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

    Proving Equivalence of Operations on Equivalence Classes

    Homework Statement Prove that if (a1, b1) ~ (a2, b2) and (c1, d1) ~ (c2, d2), then (a1, b1) + (c1, d1) ~ (a2, b2) + (c2, d2) and (a1, b1) \bullet (c1, d1) ~ (a2, b2)\bullet (c2, d2). Let [a, b] denote the equivalence class with respect to ~ of (a, b) in Z x (Z-{0}), and define Q to be the...
  2. G

    Proving Equivalence of Sets A and B

    Homework Statement 1. Suppose A-B is equivalent to B-A. show that A is equivalent to B. 2. if A,B and C are nonempty and A cross B is equivalent to A cross C then B is equivalent to C Any help would be appreciated, thanks! The Attempt at a Solution I tried constructing a bijection, but that...
  3. P

    Equivalence Relation and a Function

    Homework Statement Suppose that A is a nonempty set and R is an equivalence relation on A. PROVE that there is a function f with A as its domain such that for x and y in A, xRy (x is related to y) if and only if f(x)=f(y) Homework Equations Equivalence relations are relations that...
  4. N

    Equivalence Principle & Gravitational Time Dilation

    I'm reading in a textbook (Gravity by J. Hartle) that gravitational time dilation is implied by the equivalence principle. The following thought experiment is described. A vertical rocket at rest in a uniform gravitational field (no tidal effects) is compared to a rocket constantly...
  5. R

    Equivalence Test: Finding Appropriate Statistical Test for Different Populations

    What is an appropriate statistical test for equivalence of two population means? I'd like the null hypothesis to be that the populations are different. The problem with the t-test is that the null hypothesis says that the populations are the same. It is more appropriate in my application...
  6. L

    Proving Equivalence Relation: z~w $\in$ $\mathbb{C}$

    I'm having a bit of difficulty in showing the following is an equivalence relation: z~w iff \frac{z}{w} \in \mathbb{R} \forall z,w \in \mathbb{C}*=\mathbb{C} \backslash \{0\] clearly z~z is ok as 1 is real then i considered \frac{z}{y} \in \mathbb{R} and tried to show y divided by z is...
  7. L

    Strong Equivalence Principle Question

    My notes say that a photon traveling from a high altitude to an observer at low altitude appears to have a higher frequency i.e. gravitational blue shift. now I am having trouble getting this: the strong equivalence principle says we can substitute this "laboratory" with a laboratory that...
  8. W

    Equivalence of Unimodular (Quadratic)forms on Abelian groups

    Hi, everyone: I have been looking for a while without success, for the definition of equivalence for unimodular quadratic forms defined on Abelian groups . I have found instead ,t the def. of equivalence in the more common case where the two forms Q,Q' are defined on vector...
  9. G

    Proving Algebraic Equivalence: GCDs & Modular Arithmetic

    Homework Statement 1. Prove that if ca=cb (mod m) and gcd(c,m)=1, then a=b (mod m) 2. Prove that if a=b (mod m), then gcd(a,m)=gcd(b,m)Homework Equations The Attempt at a Solution I can't figure out how to get started on these, especially the last one. Is it just a matter of expanding definitions?
  10. C

    What is the about equivalence of ?

    Ok, well I'm sorry if this is the wrong section but it has to do with energy and nuclear physics. I have to present why we should use nuclear energy, though no one in the class knows a lot about it. I am going to be using the example of getting nuclear energy from U-235/92. I have the formula...
  11. S

    Equivalence Problem: Is A and B Equivalent?

    Homework Statement Are the statements A and B equivalent with each other? A. Suppose that n is an integer which is not a perfect square. B. If n >= 1, then \sqrt{n} is either an integer or is irrational. The Attempt at a Solution I am keen on saying that the two statements are not...
  12. C

    Equivalence relations and connected components(Please look at my calculations)

    Homework Statement Hi I have justed switched to a new subject and have some question. 1) Show that if X is a topology space then there exist an equivalence relation if and only if there exist a connected subset which contains both x and y. 2) Show that the connected components are a...
  13. S

    Dimensional Equivalence: Exploring Redundancy in Units of Measurement

    Given two units of measurements with the same physical dimensions (LMT), are they redundant? For example, Einstein's stress-energy tensor of GR uses units of pressure and energy density which are clearly distinguished in most interpretations. However both pressure and energy density have the...
  14. F

    Question about a particular equivalence relation.

    Homework Statement Determine if this is an equivalence relation. Either specify which properties fail or list the equivalence classes: A = {0, 1, 2...} R = {(m,n) | m^2 ≡ n^2 mod 3} Homework Equations m^2 ≡ n^2 mod 3 The Attempt at a Solution I've determined that it is...
  15. G

    Equivalence of inertial & gravitational mass-I need a sanity check.

    Equivalence of inertial & gravitational mass--I need a sanity check. Einstein, in his 1916 book Relativity, illustrates the equivalence of gravitational and inertial mass using the example of a braking train. The example begins with the train at rest (of course) and the scenery moving to the...
  16. L

    Electrodynamic checks of the equivalence principle

    Hello, I would like to know if there are experimental tests of the Equivalence Principle in the realm of electrodynamics. The book by http://books.google.com/books?id=BhnUITA7sDIC&pg=PP1&dq=Theory+and+Experiment+in+Gravitational+Physics&lr=#PPP1,M1" contains a lot of material about the...
  17. D

    What the equivalence class suppose to be

    I read the textbook 5 times now and I can't seem to figure out what the equivalence class suppose to be and how to find it, and i don't understand quotient set either (more importantly how to find it). I'm not familiar with any Equivalences at all if anyone can help me with it that would be...
  18. L

    Equivalence principle and gravitational neutron diffraction

    In Modern Quantum Mechanics, Sakurai say that at quantum level the gravity is no more geometrical because the schrodinger equation depend on the fraction \frac{m}{\hbar}. Can someone explain better this claim or link me some detailed paper on this argument?
  19. C

    Energy-mass equivalence question

    The book I'm reading on special relativity is a ficitonal conversation between Newton and Einstein. Einstein is explaining relativity to Newton. Its a very good book in my unprofessional opinion, but one thing I'm having problems with is energy-mass equivalence. I'm not having trouble with...
  20. J

    Vector expressions - equivalence

    Homework Statement A car is traveling at 12ms^-1. To the passenger in the car the wind appears to be blowing at 8.0ms^-1 at right angles to the road. What is the magnitude and direction of the velocity of the wind with respect to the ground. Homework Equations I can think of two...
  21. C

    Equivalence Principle Question

    I was wondering whether or not astronauts who weighed differently on Earth would weigh differently on a spaceship accelerating to simulate gravity. Do the astronauts who have more mass experience a greater force in the ship like the would on Earth or does each astronaut weigh the same on the ship?
  22. I

    Logical equivalence of statements with truth tables

    Homework Statement (S \wedge T) \wedge \neg (S \wedge T) (S \vee T) \Rightarrow (S \wedge T) Are the two (predicates?) logically equivalent? Homework Equations Not sure, but I believe that logical equivalence means that two predicates give the same output on the same input...
  23. K

    Proving the Equivalence of Two Formulas with e as Small

    Homework Statement Taking e to be small, we have been given these two formulas to prove m1 = \frac{-1+\sqrt{1-4e}}{2e} = \frac{-1+(1-2e+O(e^2))}{2e} = -1+ O(e) m2= \frac{-1-\sqrt{1-4e}}{2e} = \frac{-1-(1-2e+O(e^2))}{2e} = -1+ O(1) Homework Equations The Attempt at a Solution...
  24. P

    Equivalence Principle & Tidal Effects

    It is my understanding that the Equivalence Principle postulates that if I were standing in a closed room, I would not be able to distinguish whether the downward force that I felt was caused by (1) the presence of a massive body such as the Earth exerting a downward gravitational force or (2)...
  25. T

    Discrete math - equivalence relation

    Let A be a set. For every set B and total function f:A->B we define a relation R on A by R={(x,y) belonging to A*A:f(x)=f(y)} *belonging to - because i don't know how to make the symbole... Prove that f is one-to-one if and only if the equivalence classes of R are all singletones
  26. C

    Can someone explain equivalence of gravity and inertia?

    I've been trying to put this together in my head for a while and it doesn't come out. For example, if there are more stars on one half of the sky, does it mean that a thrown object will travel faster in that direction? If the object is at the edge of the universe, does it mean it will not be...
  27. L

    Explaining the Blue Shift in Terms of the Strong Equivalence Principle

    Ok so I was answering a question about a spacecraft accelerating through deep space with observer A in the front and observer B in the rear. I was asked to explain why a phton sent from A to B would appear blue shifted to B - simple enough I just plugged in a positive velocity to the...
  28. J

    Is the Equivalence Principle Contradictory in General Relativity?

    Hello, I can answer some basic questions about GR, but I don't teach it, and I am not proficient in it. Yesterday a student asked a question that I felt I couldn't do justice to. Can you help? I had the student email the question so I could ask around. So far I haven't gotten any useful...
  29. B

    Equivalence of Boltzmann and Clausius entropy

    I can't figure out how to show that the Clausius entropy \Delta S= \int_{T_1}^{T_2} \frac{dQ}{T} is equivalent to the Boltzmann entropy S=kln \Omega at thermal equilibrium. Does anyone know how to do this?
  30. A

    How does Uranium energy equivalence compare to coal in power production?

    One ton of Uranium used at a typical 1 GW reactor is equal to how many tons of coal being burned in a 1 GW coal fired plant?
  31. O

    What Is the Principle of Equivalence?

    Hi all. First post so please forgive the noobness. I have been listing to an audio book of "A briefer history of time" by that well known genital man and I have hit upon something I don't understand. My understanding of the Principle of Equivalence is that you will not be able to tell if you...
  32. C

    Questions deriving from the Principle of Equivalence

    Hi all, There are a few doubts I hope I can get enlightened. If we were in a lab in outer space, free of gravitational influence and the lab accelerates downwards, what will we experience? do we 'move' to the top of the lab? What I'm confused about also is, why is that when you have an...
  33. P

    Equivalence Relation: Examining the Proof

    Hi, When I was just walking through the hallway of my department I found an exercise sheet asking the student to examine the following proof. Assume R\subset M\times M is a binary, symmetric, transitive relation. Then for any a,b \in M with a\sim _R b it follows by symmetry that b\sim _R a...
  34. M

    Equivalence Classes in PxP for (1,2)

    Homework Statement On set PxP, define (m,n)\approx(p,q) if m*q=p*n Show that \approx is an equivalence relation on PxP and list three elements in equivalence class for (1,2) Homework Equations The Attempt at a Solution I will appreciate any help how to start this problem...
  35. V

    Which Relations Satisfy Specific Equivalence Conditions?

    Homework Statement Find relations that satisfying just Reflexive just Symmrtic just Transitive (R) & (S), but not (T) (R) & (T), but not (S) (S) & (T), but not (R) Homework Equations S=Z (a,b) \inR if <=> a>b (T) but, not (S) & (R). the ex is given in the class, but...
  36. H

    Mass-energy equivalence where's the arbitrary constant?

    I'm curious about the famous formula, e=mc^2 . Since mass can be measured in any of several unit systems, shouldn't the formula read e\propto mc^2 ? Thanks!
  37. B

    Can You Prove These Equivalence Relations?

    Homework Statement question 1: Define ~ on Z by a ~ b if and only if 3a + b is multiple of 4. question 2: Let A = {1,2,3,4,5,6} and let S = P(A) (the power set of A). For a,b \in S define a ~ b if a and b have the same number of elements. Prove that ~ defines an equivalence...
  38. E

    Is SR an Equivalence Relation When SR=RS?

    Homework Statement Prove that when R and S are equivalence relations, then SR is equivalance relation when SR=RS. Homework Equations The Attempt at a Solution I want to know what sets should i take for the relations ? is it ok when i take R a a subset of X x Y and S as a...
  39. atyy

    Equivalence principle & accelerated charge

    This thread discusses an interesting point Jonathan Scott brought up at https://www.physicsforums.com/showthread.php?t=264782&page=2. A naive application of the equivalence principle (EP) would suggest that a free falling charge does not radiate, and that a charge on the surface of the Earth...
  40. K

    Prove that a relation is an equivalence relation

    Please be nice to me, I'm new here. Anyway, help to solve this maths problem would be much appreciated: Homework Statement Work out a detailed proof (below) that the relation on the integers defined by p~q if and only if 7|p-q is an equivalence relation: a) the relation is reflexive b) the...
  41. S

    Equivalence principle and Acceleration

    From equivalence principle, being on Earth is equivalent to being on lift accelerating at 9.8 m/s2...so if we are in a lift and that accelerates at 9.8 m/s2 then at some time we will cross the velocity of light.. my questions are 1) is Earth accelerating ? 2) what happens if the velocity...
  42. K

    Equivalence principle and Helium balloon.

    Homework Statement Use the equivalence principle to explain the observation that a helium balloon leans forward in a forward-accelerating car. Homework Equations I don't think equations are needed. The Attempt at a Solution I'm completely confused on this question
  43. O

    Modern Algebra Problem: Equivalence Relations and Classes

    Homework Statement There's this one exam problem that I cannot solve... Here it goes: Consider the set Z x Z+. Let R be the relation defined by the following: for (a,b) and (c,d) in ZxZ+, (a,b) R (c,d) if and only if ad = bc, where ab is the product of the two numbers a and b. a) Prove that...
  44. A

    What Is the Concentration of the Stock HBr Solution?

    Homework Statement A 20.00 ml stock sample of HBr is diluted to 50 mL. If 18.76 ml of 0.1345 M KOH was required to reach the equivalence point, what is the concentration of the stock HBr solution? Homework Equations KOH + HBr --> KBr + H20 M = conc. = mol./L n = V * conc. The...
  45. C

    Is Gravitational Time Dilation Proportional to Relative Motion Time Dilation?

    Is the time dilation between two frames at different distances from a mass equivalent by some constant of proportionality to the time dilation between two frames in relative motion? For example: where r0 is the Schwarzschild radius and r is a radial coordinate greater than r0, this would be...
  46. G

    Equivalence relation in complementary subspace

    I was revising linear algebra and came across the topic of 'constructing complementary subspace given a subspace' - and since the proof (that used Zorn's lemma) of its (complementary subspace's) existence was not constructive, the author defined an equivalence relation in constructing a...
  47. S

    Logical equivalence, need mad help.

    some of you will probly find this easy but this my first assignment after learning what a logical equivalence is so I can't warp my head around it. there are 2 problems I need help with. Homework Statement Find a compound proposition logically equivalent to p V q using only the logical...
  48. S

    Does the Black Hole Event Horizon violate the Equivalence Principle?

    My understanding is that, according to the Equivalence Principle, accelerated motion is indistinguishable from that of being in a gravitational field. So then, at the Event Horizon of a black hole, where gravity is too great for light to escape, wouldn't that be equivalent to accelerated motion...
  49. N

    Weak Acid/Strong Base Titration: Explaining pH at Equivalence

    For an weak acid/strong base titration, the pH at equivalence can be estimated by pH = pKa + pHtitrant/2. Can someone explain where that equation comes from? Thanks.
  50. T

    Confused about the difference between equivalence and implication

    This is a really basic question that I don't know why I'm not getting. So from my understanding, a=b is defined as a=>b and b=>a. So say a^2=c and a=c^(1/2) so which implies which? Say I start from a=c^(1/2) I square both sides and I get a^2=c. So a=c^(1/2) => a^2=c. But if I...
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