. I think this is how to solve it? I got this example from my textbook I'm just not...
  42. marellasunny

    What is Topological Equivalence in Functions and Dynamical Systems?

    It would be helpful if someone could please explain topological equivalence of functions in simple words? I am working on dynamical systems and chaos theory.In the underlying material,topological equivalence has taken a more complex definition involving orbits.Please be kind enough to explain...
  43. A

    Equivalence principle - are all accelerations actually equivalent to gravity?

    I have read the basis of the the equivalence principle, that inertial mass = gravitational mass, and that it leads to the conclusion that any acceleration = equivalent gravitational field intensity. (I believe 'gravitational field intensity' is the same as 'gravitational acceleration'.)...
  44. H

    Mass-Energy Equivalence and Storng Nuclear Force

    This really isn't a homework question per se, but I really don't want to post in the big boys' fora. I am learning about basic modern physics at school, as the title suggests, but I am very confused on one matter. Take the tritium nucleus as an example. If tritium nucleons are separate...
  45. B

    Equivalence of Two Infinite Series

    I am having a difficult time seeing how \sum_{n=0}^{\infty} ((-1)^n + 1)x^n is equivalent to 2\sum_{n=0}^{\infty} x^{2n}
  46. O

    Equivalence of the nullspace and eigenvectors corresponding to zero eigenvalue

    Suppose a square matrix A is given. Is it true that the null space of A corresponds to eigenvectors of A being associated with its zero eigenvalue? I'm a bit confused with the terms 'algebraic and geometric multiplicity' of eigenvalues related to the previous statement? How does this affect the...
  47. S

    Help understanding equivalence relation

    Ok, I am barely beginning to understand the subject. I understand that the relation ≥ on N(naturals) is reflexive, not symmetric, and transitive. I don't understand why it is transitive though. Can someone explain? Also, I understand why x2 = y2 is reflexive and symmetric, but I don't...
  48. A

    Weak-Field Limit of Gravitational Radiation and the Equivalence Principle?

    I was just wondering if our use of a flat Minkowski background metric when looking at gravitational radiation in the weak-field limit is essentially done so that locally, for example with detectors on Earth, we can treat spacetime as being flat, but on a larger scale we use the equivalence...
  49. S

    About Relativistic Mass-Energy Equivalence

    While I was looking up E=mc^{2}, I have learned such formula only applies to stationary objects and for kinetic object, the formula is this: E_{r}=\sqrt{(m_{0}c^{2})^{2}+(pc)^{2}} Where E_{r} is relativistic energy and m_{0} is rest mass In the formula, what is p and what is (pc)^{2}...
  50. L

    Equivalence Relations in A={a,b,c,d}: Proving the Bell Number Theorem

    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...
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