Relations Definition and 540 Threads

Hi , I need help with the this exercise: a) Work out all of the canonical commutation relations for components of the operators r and p: [x,y] [x,py] [x,px] [py,pz] and so on. Answer: [ri,pj]=−[pi,rj]=iℏδij [ri,rj]=−[pi,pj]=0 , where the indices stand for x, y, or z and rx=x ry=y rz=z where...

I'm coming from a physics background, but find pure mathematics extremely interesting, so have decided to try and gain a more fundamental understanding of the subject. I've recently been reading up on relations and how one can define them as sets of ordered pairs. I am particularly interested in...

Apologies if this is in the wrong forum, but I chose to post here as the question pertains to equivalence relations and classes. Sorry if it's such a trivial question, but what is the mathematical difference between equivalence and equality? My understanding is the following, but I'm a little...

Why does the characteristic equation of a linear recurrence relation always look like an = series of constants multiplied by a number raised to n

Homework Statement If an unknown vector X satisfies the relation X · b = β X × b = c express X in terms of β, b, and c. Homework Equations X · b = |X||b|cos(θ) X × b = |X||b|sin(θ) The Attempt at a Solution I don't know where to start... :( someone pls give me a hint

Having trouble understanding the concept of transitivity. By definition: If (a,b)\in R\wedge (b,c)\in R \Rightarrow (a,c)\in R - Great. Consider the set \{a,b\}. What makes the relation \{(a,a)\} or \{(a,a),(a,b)\} transitive? How do I translate this in terms of the definition? What makes an...

<<Mentor note: Please always use descriptive thread titles.>> First of all, the title is such that it attracts most views.You see, in class our professor did some goofing around numbers and variables in the relativistic energy momentum relation: E2=(pc)2+m02c4 Since the energy required to...

Is there a way to determine the group from the commutation relations? For example, the commutation relations: [J_x,J_y]=i\sqrt{2} J_z [J_y,J_z]=\frac{i}{\sqrt{2}} J_x [J_z,J_x]=i\sqrt{2} J_y is actually SO(3), as can be seen by redefining J'_x =\frac{1}{\sqrt{2}} J_x : then J'_x , J_y and...

Hi everyone. I tried searching this up, but I could not come with anything conclusive. I was reading a medical ethics book that leaves this question as "food for thought": You live with your stepmother. You study medicine. You work very hard in the library, go to your classes and take air...

Hello! (Wave)Let $e,b \in \mathbb{Z}, d \neq 0$. How could we prove the following? Could you maybe give me a hint? If $d>0$ then $e \text{ div } d = \lfloor \frac{e}{d} \rfloor$ $$$$ If $d<0$ then $e \text{ div } d = \lceil \frac{e}{d} \rceil $ Could we show the above, using the...

Homework Statement Show that the real and imaginary parts of the following susceptibility function satisfy the K-K relationships. Use the residue theorem. $$ \chi(\omega) = \frac{\omega_{p}^2}{(\omega_0^2-\omega^2)+i\gamma\omega} $$ Homework Equations The Kramers-Kronig relations are $$...

Homework Statement Show that: (\frac{∂T} {∂V})_S,_n=-(\frac {∂P} {∂S})_V,_n Homework Equations dU=TdS-PdV+μdn The Attempt at a Solution \frac {∂} {∂S} (\frac{∂U} {∂V})_S,_n=-(\frac {∂P} {∂S})_V,_n \frac {∂} {∂V} (\frac{∂U} {∂S})_V,_n=(\frac{∂T} {∂V})_S,_n I tried to isolate T and P...

Hey! (Nerd) Given the relation: $R=\{ \langle\{ \{ \varnothing \} \}, \varnothing \rangle, \langle \varnothing, \{ \varnothing \}\rangle, \langle \{ \varnothing \},\{ \{ \varnothing \} \}\rangle \}$, I want to calculate $R^{-1}[\{ \varnothing \}], R \circ R, \mathcal{P}R$. That's what I have...

Hi! (Wave) If I want to prove that $A \cap B=A \text{ iff } A \subset B \text{ iff } A \cup B=B$. Do I have to prove the following: $A \cap B=A \rightarrow A \subset B$, $A \subset B \rightarrow A \cap B=A, A \subset B \rightarrow A \cup B=B, A \cup B=B \rightarrow A \subset B $ and $A \cup B=B...

Suppose that R1={(2,2),(2,3),(2,4),(3,2),(3,3),(3,4)}, R2={(1,1),(1,2),(2,1),(2,2),(3,3),(4,4)}, R3={(2,4),(4,2)} , R4={(1,2),(2,3),(3,4)}, R5={(1,1),(2,2),(3,3),(4,4)}, R6={(1,3),(1,4),(2,3),(2,4),(3,1),(3,4)}, Determine which of these statements are correct. Check ALL correct answers...

Question: Find the equivalence classes and the number of equivalence classes of the following relations. A is the set of all possible strings of 3 or 4 letters in alphabet {A, B, C, D}, and (x, y) ∈ R if and only if x and y have the same first letter and the same third letter. My attempted...

Homework Statement Suppose that R is a partial order on A, B1 ⊆ A, B2 ⊆ A, x1 is the least upper bound of B1, and x2 is the least upper bound of B2. Prove that if B1 ⊆ B2 then x1Rx2. Homework EquationsThe Attempt at a Solution I split the proof into two different cases: case 1: x_1 is an...

So lately I've been thinking about whether or not it'd be possible to have the commutation relation [x,p]=i \hbar in a Hilbert Space of finite dimension d. Initially, I was trying to construct a lattice universe and a translation operator that takes a particle from one lattice point to the...

The problem asks for an approximation for the torque needed to rotate a disk that is separated from a stationary boundary by a viscous fluid, given that \tau = \mu \frac{U}{H}. I did it first using like this: T = F R F = \tau A = \mu \frac{U}{H} \pi R^2 where U = \Omega R And thus T = \mu...

Homework Statement Suppose R is a relation on A, and define a relation S on P (A) as follows: S = {(X, Y ) ∈ P (A) × P (A) | ∀x ∈ X∃y ∈ Y (xRy)}. For each part, give either a proof or a counterexample to justify your answer. (a) If R is reflexive, must S be reflexive? (b) If R is symmetric...

Hi, I have a question If I am not mistaken, the Maxwell relations of theromdynamics -----for example: ∂G/∂T) = -S ; ∂G/∂P) = V ----- are valid only for reversible processes. On the other hand, dG = ∂G/∂T)*dT + ∂G/∂P)*dP + ∑ μi dni is valid for any process. This means that dG = -SdT + VdP...

On a post involving the proof of the Fourth Isomorphism Theorem for vector spaces (in which I was immeasurably helped by Deveno) I have become aware that my knowledge of sets and functions was not all it should be when it comes to things like inverse images, left and right inverses and the like...

A single pair (male and female) of rabbits is born at the beginning of the year. Assume the following: 1) Each pair is not fertile for their first month bet thereafter give birth to four new male/female pairs at the end of every month 2) no rabbits die a) let r_{n} be the number of pairs of...

Hi all, I haven't been able to find an answer online but this seems like a pretty basic question to me. What are the commutator relations between the position/momentum operators and the field operator? I'm not even certain what the commutation relations between X/P and a single ladder operator...

Hi, I'm currently stuck on a few questions regarding binary relations as I'm unsure on how to prove their properties. R is defined on N by aRb if and only if a <= b and b <= a+5 Is R reflexive, symmetric, antisymmetric, transitive? S is defined on Z (union) {x + 1/2 : x is an element of all...

We know that for constant pressure thermodynamic processes which only expansion work is possible, dH=dQp. My question is, is it necessary that both work W and Qv be reversible to arrive this relation? What if the heat transfer is irreversible? Similar question for dU=dQv, does the heat...

Hello! (Cool) I want to find the Taylor series of $f(x)=\arctan(x), x \in [-1,1], \xi=0$ $$f'(x)=\frac{1}{1+x^2}$$ According to my notes: $$\frac{1-(-t^2)^n}{1+t^2}=1-t^2+ \dots + (-1)^{n-1} t^{2n-2}$$ So, $\frac{1}{1+t^2}=1-t^2+ \dots +(-1)^{n-1}t^{2n-2}+\frac{(-t^2)^n}{1+t^2} \Rightarrow...

This is an exercise of Special Relativity the professor asked last week. Sorry for the long post, I hope you don't get bored reading it, also, this is my first post here :shy: Homework Statement Defining the 4-force that acts on a particle as the proper-time variation of the 4-momentum...

Homework Statement Solve the recurrence relation an = 3an−1 −2an−2 +3, a0 = a1 = 1.Homework Equations an = general solution + particular solutionThe Attempt at a Solution I started with finding the general solution, which was easy. it ended up being A12n + A0 now I am having trouble solving...

This is not a homework question, I just can't find a good resource on this topic. I am working in quantum mechanics on commutator relations. My book (Griffiths) lacks information on how to manipulate the commutator relations. For instance, when I have [AB,C], when can I make it A[B,C]? Or...

Hi, I'm reading a book on sets and it mentions a set B = {1,2,3,4} and it says that R3 = {(x, y) : x ∈ B ∧y ∈ B} What does that mean? Does that mean every possible combination in the set? Also the book doesn't clarify this completely but for example using the set B say i had another...

Hi, I'm having trouble understanding how to determine whether or not a binary relation is reflexive, symmetric, antisymmetric or transitive. I understand the definitions of what a relation means to be reflexive, symmetric, antisymmetric or transitive but applying these definitions is where I...

I am doing a problem on the "super symmetric harmonic oscillator" Defined by.. \hat{H}=\hat{H}_b+\hat{H}_f= \hbar \omega \left( \hat{b}^{\dagger}\hat{b}+\hat{f}^{\dagger}\hat{f} \right) I am given the operator.. \hat{Q}=\sqrt{\hbar \omega} \hat{b}^{\dagger} \hat{f} and asked to show...

Hi guys! First time poster, long time lurker! I can't make any sense out of equivalence relations:confused: These kinda questions crop up every year on the exam and I was wondering if someone could help me understand the concept behind them. (i)Show that relation R defined on the of the set S =...

HelloI have learned about maxwells relations and can derive them. I noticed that we had made an assumption of a closed system. We just learns about chemical potential and the fundamental relations for an open system. I had a thought experiment that there may exist maxwells relations for an open...

Homework Statement Let ## H = \{ 2^{m} : m \in Z\}## A relation R defined in ##Q^{+} ## by ##aRb ##, if ## \frac{a}{b} \in H## a.) Show that R is an equivalence Relation b.) Describe the elements in the equivalence class [3]. The Attempt at a Solution For part a, I think I am able to solve...

i don't have a specific question. i just need an explanation on what this topic is about. i am not understanding it

If I have a relation which is not only antisymmetric (##aRb\rightarrow{}\neg{}bRa##) but it also has a property that ##aRb\land{}bRc\rightarrow{}\neg{}cRa##. How can I be sure that this property holds for any string like that? So that ##aRb\land{}bRc\land{}cRd\rightarrow{}\neg{}dRa## without...

I am having difficulty understanding the following problem. I feel it should be very simple but am unsure how to interpret it. A relation ##R## is defined on ##N## by ##aRb## if ##\frac{a}{b} \in N##. For ##c, d \in N##, under what conditions is ##c R^{-1} d##? (Exercise 8.6 from Chartrand...

Hi all, this is a problem that I want to start early on, but I'm not sure how to show my work for this. If i say the theorems it just feels repetitive. So I guess my question is, is using theorems enough to answer this? thank you! Homework Statement determine the properties of each...

Hey! :) Let $f,g: [a,b] \to \mathbb{R}$ integrable functions.Show that: $\int_{a}^{b}(f+g)=\int_{a}^{b}f+\int_{a}^{b}g$ We suppose the subdivision $P=\{a=t_0<t_1<...<t_n=b\}$ of $[a,b]$. Let $t \in [t_k,t_{k+1}]$. $$f(t) \leq sup f([t_k,t_{k+1}])$$ $$f(t) \geq inf f([t_k,t_{k+1}])$$ $$g(t)...

Hi, I have a transitive relation and wana build a complete set of pairs that reflect all (direct/indirect) relations among the pairs. Ex.: suppose I have this relation R = { (1,2), (2,3), (3,5), (5,7), (3,4) } I wana to produce this relation R oper R = { (1,2), (1,3), (1,4), (1,5)...

I'm having trouble with the following: Let R be a relation on A. Prove that if Dom(R) \bigcap Range(R) = ø, then R is transitive. I took the negation of the "R is transitive" to try proof by contrapositive (as the professor suggested), and have the following: \exists x,y,z \in A s.t. (x,y)...

Hi all, I am wandering if I can apply the Kramers-Kronig (KK) relations to the complex wavenumber k(ω) = k'(ω) + i k"(ω). I have a measurement that easily gives me k'(ω) for a certain range of frequencies, but where k"(ω) is unreliable. I would like to use KK to find k" from k'. According...

If f(x)= 2x+5, g(x)=0.5 and h(x)=3-1 find: fg(x), gf(x), fh(3) fg(x) fg(x)= 2(0.5x)+5 fg(x)= x+5 gf(x)= 0.5(2x+5) = x+2.5 fh(3) fh (x) =2(3-1)+5 = 6-2+5 = 4+5 this last part of the question been puzzling me... could I get a little...

What are the uncertainty relations for the following: 1. position and energy? 2. position and time?

I'm doing something horribly wrong in something that should be very easy. I want to show that: [L^2, x^2] = 0 So: [L^2, x x] = [L^2, x] x + x [L^2, x] L^2 = L_x^2 + L_y^2 + L_z^2 Therefore: [L^2, x] = [L_x^2 + L_y^2 + L_z^2, x] = [L_x^2, x] + [L_y^2, x] + [L_z^2, x] = L_y [L_y...

Homework Statement Show that if a set has 3 elements, then we can find 8 relations on A that all have the same symmetric closure. Homework Equations Symmetric closure ##R^* = R \cup R^{-1} ## The Attempt at a Solution If the symmetric closures of n relations are the same then...

Homework Statement I've actually got a couple questions, I'll provide an example for each question, but I'm not really looking for an answer to the example, but an explanation of the concept. I have very little to go on from class notes. We've had some inclement weather in these parts leading...

Hey everyone, I have three problems that I'm working on that are review questions for my Math Final. Homework Statement First Question: Determine if R is an equivalence relation: R = {(x,y) \in Z x Z | x - y =5} and find the equivalence classes. Is Z | R a partition? Homework...