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. G

    Weight, Acceleration, Gravitational Equivalence

    When you are in a box it is difficult to know the difference between if the box is accelerating in space or you are standing in the box on the surface of a planet. That is correct is it not? In both the accelerating box and on the surface of the planet you can weigh yourself and if the gravity...
  2. N

    Simplifying a logical equivalence statement without a truth table

    Homework Statement [(p->r) ^ (q->r)] -> (p ^ q) -> r Homework Equations anything but a truth table! laws such as (p->q)= ~(p^~q) or (p->q)=(~q->~p) might help
  3. W

    Discrete Math - equivalence laws

    I need to show that P<->Q is logically equivalent to ( P ^ Q ) v ( ~P ^ ~Q) So far I have P <-> Q is equivalent to ( ~P v Q ) ^ ( ~Q v P ) by a example I have no idea where to go from here
  4. M

    Proving an equivalence relation

    Homework Statement On the set of integers, define the relation R by: aRb if ab>=0. Is R an equivalence relation? Homework Equations The Attempt at a Solution R is an equivalence relation if it satisfies: 1) R is reflexive Show that for all a∈Z, aRa. Let a∈Z. Then if a is...
  5. E

    An equivalence? An equivalence??

    An equivalence?? The two statements : 1)A sequence diverges to infinity . 2)The sequence is unbounded from above. Are equivalent?? If yes how do we prove it? If no give a counter example
  6. E

    Quantum physics and equivalence principle

    I read some articles, where equivalence principle is no more valid in quantum area. One example are neutrinos, changing colours (electron, muon, tauon neutrino). But formulae are not derived from fundaments, so I do not understand, where it is the catch. I read also about COW experiment...
  7. G

    Equivalence Principle: Mass Increase With Time?

    An uncharged, isolated particle of mass m is subjected to a constant force that increases its speed relative to an IRF. Its mass presumably increases as long as the force acts. Does the mass of an identical particle, held at rest in a gravitational field also increase with the passage of time...
  8. M

    Are SI Units Equivalent in Differential Equations?

    Hi all, I would like to know if any of you know about anything the equivalence of SI units for differential equations? For example, for the equation E=mc2 SI units for RHS must equal LHS. I am wondering if this would apply to differential equations? I recently came across a journal paper with...
  9. S

    Deriving Bessel Function Equation with Basic Relation

    Homework Statement Known formula:J_0(k\sqrt{\rho^2+\rho'^2-\rho\rho'\cos\phi})=\sum e^{im\phi}J_m(k\rho)J_m(k\rho') I can't derive to next equation which is e^{ik\rho\cos\phi}=\sum i^me^{im\phi}J_m(k\rho) Homework Equations Can anyone help me? Thanks a lot! The Attempt at a Solution
  10. P

    Equivalence Classes Homework: (a) & (b)

    Homework Statement \forall (a,b), (c,d) \in (Z^2), (a,b)D(c,d) \leftrightarrow a\equiv c\mod\2\and\b\equiv d mod 3 *edit* Sorry the b = d mod 3 is all part of the same line. (a) List four elements of the equivalence class [{5,3}] (b) How many equivalence classes of D are there in total...
  11. T

    Weak and strong equivalence- what is the difference really?

    Hi! My lecture notes make me really confused as to what is the difference between weak and strong equivalence. I also read about it on wikipedia, but I'm still not sure. Can anyone give an example how in real life (thought experiment?) of what weak and strong equivalence is.
  12. S

    Closed set equivalence theorem

    Homework Statement Hi guys, this problem gave me some trouble before, but I'd like to know if I have it worked out now... "If S = S\cupBdyS, then S is closed (S_{compliment} is open) Homework Equations S is equal to it's closure. The Attempt at a Solution 1. Pick a point p in...
  13. P

    Determine if the following is an equivalence relation

    Homework Statement Determined if the following is an equivalence relation, if so describe the equivalence class. The relationship C on a group G, where aCb iff ab=ba Homework Equations The Attempt at a Solution So i know there's 3 things to check: reflexive condition, symmetric...
  14. S

    Einstein's box of phonons and energy-mass equivalence

    If Einstein's box (http://galileo.phys.virginia.edu/classes/252/mass_and_energy.html ) is full of phonons instead of photons, the momentum is Mv=E/c' and time is t=L/c' where c' is the velocity of a phonon. The result should now be E=mc'^2 rather than E=mc^2. E=mc^2 is tenable to photons...
  15. S

    Einstein's box in media and mass-energy equivalence

    Einstein proposed a very simple derivation to E=mc^2 in 1940s which is well-known as Einstein’s box and a brief introduction is in http://galileo.phys.virginia.edu/classes/252/mass_and_energy.html . If this event occurs in media instead of vacuum, the light speed should be u=c/n rather than c...
  16. A

    Titration question 2 equivalence points. Am i right?

    Titration question 2! equivalence points. Am i right? Homework Statement this is the originial question. A 0.149 g sample of a weak acid (HA) requires 16.80 mL of 0.110 M NaOH to completely reach the equivalence point. What is the Molar Mass of the acid? and i got 86.63 g/mol. I have...
  17. G

    Equivalence principle to other forces?

    Why is it that to derive general relativity you use the equivalence principle on gravity and not electromagnetism for example?
  18. A

    Equivalence Relation: Can't Understand x~x+1 on Real Numbers

    Hey I can't see how x~x+1 is an equivalence relation on the real numbers? I don't understand what the relation is. Can anyone help?
  19. M

    Interpreting equivalence classes geometrically

    Homework Statement For (x, y) and U, v) in R2, define (x,y)~(u,v) if x2+y2 = u2+v2. Prove that ~ defines an equivalence relation on R2 and interpret the equivalence classes geometrically. Homework Equations (none) The Attempt at a Solution The first part is easy. I proved...
  20. TurtleMeister

    Question about equivalence principle

    First, I would like to say that this is my first post in this forum and that my knowledge of GR is weak. So I am hoping that my question can be answered in layman terms. If I hold a 1kg object in my hand, I can easily feel it's inertia if I move it from side to side. I could also measure it's...
  21. M

    How Do I Define an Equivalence Relation on a Subset?

    If I have a subset, how do I define an equivalence relation. I understand it has to satisfy three properties:transitive, symmetric and reflexive, but I'm not sure how to give an explicit definition of the equivalence relation, for example on I where I=\{(x,y) : 0 \le x\le 1 \ \& \ 0 \le y \le 1\}
  22. D

    Equivalence Relations on Integers with a Unique Property

    This is a question from A consise introduction to pure mathematics (Martin Liebeck) Hi guys, just stuck on one problem was wondering if someone could lend me hand. Let ~ be an equivalence relation on all intergers with the property that for all "m" is an element of the set of intergers ...
  23. A

    Proving equivalence classes bijective to the set of points on the unit circle

    Homework Statement De fine a relation on R as follows. Two real numbers x, y are equivalent if x - y \epsilon Z . Show that the set of equivalence classes of this relation is bijective to the set of points on the unit circle. Homework Equations N/A? I don't think there are any special...
  24. D

    Equivalence Relations on Integers: Proving Equivalence for All Elements

    This is a question from A consise introduction to pure mathematics (Martin Liebeck) Hi guys, just stuck on one problem was wondering if someone could lend me hand. Let ~ be an equivalence relation on all intergers with the property that for all "m" is an element of the set of intergers ...
  25. I

    Binary relations: weak order, strict partial order, equivalence

    Homework Statement I'm totally lost about this. I know the properties of binary relations (or at least I think I know them, what it means to be transitive, complete etc). This exercise asks me to show that P and I are strictly partial and equivalent respectively when P and I are defined...
  26. U

    Help with energy-mass equivalence

    Of all things I'm concerned, this one should be the most confusing ever. I'll tell how... E=mc2 That is, energy is an object's mass times the speed of light squared. Mass is w*g, and g is 9.81 for so long you're on Earth. Then, the speed of light is 299,792,458 m/s*299,792,458 m/s, which...
  27. C

    Proving limit equivalence statements.

    Homework Statement Is it always true that lim f(x) x-->infinity = lim f(1/t)t --> 0 lim f(x)x--> 0 = lim f(1/n)n-->infinity The Attempt at a Solution How can you begin to prove or disprove these statements if you don't know what f is doing to x. In other words, lim f(x) could not...
  28. N

    Current Proportionality and Thevenin Equivalence

    Homework Statement For the linear circuit shown, given that the current I in the 0.9 K\Omega is 10mA when Vs = 100V: (a) Predict I using the proportionality property for: Vs = 25V, -12V, and 145 V respectively. (b) Use Thevenin’s theorem across terminals a and b. Find I in terms of Vs and...
  29. M

    Proving Equivalence of Sets - One-to-One and Onto Function

    Homework Statement Hi, new to the Physics Forum and desperately need some help with a math analysis problem... Prove that {x|x>1} and {x|0<x<1} are equivalent sets by writing a function and show that it is one-to-one and onto. Homework Equations The Attempt at a Solution
  30. J

    Is a Symmetric and Transitive Relation Always Reflexive?

    Statement: Prove or Disprove: A relation ~ on a nonempty set A which is symmetric and transitive must also be reflexive. Ideas: If our relation ~ is transitive, then we know: a~b, and b~a \Rightarrow a~a. Therefore our relation ~ is reflexive, since b~c and c~b \Rightarrow b~b, and c~a...
  31. B

    Equivalence Relations on Set S: Description and Number of Classes

    Hello! I'm a bit lost on these questions pertaining to equivalence relations/classes. If someone could run me through either, or both, of these questions, I'd be very thankful! I'm completely lost as to what to do with the z in terms of set S... Homework Statement Show that the given...
  32. P

    Equivalence between a Black Hole and travelling at the speed of light

    I am not sure that I am using the correct term of "equivalence" for this question, but here goes. If I were traveling at the speed of light and tried to shine a flashlight forward, would it send out a light beam? Would the same thing not happen if I were standing on the event horison of a Black...
  33. G

    Equivalence Relation: Is (x,y) \propto (a,b) ∝ x^2 + y^2 = a^2 + b^2?

    Homework Statement is the relation \propto deifined on RxR defined by (x,y) \propto (a,b) \Leftrightarrow x^{2} + y^{2} = a^{2} + b^{2} an equivalence relation? The Attempt at a Solution I know i have to show that they hold true for the 3 properties 1 reflexive 2 symetric 3...
  34. quasar987

    Equivalence of Norms in Separable Hilbert Space

    Let H be a separable Hilbert space and let {e_k} be a Hilbert basis (aka total orthonormal sequence) for H. Then |||u|||_1:=\sum_{k=1}^{+\infty}\frac{1}{2^k}|(e_k,u)| is a norm. If {f_k} is another Hilbert basis, we get another norm by setting...
  35. V

    What is the difference between the equivalence principle and Newton's third law?

    The equivalence principle is a straightforward application of Newton's third law, isn't it? There's nothing new in it
  36. J

    Proving R is an Equivalence Relation on R^2

    A relation R on R^2 is defined by (x_{1},y_{1})\mathit{R}(x_{2},y_{2})\;\;\;if\;\;\;x_{1}^{2}+y_{1}^{2}=x_{2}^{2}+y_{2}^{2} How do you show that R is an equivalance relation?
  37. R

    Proving Set Theory: A⊆B Equivalence

    Hi, Im only starting to learn about naive set theory from a book , so pardon me if my answer to the question is really obvious.. Prove that .. A\subseteqB , if and only if A\capB =A,if and only if A\cupB=B, if and only if A-B=empty set.. I was thinking of using venn diagrams to...
  38. D

    Weak Equivalence Principle (WEP)

    As most of you probably know, the WEP states that the intertial mass and gravitational mass of any object are equal. This principle has base in Galileo's observations, that all free-falling objects have a constant acceleration. What I would like to get clear is the order of arguments that leads...
  39. W

    Equivalence Principal, Acceleration, Curved Space

    According to the Equivalence Principal, the effects of Gravity are locally indistinguishable from those of Acceleration. QUESTION: Since Gravity curves Space, a la the Flamm Paraboloid, does acceleration do the same ? Acceleration does impose a comparable Time Dilation effect, from the...
  40. N

    Equivalence of the 'expanding space' & 'kinematic' paradigms

    The Equivalence Principle suggests the possibility that a 'kinematic' paradigm might produce exactly the same cosmic observations as the standard 'expanding space' paradigm. In my view, functional equivalence demands that the 'kinematic' paradigm make the same observational predictions as the...
  41. D

    Definite Integral and Summation Equivalence

    Can someone give me an explanation or possibly a proof that \int^{a}_{b}f(x)dx= \displaystyle\lim_{m\to\infty}\sum^{m}_{k=1}f(x^{*}_{k})\Delta x
  42. M

    Is equivalence principal really true?

    Is it possible to eliminate Gravity completely even locally as equivalence principal says? I think this is not possible,since in a curved spacetime, Riemann tensor cann't be made zero in any frame. Then in what sense equivalence principal(which says that for a freely falling observer in a...
  43. U

    Is there an analytical explanation for Einstein's famous equation E=mc^2?

    I obtained an analytical description of time dilation and length contraction which describes why these occur at high speeds. However there was no such analysis of Einstein's famous equation E=m(c squared). Instead, a derivation that I found in http://www.karlscalculus.org/einstein.html starts...
  44. S

    Confused about Equivalence Principle

    Suppose I set up two labs, Alice on Earth and Bob in a region of intergalactic space (or maybe in a void) where the Riemann tensor is zero. I accelerate Bob's lab at 32.2 ft/sec/sec. Both measure the Riemann tensor using identical equipment. Alice will get a nonzero value for some components...
  45. J

    Equivalence classes for a graph

    Is there a way to figure out the number of equivalence classes for all graphs of order n, size k?
  46. JK423

    An idea on explaining the weak equivalence principle

    An idea crossed my mind on how to explain the weak equivalence principle(WEP) without using the gravitational law: F=G m1m2/r2 Part of what WEP sais is that assuming we are in a uniform gravitational field, if we let two objects of different masses, they will fall with exactly the same speed...
  47. S

    Is <-> an Equivalence Relation on Propositions?

    Homework Statement We have an equivalence relation such that A <-> B. Prove that the equivalence relation is true. The Attempt at a Solution Let P: A -> B Q: B -> A Let's prove the relation by contradiction. Assume \neg A -> \neg B The previous assumption is the same as Q. Thus, we have...
  48. T

    On equivalence of QFT and Quantum Statistical Physics

    Does fact that QFT in imaginary time is equivalent to QSP represents the proof that many-particle quantum physics is equivalent to quantum theory of fields? To elaborate a little, I had some discussion with some engineers, and when I was explaining them Standard Model I had to invoke concepts...
  49. G

    Equivalence Relation: Proving Properties on a Set S

    let the relation \propto on a set S have the properties (i) a \propto a for every a \in S (II) if a \propto b and b \propto c then c \propto a show that \propto is an equivalence relation on S. Does every equivalence relation on S satisfy (i) and (ii) I'm not sure where to start this i know...
  50. 3

    Empty relation and equivalence classes

    These two terms would be the simplest terms in the set theory as they are on the first chapter in my textbook. But why can't i understand it? In my textbook it says A relation R in a set is a set of ordered pairs, so any subset of a set of ordered pairs will be a relation. This includes the...
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