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. S

    Proof of Vector Equivalence: a‧b=a‧c

    1.Determine if it is true that for any vectors a, b, c such that a is not equal to 0 and a‧b = a‧ c, then b = c. i tried to let a‧b-a‧c=0 then a‧(b-c)=0 but i found it's not meaningful so how can i solve it =[ thz
  2. Fredrik

    Equivalence classes of Cauchy sequences

    \mathbb R can be defined as "any (Dedekind-)complete ordered field". This type of abstract definition is a different kind than e.g. the "equivalence classes of Cauchy sequences" construction. I prefer abstract definitions over explicit constructions, so I would be interested in seeing similar...
  3. J

    General Relativity Basics: The Principle of Equivalence

    What I have heard *about* the principle of equivalence is a great and grave over generalization; primarily that gravity is equivalent to acceleration. I would be prepared to acknowledge that it is highly likely that the behavior of free-falling bodies in the region where F=m*g would be...
  4. B

    Module Equivalence: Understanding Ann(M)

    Module "equivalence" There is a problem in a book I'm not quite understanding. Let M be an R-module and let I=Ann(M). Show that M can be regarder as an R/I-Module where scalar multiplication is given by the rule m(I+r)=mr I don't understand what they mean by "regarded as". Am I suppose to...
  5. E

    Is R an Equivalence Relation on Functions to [0,1]?

    Homework Statement Given is the set X. The set of functions from X to [0,1] we call Fun(X,[0,1]). On this set we consider the relation R. An ordered pair (f,g) belongs to R when f^{-1}(0)\setminus g^{-1}(0) is a countable set. a) Prove that R is transitive. b) Is R an equivalence relation...
  6. H

    Equivalence Classes: Unique Unit Circle Rep.

    Homework Statement Let S := (\Re x \Re \ {(0,0)}. For (x,y), (x',y') \in S, let us say (x,y) ~ (x',y') if there exists a real number \lambda > 0 such that (x,y) = (\lambdax',\lambday'). Show that ~ is an equivalence relation; moreover, show that each equivalence class contains a unique...
  7. B

    Proving Equivalence of Euler-Macheroni Constant

    Hi Everyone, I just registered for PF today because this problem was driving me nuts and I was hoping to get some help. It comes from pg. 5 of Peter Miller's "Applied Asymptotic Analysis" and goes like this: The Euler gamma constant has one definition as \gamma := \int_0^\infty...
  8. D

    Equivalence between power sets

    Homework Statement Part a: Show that X \subseteq Y and X \subseteq Z if and only if X\subseteq Y \cap Z, for sets X,Y,Z. I have done this. Part b: Use the equivalence from part a to establish the identity P(A) \cap P(B)= P(A \cap B), where P is the power set. Homework Equations...
  9. D

    Equivalence relation on the Cartesian plane

    Homework Statement A relation p is defined on R^2 (fancy R, as in Reals) by (a,b)p (c,d) if a+d=b+c Show that p is an equivalence relation. b) Consider R^2 to be the Cartesian Plane. Describe p's equivalence classes geometrically. (Consider which points will be in the particular...
  10. A

    Understanding Equivalence Classes in Integer Sets

    Homework Statement Definition: If A is a set and if ~ is an equivalence relation on A, then the equivalence class of a\inA is the set {x\inA l a~x}. We write it as cl(a)Let S be the set of all integer. Given a,b \in S, define a~b if a-b is an even integer. so, the equivalent class of a...
  11. Buckethead

    Equivalence Principle question

    According to Einstein's Equvalence Principle inertial mass and gravitational mass are interchangable. If we lived in a universe where these two masses were not equal, how would this translate into everyday experience? For example, if gravitational mass were twice the value of inertial mass...
  12. R

    Is x Equivalent to y in Congruence Class Equivalence?

    Homework Statement "Prove that if x is an element of [y] then [x] = [y]"
  13. R

    Proving Equivalence Classes in Modular Arithmetic

    Homework Statement Suppose [d], [b] \in Z sub n.
  14. L

    Equivalence Relations on [0,1]x[0,1] and Hausdorff Spaces

    We have a equivalence relation on [0,1] × [0,1] by letting (x_0, y_0) ~ (x_1, y_1) if and only if x_0 = x_1 > 0... then how do we show that X\ ~is not a Hausdorff space ?
  15. R

    Understanding Mass-Energy Equivalence: A Guide to Fdx and dm in E=mc^2

    Hi I was wondering if anyone could help me with this equation. Fdx &= dm c^2 First of all, excuse me for my limited knowledge of calculus, but how exactly can you just use the numerator of a derivative? What do Fdx and dm mean if they are not in respect to anything? Do they simply mean a...
  16. C

    Lorentz boost and equivalence with 3d hyperbolic rotations

    I was thinking that if i have for example a boost in the direction of x, then the boost should be equivalent to an hyperbolic rotation of the y and z axes in the other direction. I don't know if it's true or not. Then I want to know if somebody knows this result or why is false? I was...
  17. A

    What is the Equivalence Class for the given Equivalence Relation?

    Homework Statement Find the equivalence class [2] for the following equivalence relations: a) R: Z <-> Z, where xRy, iff |x| = |y| b) T: N <-> N, where xTy, iff xmod4 = ymod4 N means natural numbers etc...there wasnt the correct symbols in the latex reference Homework Equations ?? The...
  18. D

    Matrix Row Equivalence: Understanding Non-Singular Matrices"

    Every matrix is row equivalent to a unique matrix in echelon form? False, a matrix is row equivalent if it is non-singular. Is the above correct reasoning for the initial statement.
  19. D

    Proving Equivalence Relations for Real Numbers x, y, z in R

    x,y,z\in\mathbb{R} x\sim y iff. x-y\in\mathbb{Q} Prove this is an equivalence relation. Reflexive: a\sim a a-a=0; however, does 0\in\mathbb{Q}? I was under the impression 0\notin\mathbb{Q} Symmetric: a\sim b, then b\sim a Since a,b\sim\mathbb{Q}, then a and b can expressed as...
  20. J

    Boolean Equation Equivalence Problem

    Homework Statement Part a was to prove the equivalence of the two equations using a truth table. Done. Part b is to prove the equation using the 10 properties of boolean logic, as seen in http://en.wikipedia.org/wiki/Boolean_logic#Properties" We are trying to prove: (p∨q)∧(¬q∨r) ⇔...
  21. H

    Testing the Equivalence Principle with Spectral Anisotropies

    It suddenly occurred to me that I've never heard of a test of the Equivalence Principle itself - such as something like ... An accelerating laboratory [in space] can see very obvious effects in external astronomical observations. Depending on the rate of acceleration, there would be...
  22. jaketodd

    Equivalent Trajectories in Relativity: Observer Effects

    In relativity, can two sets of trajectories, carried out at different times, be considered equivalent if they only differ by when they change directions as they traverse their sets of trajectories? They traverse the same trajectories. The only difference is the rate at which they traverse the...
  23. X

    Equivalence of mass, energy and gravity

    Hi guys, I'd like to hear what you think about a little thing I did talking about the equivalence of mass, energy and gravity. I used planet Earth as an example. Here's the link to the .docx Word file: http://www.angelfire.com/bug/chaos1/Gravity_Rotation.docx" My premise is that the more...
  24. N

    Proving Numerical Equivalence of Real Number Intervals with S-B Theorem

    Homework Statement Using the Schroeder-Bernstein Theorem, prove that any two intervals of real numbers are numerically equivalent. Homework Equations Schroeder-Bernstein Theorem: Let A and B be sets, and suppose that there are injections from A into B and B into A. Then, there exists a...
  25. F

    Linear Algebra: Equivalence of Linear Transformations

    Homework Statement 1) two linear transformations B and C are equivalent iff there exist invertible linear transformations P and Q such that PB=CQ 2) if A and B are equivalent then so are A' and B' in dual space 3) Do there exist linear transformations A and B such that A and B are equivalent...
  26. H

    Deriving the Schwarzschild metric just by using the equivalence principle

    I've read a few papers about derivation of the Schwarzschild metric by using the equivalence principle ( http://cdsweb.cern.ch/record/1000100/files/0611104.pdf" )... but I couldn't understand them completely they assume , According to Einstein’s equivalence principle, that the influence of...
  27. D

    Smallest Equivalence Relation on Real Numbers: Proving with Line y-x=1

    1) Recall that an equivalence relation S on set R ( R being the reals) is a subset of R x R such that (a) For every x belonging to R (x,x) belongs to S (b) If (x,y) belongs to S, then (y,x) belongs to S (c) If (x,y) belongs to S and (y,z) belongs to S then (x,z) belongs to S What is the...
  28. P

    What is the volume of base added at second equivalence point?

    Homework Statement You have a solution that is buffered at pH =2.0 using H3PO4 and H2PO−4 (pKa1 = 2.12; pKa2 = 7.21; pKa3 = 12.68). You decide to titrate this buffer with a strong base. 15.0 mL are needed to reach the first equivalence point. What is the total volume of base that will have...
  29. D

    Equivalence Relation in Math: Proving Transitivity

    \forall a,b\in \mathbb{Z} a\sim b iff. \left\vert a-b \right\vert \leq 3 I have already shown reflexive and symmetric but not sure on how to show transitive. I know the definition.
  30. I

    Why is the Equivalence Principle True?

    The equivalence principle states that an accelerating observer who has no external information (view of fixed stars, etc) can in principle not perform an experiment to determine whether he is either undergoing linear acceleration or at rest in a gravitational field. This leads to the equivalence...
  31. N

    Three Questions About Energy-Mass Equivalence

    I read somewhere that Einstein found out in 1905 that as an object approached the speed of light (c), the object's mass would increase. He also found out that as the object goes faster and approaches "c", it also gets heavier. In fact, at "c" itself, an object's mass and energy should both be...
  32. JK423

    Is Negative Total Energy Possible in Mass-Energy Equivalence?

    Suppose you have two particles of mass 'm'. Their combined rest energy will be: E_rest=2mc^2 Its said that when they interact (gravitationaly), they're total energy will decrease due to the negative gravitational potential energy. The rest of the energy is stored in the field (they say)...
  33. M

    Describing Equivalence Classes

    Hey guys! So I am having trouble understanding equivalence classes. How are they determined?? Anyways here is my problem! Homework Statement Let A and B be two sets, and f: A-->B a mapping. A relation on A is defined by: x~y iff f(x) = f(y) a) Show ~ is an equivalence relation b)...
  34. K

    Understanding Equivalence Relations in Real Numbers and Vector Spaces

    Homework Statement I have got myself very confused about equivalence relations. I have to determine whether certain relations R are equivalence relations (and if they are describe the partition into equivalence classes, but I'll worry about that once I understand the first part). Here are...
  35. S

    Is RxS an Equivalence Relation on ExF?

    Homework Statement I need a little help in understand this question: Let E and F be two sets, R a binary relation on the set E and S a binary relation on the set F. We define a binary relation, denoted RxS, on the set ExF in the following way ("coordinate- wise"): (a,b) (RxS) (c,d) <-->...
  36. G

    Mass Energy Equivalence and Kinetic Energy

    If a particle is moving through free space (no forces acting upon it) should its kinetic energy equal its mass energy equivalence or am I getting confused. In other words is an object's kinetic energy absorbed within its mass? Is the following true? E_{kinetic} = \frac{p^{2}}{2m} = mc^{2}...
  37. N

    Proving ~ is an Equivalence Relation of A

    Homework Statement Consider the set A={(u,v,w) in R^3 : u^2+v^2>0} and define a relation ~ on A by (u,v,w)~(u',v',w') IFF there exists a "k" in R, where k doesn't equal 0: (u',v',w')=(ku,kv,kw) Prove that ~ is an equivalence relation of A Homework Equations I honestly don't know...
  38. L

    Linear Algebra Vector Spaces: Prove equivalence

    Homework Statement Prove that the following are equivalent: 1. N(A)=0 2. A is nonsingular 3. Ax=b has a unique solution for each b that exists in R^n. Homework Equations The Attempt at a Solution I think you prove this by showing that 1 implies 2, 2 implies 3, & 3 implies 1...
  39. Rasalhague

    Equivalence of mass and energy

    Some naive questions about the meaning of this expression. In what follows, I'll use the word mass for "rest mass", the magnitude of the energy-momentum 4-vector. (Answers in terms of "relativistic mass" are fine, just let me know what definitions you're using. Taylor & Wheeler in Spacetime...
  40. C

    Is the equivalence principle good for anything?

    I heard that there is now consensus in literature that a charged particle will take a DIFFERENT path around a neutral object than an uncharged particle. The reason is that the charged particle will radiate. I have even heard physicists try to wave this away as "well a charged particle needs...
  41. R

    Equivalence relations and equivalence classes

    Hey! Hoping you guys could help me with a small issue. No matter how hard I try, I don't seem to fully understand the notion of an equivalence relation, and henceforth an equivalence class. What I do understand that, in order to have and equivalence relation, it is defined to satisfy three...
  42. L

    Equivalence Relation Homework: Is R on X Reflexive, Transitive, Symmetric?

    Homework Statement Let X be Z*Z, i.e. X is the set of all ordered pairs of the form (x; y) with (x, y) are integers. De fine the relation R on X as follows: (x1^2, x2^2)R(y1^2, y2^2) = (x1^2 + x2^2) = (y1^2 + y2^2) Homework Equations By definition, an equivalence relation bears the...
  43. P

    Equivalence Relation: Proving R and Finding Class

    Moderators note: thread moved to homework area. Please note, homework assignments or textbook style exercises for which you are seeking assistance are to be posted in the appropriate forum in our https://www.physicsforums.com/forumdisplay.php?f=152" area. This should be done whether the...
  44. P

    Exploring Numerical Equivalence of Sets: A Brief Definition

    1. Define numerical equivalence of sets 2. I'm not sure how in depth the definition needs to be, how is my current def? 3. X is numerically equivalent to Y if \existsF:X\rightarrowY that is bijective or there are two injective functions f:X\rightarrowY and g:Y\rightarrowX
  45. N

    Is R an Equivalence Relation in R x R Based on a Quadratic Equation?

    In R x R , ley (x,y) R (u,v) if ax^2 +by^2=au^2 + bv^2, where a,b >0. Determine the relation R is an equivalnce relation. Prove or give a counter example
  46. P

    Equivalence Classes of Continuous Functions with a Common Value at x=4

    Homework Statement Identity if it is an equivalence relationship and describe the equivalence class. The relationship T on the set of continuous functions mapping R to R, where fTg iff f(4)=g(4) Homework Equations The Attempt at a Solution It is an equivalence relationship just...
  47. N

    Quantification logic and equivalence relations

    I wasn't sure whether to post this in the algebra forum or here, but it seems that this is more of a logic question so I'm going with here. I am trying to understand whether there is a difference between the following two definitions of an equivalence relation: Definition 1: A binary relation...
  48. 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...
  49. 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
  50. 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
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