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. entropy1

    B Equivalence of entanglement and decoherence?

    Just taking an advance in what I want to learn someday: I understand that decoherence and entanglement are more or less equivalent. So, I take it decoherence is in principle a process of entanglement. Consider two particles A and B who are entangled. if A decoheres by interacting with particle...
  2. M

    MHB Can Logarithmic Functions Be Expressed as Infinite Series?

    Hi, I'm stuck on the following proof: \log[3] = \frac 1{729} \sum_{k=0}^\infty \frac 1{729^k} \left[\frac{729}{6k+1}+\frac{81}{6k+2}+\frac{81}{6k+3}+\frac 9{6k+4}+\frac 9{6k+5}+\frac 1{6k+6}\right] Manipulating and converting summands to integrals of the form $x^{-(6k+n)}$ over {x,0,3} seems...
  3. S

    A Theories that respect one form of the equivalence principle but violate another

    This extract is taken from Sean Carroll's textbook. It talks about a theory that respects the Weak Equivalence Principle but violates the Einstein Equivalence Principle. we could imagine a theory of gravity in which freely falling particles begin to rotate as they moved through a gravitational...
  4. C

    I How does SU(5) explain proton-electron charge equivalence?

    Hi, In my notes it says that SU(5) operates on a five component vector. Fine. But the example given is (d d d e v) ie. electron, neutrino and 3 down quarks (one for each colour) it says this explains why the proton charge is the equal and opposite to the electrons but why? A proton is uud...
  5. kamhogo

    Equivalent resistance between a and b -- Complex circuit

    Homework Statement Homework Equations Req = [ ( 1 / R1) + ( 1 / R2) +. ...]^-1 (Parallel) Req = R1 + R2 +. . . (Series ) The Attempt at a Solution I tried to simplify the circuit by spreading it out but I guess something is wrong with my simplification since I can't arrive at the correct...
  6. M

    B Einstein’s equivalence principle for a collapsed star

    The Equivalence Principle says depicting gravity as a field associated with matter is equivalent to picturing it as twisting of space. Is there a space-twisting equivalent of a collapsed star as a shell of condensed matter with interior of intense gravitational field energy (found in the field...
  7. RoboNerd

    Titration error affecting the PH of the endpoint

    Homework Statement Hi everyone. 1) If I perform a titration with a buret not rinsed in the titrant, then how will the pH of the endpoint be affected? 2) Also, if the volumetric pipet used to measure the analyte is not rinsed with the analyte, how will that affect the pH of the endpoint...
  8. E

    Does the mass of water increase when it is cooled to form ice?

    Homework Statement If mass-energy equivalence is taken into account, when water is cooled to form ice, the mass of water should increase, decrease or remain unchanged? Homework Equations E = mc2 The Attempt at a Solution Since the internal energy of ice is less than water, it's mass should...
  9. E

    Equivalence factor or n factor (Equivalent weight)

    Homework Statement Equivalence factor of the salt is FeSO4.Cr2(SO4)3.24H2O is Ans is 8. Homework EquationsThe Attempt at a Solution Equivalence factor of an acid or base is its basicity or acidity, and for a compound undergoing redox reaction, it is the number of electrons exchanged per mole...
  10. NoName3

    MHB Proving Equivalence of $g' \in Hg$, $g'g^{-1} \in H$, and $Hg = Hg'$

    Let $H$ be be a subgroup of a group $G$. Let $g'$ and $g$ elements of $G$. Prove that the following are equivalent: $(a)$ $g' \in Hg$, $(b)$ $g'g^{-1} \in H$, and $(c)$ $Hg = Hg'$. $g' \in Hg$ means $g' = hg$ for some $g \in G$ and $h \in H$. And $g' = hg \implies g'g^{-1} = hgg^{-1} = h$. But...
  11. R

    Resistance equivalence and circuit current

    Homework Statement a) find the equivalent resistance of the circuit below b) each current in the circuit c) the potential difference across each resistor d the power dissipated by each resistor. Homework Equations V=IR R=(1/R)^-1 The Attempt at a Solution Not sure if I did the resistance...
  12. RJLiberator

    Isomorphism is an equivalence relation on groups

    Homework Statement Prove that isomorphism is an equivalence relation on groups. Homework Equations Need to prove reflexivity, symmetry, and transitivity for equivalence relationship to be upheld. **We will use ≅ to define isomorphic to** The Attempt at a Solution Let G, H, and K be groups...
  13. arupel

    I Exploring Mass-Energy Equivalence: What Causes Different States?

    I am not sure exactly what E = mc^2 means. 1. Does it simply means if we were able to convert mass into energy this is the amount of energy. 2. That mass and energy are different states of the same thing. An example would be water which can either be a solid (ice), a liquid (water) or a...
  14. Kingyou123

    Equivalence Relations in Mathematics: Understanding Transitivity and Symmetry

    Homework Statement Homework Equations a/b=c/d The Attempt at a Solution I started out with transitivity and I figure it's transitive because a/b=e/f = af=be=(a,b)R(e,f). It can't be symmetric because ad does not equal bc, 1*4 does not equal 2*3 I'm I correct or completely wrong?
  15. B

    What equivalence of vacuum in pressured system is possible?

    If I have a volume of gas in a sealed container with a special one way valve in it, and I heat the gas inside the container, the pressure increases and some gas escapes through the valve to the outside. I now have less atoms of gas inside the container, which, when cooled back to the starting...
  16. F

    Equivalence principle and mass

    Hello Forum, What is the whole reasoning and importance of discovering that the inertial mass, which is introduced as the resistance to change the state of motion and the gravitational mass, which promotes motion (the larger it is the larger the attractive force). The equivalence principle...
  17. K

    Proof involving partitions and equivalence class

    Homework Statement Show that every partition of X naturally determines an equivalence relation whose equivalence classes match the subsets from the partition. Homework Equations ( 1 ) we know that equivalence sets on X can either be disjoint or equal The Attempt at a Solution Let Ai be a...
  18. D

    Non-Euclidean geometry and the equivalence principle

    As I understand it, a Cartesian coordinate map (a coordinate map for which the line element takes the simple form ##ds^{2}=(dx^{1})^{2}+ (dx^{2})^{2}+\cdots +(dx^{n})^{2}##, and for which the coordinate basis ##\lbrace\frac{\partial}{\partial x^{\mu}}\rbrace## is orthonormal) can only be...
  19. RJLiberator

    Addition/Multiplication of Equivalence classes, Well defined

    Homework Statement For the set ℝ, define ~ as x~y whenever |x| = |y|. Define the addition and multiplication of equivalence classes as: [x]+[y] = [x+y] [x]*[y] = [xy] a) Show that the multiplication of equivalence classes is well defined. b) Give an example that illustrates that the addition...
  20. RJLiberator

    Is This a Valid Equivalence Relation on ℚ?

    Homework Statement For each of the relations defined on ℚ, either prove that it is an equivalence relation or show which properties it fails. x ~ y whenever xy ∈ Z Homework EquationsThe Attempt at a Solution Here's my problem: I am starting off the proof with the first condition of...
  21. RJLiberator

    Equivalence Relations Questions

    Homework Statement For the set ℤ, define ~ as a ~ b whenever a-b is divisible by 12. You may assume that ~ is an equivalence relation and may also assume that addition and multiplication of equivalence classes is well defined where e define [a]+[ b ] = [a+b] and [a]*[ b ] = [ab] for all [a],[ b...
  22. H

    MHB Understanding R-1/R and its Equivalence

    Hey, What is r-1/r! And sow it is equal to 1/(r-1) -1/r! Thanks in advance.
  23. Dema

    Equivalence of formulas for time reduction factors in GR

    Hello, I’d have a question regarding two apparently different formulas for the time reduction factors for observers in orbit around a rotating black hole, as reported in this nice thread: Equation for time dilation of body in orbit around Kerr black hole? The first one is: $$A=\sqrt{g_{tt} +...
  24. barbara

    MHB Proving an Equivalence Relation on Real Numbers

    I know that 1. To show the relation is reflexive, we need to show that for any x, using the definition of R, we have xRx. The definition of R means that we must have |x - x| is even.2. To show that R is symmetric, we would have to show that if xRy then yRx. In the context of the definition...
  25. Odious Suspect

    The equivalence of a set and its permutations.

    The following is from an introduction to groups. It is not clear to me why the authors bothered to introduce the subset ##\mathfrak{Q}\subseteq \mathfrak{R}## and a subset ##\mathfrak{K}\subseteq \mathfrak{S}^{\mathfrak{R}}## into the discussion. (3) seems to follow trivially from the...
  26. bcrowell

    A Andrei Lebed, equivalence principle

    Andrei G. Lebed, "Breakdown of the equivalence between gravitational mass and energy for a composite quantum body," http://arxiv.org/abs/1404.4044 [corrected a mistake in the following paragraph] He seems to argue that hydrogen atoms moving from one region of space to another, with a different...
  27. D

    Equivalence Classes Homework Solutions

    Homework Statement Hi,I am stuck on my maths assignment. Describe equivalence classes for the following equivalence relations on the given set S. (i) S = R, and a ∼ b ⇐⇒ a = b or −b. (iii) S = R, and a ∼ b ⇐⇒ a^2 + a = b^2 + b. (v) S is the set of all points in the plane, and a ∼ b means a...
  28. H

    Are all physical quantities an equivalence relation?

    Consider this self-evident proposition: "If object A has the same mass as object B and object C separately, then object B has the same mass as object C." Why isn't this stated as a law, but the zeroth law of thermodynamics is? Is there a physical quantity u such that the u of A is equal to the...
  29. S

    Einstein equivalence principle

    I can not understand what exactly this principle says.while reading a book about it ,it was written that einstein abolished gravity by saying the lift is free falling can some explain clearly what does that mean?
  30. H

    Is the equivalence principle related to the principle of relativity?

    Here is what I understand of the equivalence principle. However, given that the concept is still a bit cloudy to me, please tell me if I am wrong: The equivalence principle is a phenomena where all inertial objects experience the same laws of physics. In other words, if you were going at...
  31. M

    Proving Equivalence of {5n | n = 1,2,...} and ℤ

    How would I go about showing that the set {5n | n = 1,2,...} is equivalent to ℤ. I previously attempted to define a function f as f(n) = 5n/2 , when n is even AND (-1)[(5n-1)/(2)] , when n is odd. I then went on to show that the chosen function is 1-1 and onto, but my professor said this was...
  32. E

    I What is the Principle of Equivalence and how was it determined?

    Equivalence principle says that gravitational forces are equivalent physically to inertial forces. Can someone explain what is meant by that and how was it concluded?
  33. J

    Show group equivalence relation associated with normal subgroup

    Homework Statement Let ##G## be a group and ##\sim## and equivalence relation on ##G##. Prove that if ##\sim## respects multiplication, then ##\sim## is the equivalence relation associated to some normal subgroup ##N\trianglelefteq G##; i.e., prove there is a normal subgroup ##N## such that...
  34. Master replies:

    Concerning equivalence of sini (sinus pl.(?))

    If sin(x)=sin(x') ,where x is some angle and x' a angle of another triangle, does it then follow that sin(x-90°)=sin(x'-90°)=cos(x)=cos(x')?
  35. C

    Why Did My Logical Equivalence Proof Fail?

    I took a quiz that I was very confident in and just got the scores back today -- I did terribly (50%). Anyway, I am trying to understand where my mistake is below. I went over this three times and I cannot figure out why it's wrong (it looks right to me). To Prove: ~(p ∧ r) ∨ ~(q ∨ r) ≡ p ∧ r →...
  36. bcrowell

    Runaway solutions and the equivalence principle

    The references below describe two interrelated relativistic issues that come up when you try to describe radiation from point charges: 1. We normally expect that an accelerating charge would radiate, but this would seem to violate the equivalence principle. 2. Classical E&M can't describe point...
  37. H

    Exploring the Equivalence of Category Theory and Versatility

    I am a final year math major thinking of taking Category Theory (CT). The problem is people keep telling me its useless. Why? I have a biology degree and decided to get a second degree in mathematics. My motivation for studying mathematics is related partly to an idea I stumbled across in high...
  38. Dadface

    The Equivalence of Mass and Energy

    Hello I've been trying to get confirmation, or otherwise, that the total mass of an atom can be considered to be equal to the sum of the masses of the individual atomic particles it contains plus the mass equivelent of the internal energy, (PE+KE), of the particles. I have also been trying to...
  39. P

    Equivalence of inertial and gravitational mass

    Is there any credible hard evidence that this equivalence extends to all moving bodies? We accept on good grounds that the apparent mass of moving objects is enhanced by motion, to a measurable degree that increases indefinitely as observed speeds of relative motion approach c. Likewise...
  40. A

    MHB Test Equivalence: Med Student Seeking Help

    Hey you guys! I'm a med student from Sweden in need of some help! I'm doing a summer course type-a thing in a lab, and I have to present some data from a project I've been doing here. Now, I've tested several genes' expression and I've compared those of an experimental group to a control group...
  41. D

    What is the exact definition of equality?

    I've been grappling with the idea in my head as to how I would explain to someone exactly what equality between two mathematical objects actually means. This maybe a very stupid question, so apologies in advance, but if I'm honest I struggle to come up with an answer that doesn't involve using...
  42. fyyfifowffums

    Equivalence ratio in partially premixed flame system

    Hi, I'm learning the basics of plasma assisted combustion so I've been reading up on a few articles related to it. My question is regarding this article http://enu.kz/repository/2011/AIAA-2011-971.pdf Using a setup consisting of a pair of counterflow burners, the composition of the oxidizer...
  43. Andre' Quanta

    Equivalence Principle: Einstein vs Strong

    1-What is the different between the Einstein equivalence principle and the Strong equivalence principle? 2-Is it possible to formulate the Einstein equivalence principle in this way? "For every point in space time, always exist a chart / reference frame in which the metric in that point is...
  44. W

    Equivalence of Databases (Notion of)?

    Hi All, Say we have two databases D1, D2, with the same number of entities and with equal functionality diagrams (i.e., the functional dependence graphs are graph-isomorphic). Is there a notion in database under which D1, D2 are considered to be "the same" database , or some term equivalent to...
  45. Keen94

    Truth Set for the given Equivalence

    Homework Statement Find the truth set of the given equivalence. Assume U=ℝ #56. (x2=1)↔[(x=1)∨(x=-1)] Source: Principles of Mathematics by Allendoefer and Oakley section 1.10 Homework Equations {x I px↔qx}=(P∩Q)∪(P'∩Q')[/B] The Attempt at a Solution P={x I px}={x I x2=1}={x I x=1 or -1)...
  46. S

    What is the correct value for the mass-energy equivalence of an alpha particle?

    Homework Statement Calculate the energy equivalent of the mass of an alpha particle. alpha particle (amu) = 4.00150618 u 1 amu = 1.660566*10-27 kg 1 amu = 931.5020 MeV alpha particle (kg) = 6.64465675 * 10-27 kg Homework Equations e = mc2 The Attempt at a Solution 931.5020 MeV *...
  47. W

    Can every countable number be represented in all numeral systems?

    Hi. This might be a stupid question (I'm studying engineering :p), but how do you prove that all numeral systems (binary, ternary etc.) can represent every countable number? I guess you will need to prove that any number ##N## can be written as ##N= S^0 n_0 + S^1 n_1 + S^2 n_2 + ...## where...
  48. S

    MHB Logic equivalence and implication, question

    Hey guys I am having a bit of a difficult time with this question, if some one could help me out it would be appreciated, thanks. Consider the following argument. "If the weather is fine, and the train is early, then the dog will sit on the tuckerbox. The train will be early, (but) also there...
  49. B

    Equivalence principle - light beam through a rocket

    An observer outside the rocket sees a light beam through a rocket that's accelerating. How will an observer inside the rocket see the light beam? (The problem is showed in the picture below) My friends and I had this problem at school today, and we couldn't decide which is right. Here are our...
  50. TrickyDicky

    Equivalence between Weyl relations and CCR

    Due to the fact that the operators in the canonical commutation relations(CCR) cannot be both bounded, in order to prove the Stone-von Neuman theorem one must resort to the Weyl relations. Now the Weyl relations imply the CCR, but the opposite is not true, the CCR don't imply the Weyl relations...
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