Equivalence Definition and 721 Threads

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...

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...

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...

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...

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...

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...

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...

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...

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...

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...

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...

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.

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...

: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...

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

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)...

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...

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...

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)) |...

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...

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...

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?

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...

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...

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...

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...

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."...

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...

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...

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...

Homework Statement Let S be a finite set and denote by 2^{S} = {A|A ⊆ S} the set of all subsets of S. Define a relation ∼ on 2^{S} by A ∼ B if and only if A and B have the same number of elements. (a) Show that ∼ is an equivalence relation on 2^{S}. (b) Let S = {1, 2, 3, 4}. List the...

Homework Statement All the capacitors have the same value. Need to find the capacitance between ab. Homework Equations The Attempt at a Solution

I have two questions: i) Does a distinct equivalence relation on a set produce only one possible partition of that set? ii) Can multiple (distinct) equivalence relations on a set produce the same partition of that set? In other words, given a set S and two distinct equivalence relations ~...

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

\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...

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...

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...

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...

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...

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...

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...

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...

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...

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...

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

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

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 ?

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...

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...

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...