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. K

    Exploring Mass-Energy Equivalence and Its Impact on Gravitational Fields

    I understand energy mass equivalence, but when mass is changed to energy what happens to it's gravitational field?
  2. T

    Equivalence priciple, light bend

    in my textbook it say that a frame which move at constant velocity in deep space and a frame under free fall in uniform gravitational field will be Equivalent. There is no experiement that observer can do to see whether he is in which of the two frame. But it later say that light bent in...
  3. I

    Do tidal forces mean the Equivalence Principle is BS?

    From another thread this question came up: What about tidal forces? Although they could be incredibly small (and I don't think anyone denies they exist) you could conceive of a test in a closed room that detects tidal forces (if it is gravity) and when it doesn't detect them it is uniform...
  4. Saitama

    Chemical Equivalence: Learn the Concept Easily

    Hello everyone! I am not able to understand Chemical Equivalence? The bookish language is very hard to understand... If someone could explain me the Concept Of Equilvalence, it would be very helpful for me... Thanks, Pranav
  5. G

    Charge and the Equivalence Principle

    A charged particle is held at rest inside a box in gravity-free space. The box is accelerated uniformly. The charge theoretically radiates electromagnetic power. Now picture the same box held at rest in a gravitational field. Does the charge radiate?
  6. J

    Equivalence Principle: Are there any restrictions

    Hello all. I've been thinking a lot about the Equivalence Principle as it has been taught to me: "The effects of the acceleration (vector) a of a referential are indistinguishable of those of a gravitational field (vector) g=-a". The same book has the phrase: "There is no experience that...
  7. R

    Help with equivalence principle please.

    I am just trying to understanding of the equivalence principle and was hoping someone would be kind enough to sense check my thoughts below. 1) If I was in a rocket ship accelerating at a rate of 1g relative to the earth, the clock on my ship would tick at the same rate as a clock on the...
  8. e2m2a

    Principle of Equivalence Re-examined

    Many textbooks state the principle of equivalence as such: There is no experiment that can be done inside a small closed space, such as a box, that would allow an observer inside the box to distinguish between being in a uniformly accelerated box or being in a box that is at rest on the surface...
  9. 0

    Partition for the equivalence relation of a parabola

    Homework Statement Let f: R -> R, x -> x^2 What does the partition for the equivalence relation of this function look like? Homework Equations The Attempt at a Solution Uh...I have no idea. Sorry, the book only has examples of like integers from modulo n, if anybody could...
  10. A

    Equivalence of sets proof assistance

    Homework Statement Suppose there exist three functions: f:A\stackrel{1-1}{\rightarrow}B g:B\stackrel{1-1}{\rightarrow}C h:C\stackrel{1-1}{\rightarrow}A Prove A\approxB\approxC Do not assume the functions map onto their codomains. Homework Equations The Attempt at a Solution I took a...
  11. TrickyDicky

    Weak equivalence principle and GR

    In another thread, it's considered a generally accepted fact that the WEP is not valid anymore the way it was initially postulated: I find this surprising, because such a change in the principles of the theory should be more stressed in introductory GR textbooks, and kind of disturbing...
  12. C

    Spontaneous fission rate to half-life equivalence

    Homework Statement The spontaneous fission rate for U-238 is roughly 1 fission per gram per 100 seconds. Show that this rate is the equivalent to a half-life for spontaneous fission of ~5.5x1015 years Homework Equations none that I can find in my book The Attempt at a Solution I'm...
  13. L

    Equivalence classes for an particular relation question

    Homework Statement [PLAIN]http://img15.imageshack.us/img15/1/unledjs.png Homework Equations The Attempt at a Solution Hi, If anyone could help me with this I would be very glad! I have said that M=(aij) and M^T=M^-1 therefore if e1 relates v, where v=(x,y,z) then...
  14. I

    Equilibrium and Equivalence Calculation [Very Difficult]

    Homework Statement 1. What volume of .022M HCl must be added to the 110ml buffer solution to completely react with the ammonia (NH3) to reach equivalence. Then, what is the pH of this at the equivalence point. Buffer solution (equilibrium): NH3 + H20 ⇌ NH4+ + OH- Other Info: .045M NH4+ and...
  15. A

    Equivalence of two sine arguments

    Homework Statement Hecht in his optics mentions that Asin(kx-wt+pi) is equivalent to Asin(wt-kx) w=greek omega Homework Equations What is the fundamental reason behind this? The Attempt at a Solution I have a hunch it's plain trigonometry applied, but none of the things that I can...
  16. C

    Astrophysics: nuclear reactions orbital period spectral class equivalence principl

    Homework Statement 1. Suppose you are told that a star has been observed with a UBV color index of B-V=1.6 and that interstellar reddening is negligible. In addition, its apparent visual magnitude is 9.8. Detailed spectroscopy also reveals that the star has all the characteristics of a main...
  17. J

    Multiplication Properties of Equivalence Classes

    Homework Statement Prove or disprove and salvage if possible: for [a], [b] ∈ Zn for a positive integer n, if [a]·[b]=[0], then either [a]=[0] or [b]=[0]. The Attempt at a Solution I've managed to disprove the statement: Let n=6,[a]=3,and[b]=[4]. The[a]·[b]=[ab]=[3·4]=[12]...
  18. N

    Proving that an equivalence relation is a bijection

    Homework Statement Let (a, b), (c, d) be in R x R. We define (a, b) ~ (c, d) iff a^2 + b^2 = c^2 + d^2. Let R* = all positive real numbers (including 0). Prove that there is a bijection between R* and the set of all equivalence classes for this equivalence relationship. Homework Equations...
  19. G

    Partitions, Equivalence Classes and Subsets

    Homework Statement Suppose A_{\lambda}, \lambda in L, represents a partition of the nonempty set A. Define R on A by xRy <=> there is a subset A{\lambda} such that x is in A{\lambda} and y is in A{\lambda}. Prove that R is an equivalence relation on A and that the equivalence classes of R are...
  20. J

    Understanding the Mass-Energy Equivalence Concept behind E=mc2

    For E=mc2 I'm having trouble understanding intuitively how every kilogram of m conveniently is associated with a neat c2 joules since as far as I know neither kg or joules were formulated with c in mind. I've seen that the mathematical derivation works out but I can't quite put it together on a...
  21. Z

    Is the Equivalence Relation on Complex Numbers Related to Determinants?

    I was checking that the following is an equivalence relation on \mathbb{C} xRy iff x\bar{y}=\bar{x}y It is an equivalence relation and so by letting x=a+bi and y=c+di, then it is equivalent to a/b=c/d so I was viewing it as partitioning points in \mathbb{C} by drawing lines through the...
  22. Matterwave

    Equivalence Principle: Questioning Inertia & Gravity

    Hey, so I have a question. The equivalence principle, the way it has always been taught to me, states that the "gravitational mass" is equal to the "inertial mass". Or, in other words, that the amount of inertia an object has really in some way "equal" (or proportional) to the amount of...
  23. Q

    How to prove the inclusion is a homotopy equivalence?

    How to prove the inclusion is a homotopy equivalence? Homework Statement A deformation retraction in the weak sense of a space X to a subspace A is a homotopy f_t: X\rightarrow X such that f_0=Id_x, f_1(X)\subset A, and f_t(A)\subset A for all t. Show that if X deformation retracts to A in...
  24. K

    F is a field, prove congruence defines an equivalence relation?

    Let F be a field. Given p(x) in F[x], prove that congrence modulo p(x) defines an equivalence relation on F[x]. congruence modulo p(x) means a(x)=b(x) (mod p(x)) for a(x), b(x) in F[x]. Well, I know an equivalence relation on the integers means: a~a if a~b then b~a if a~b and b~c, then...
  25. W

    Energy equivalence between linear and rotational motion

    Technically this is a homework question because it's from an assignment I'm doing as practice for my exam tomorrow. Imagine a rod standing on a table, the base of the rod is attached to the table with a hinge, so that the rod is able to swing between standing position and parallel with the...
  26. S

    Function Equivalence: Proving Equality of Functions in F(S,F)

    Homework Statement Two functions in F(S,F) (or going from the vector space S to the vector space F) are equal if and only if they have the same value at each element of S. True or False? Homework Equations How can you prove: if two functions, x and y, are equal then they have the same...
  27. Y

    Equivalence relation on Vector Space

    Let W be a subspace of a vector space V. We define a relation v~w if v-w is an element of W. It can be shown that ~ is an equivalence relation on V. Suppose that V is R^2. Say W1 is a representative of the equivalence class that includes (1,0). Say W2 is a representative of the equivalence...
  28. R

    Proving R is an Equivalence Relation: Steps and Explanation

    1. Let R be a relation on X that satisfies a) for all a in X, (a,a) is in R b) for a,b,c in X, if (a,b) and (b,c) in R, then (c,a) in R. Show that R is an equivalence relation. 2. In order for R to be an equivalence relation, the following must be true: 1) for all a in X, (a,a) is...
  29. S

    Can someone explain the concept of Mass-Energy Equivalence in simple terms?

    Homework Statement My textbook is only confusing me further and I need to understand this for a presentation in front of the class! The chapter is entitled Mass-Energy Equivalence, with sub titles Relativistic Momentum and Relativistic Energy. I don't understand relativity, I'm reading the...
  30. C

    Equivalence Principle Explained - What is it?

    The Equivalence principle says or at least this is what i learned , Is that being in free-fall is the same as being out in space , But in free fall if you shined a laser up it would get Doppler shifted and gravitationally red shifted but out in space it would not . Or do i have something wrong.
  31. FeDeX_LaTeX

    What is a 'Dollop'? Measurement Equivalence Explained

    Add One "Dollop" Hello; I have tried to be as accurate as possible with my measurements while cooking and I have done fine so far, but what exactly is one 'dollop'? I have been told to add one 'dollop' of mayonnaise and I have no idea what this means. I am assuming it is not a small...
  32. G

    Bell's impossibility theorem, equivalence classes: SOS

    :smile: : 1. I am an engineer seeking to fully comprehend Set Theory, Logic, Probability; especially as it relates to equivalence/classes, class invariants, etc., in the context of an essay that I have posted at http://quantropy.org/12/ [6 pages, 194 Kb, 31 references]. 2. The essay relates to...
  33. A

    Can Energy be Converted into Mass?

    So, there are tons of examples of mass being converted to energy, but can energy be converted to mass? Thanks
  34. R

    Are These Predicate Logic Equivalences Valid?

    Homework Statement state whether the equivalences are valid for P and Q (latex is screwing up, wherever a letter has been made into superscript it should be normal and there should be a ^ in front of it). 1.. poop \exists x [ P(x) ^ \wedge p Q(x) ] \equiv \exists x P(x) \wedge \exists x Q(x)...
  35. G

    Understanding Equivalence Relations & O, o, ~ Notation

    Sorry for such a basic question, but I don't know what they mean by the O, o, and ~ in a book I am reading. I'll write out the whole thing to show what I am asking about as well as to give context. Those symbols appear in ii) and iii) below. Also note that I wrote them here as having a...
  36. W

    Equality of Equivalence Classes

    Homework Statement Suppose is an equivalence relation on a set S. If a \sim b for some a,b \in S,then E_{a}=E_{b}Homework Equations The Attempt at a Solution Assume a \sim b for some a,b \in S. Pick x \in (a,b). For a \in S the equivalence class of a can be written as \{x \in S | a \sim...
  37. R

    Equivalence Relations on a Plane - Proofs, Cases, and Geometric Interpretations

    Homework Statement For each of the relations on the set R x R - (0,0) (ie. no origin) : - prove it is an equivalence - give the # of equivalence cases - give a geometric interpretation of the equivalence cases assuming an element of R x R is a point on a plane a) {((a,b),(c,d)) |...
  38. U

    Consequences of the Equivalence Principle

    I was reading a book on SR and GR and it used the example of a falling elevator with a light beam traveling through it. Considering this setup leads to the conclusion that light bends in a gravitational field. My question is, would light bend as a result of any kind of acceleration given the...
  39. Q

    Understanding Equivalence Classes in the Plane

    Homework Statement Define two points (x_{0}, y_{0}) and (x_{1}, y_{1}) of the plane to be equivalent if y_{0} - x_{0} ^2 = y_{1} -x_{1}^2. Check that this is an equivalence relation and describe the equivalence classes. Homework Equations The Attempt at a Solution I can...
  40. G

    How Does the Principle of Equivalence Lead to Gravity Curving Space?

    I understand the principle of equivalence (e.g. thought-experiments with lifts etc), but how come that from it one can arrive at the result that gravity acts curving the space? Where can I find a step-by-step reasoning illustrating this link?
  41. J

    How Do You Prove Equivalence of Two Polynomials?

    Can somebody prove the equivalence statement of two real polynomials in one variable x for me? My Math teacher just told us to remember it as a definition and so I didn't get any proof for it; I attempted to prove it myself and ended up confusing myself with a lot of symbols. So, can somebody...
  42. G

    Mass equivalence and instantaneous gravitational field

    Here is Max Born explaining the crash of a train, in which the train is regarded to be at rest: There are two peculiar features of this gravitational field: 1. Causation. The field appears coincidentally with the collision of the train with an obstacle. If a passenger had pulled the...
  43. W

    Proving Equivalence Class Intersection and Equality

    Homework Statement I'm trying to prove that "if R is an equivalence relation on a set A, prove that if s and t are elements of A then either [s] intersect [t] = empty set, or, [s]=[t]" Homework Equations The Attempt at a Solution I know that if you were to start trying to solve...
  44. W

    Equivalence Relation (discrete structures)

    Homework Statement Let A be a set prove relation R, defined as R : = {<c,d> \in A x A | T } is an equivalence relation on A. What's the quotient A / R ? 2. Relevant Info the x in A x A means the Cartesian product T means "always true" set theory briefly covered in class The Attempt...
  45. A

    What Is the Fortuin's Method in Potentiometric Titration?

    in my potentiometric titration instrument manual the manufacturer wrote that the equivalence point locationis determined using procedure based on the Fortuin's method ! i searched the web for this Fortuin's method all what i can gain is that it is mathematical method ! can anybody tell me...
  46. H

    Why is reflexive property necessary? equivalence relations

    Homework Statement Provide an example that shows why the reflexive property is not redundant in determining whether a relation is an equivalence relation or not. For example, why can't you just say, "If xRy then yRx by symmetric property, and then using transitive property you get xRx."...
  47. D

    Equivalence of (P -> R) V (Q -> R) and (P ∧ Q) -> R

    I can see how the equivalence can formulated with (P -> R) V (Q -> R) = (¬P V R) V (¬Q V R) = (¬P ∧ ¬Q) V R = ¬(P ∧ Q) V R = (P ∧ Q) -> R (Sorry, I would've written this in LaTeX if I were more competent.) although I still it counter-intuitive and, at a glance, first thought it was...
  48. P

    Equivalence relations and classes problem.

    Homework Statement Let X = {a,b,c,d}. How many different equivalence relations are there on X? What subset of XxX corresponds to the relation whose equivalence classes are {a,c},{b,d} Homework Equations N/A The Attempt at a Solution So I wrote out all the possible "blocks"...
  49. K

    Final part of determining an equivalence relation

    Homework Statement For a,b elements of the real numbers, define a~b if \left|a-b\right|\leq 1. Determine if we have an equivalence relation. Homework Equations The Attempt at a Solution I've already done the first two parts of determining. it's only the last part that I'm having...
  50. S.Daedalus

    Are dualities not just an expression of equivalence in physical theories?

    By 'equivalence', I mean of the computational kind -- i.e. in the same way any universal computer can emulate any other. First of all, hi there, I'm not sure I put this question in exactly the right forum, but it seems to me that most dualities currently being discussed fall under the...
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