Equivalence Definition and 721 Threads

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.

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...

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) ⇔...

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...

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...

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...

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...

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...

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...

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...

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...

\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.

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...

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...

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)...

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)...

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...

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) <-->...

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}...

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...

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...

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...

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...

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...

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...

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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

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

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...

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...

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...

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

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

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...

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

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...

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...

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...

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

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...

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.

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...

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...

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...

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...

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...

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

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?

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...

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...