Relations Definition and 540 Threads

<Moderator's note: Moved from a technical forum and thus no template. Also re-edited: Please use ## instead of $$.> If ##R_{1}## and ##R_{2}## are relations on a set S with ##R_{1};R_{2}=I=R_{2};R_{1}##. Then ##R_{1}## and ##R_{2}## are bijective maps ##R_{1};R_{2}## is a composition of two...

Homework Statement Prove the following relation for which clausius equation holds : Cs=Cp-αV(ΔH/ΔV) Where Cs=∂q/∂T at constant S and is the heat capacity in the coexistence line of 2 phases Homework Equations dq=dU+dW dP/dT=ΔH/(ΔV*T) The Attempt at a Solution I do not fully understand why q...

Homework Statement [/B] I'm stuck on part c of the attached problem: Homework Equations $$C_P - C_V = \left[P + \left( \frac {∂U}{∂V} \right)_T \right]\left( \frac {∂V}{∂T} \right)_P$$ $$P + \left( \frac {∂U}{∂V} \right)_T = T \left( \frac {∂P}{∂T} \right)_V$$ $$\left(P + \frac {a}{V^2}...

Hello. In fields and waves and transmission lines We have a attenuation constant formula with power relations: 1- Alpha = 1/2R0(R+G|Z0|^2) 2-Also we can calculate attenuation constant from these : Z0= radical[(R+jwL) / (G+jwc)] gamma = radical[(R+jwL)(G+jwc)] I want show alpha from 1...

If we have a relation, ##R##, and it's inverse, ##R^{-1}## they behave such that a point on ##R##, say (a,b), corresponds to the point (b,a) on ##R^{-1}## This is a reflections across the line y=x. This relation does not mean that ##R^{-1}## is a function. For example, Let ##R## be...

For the free boson, the field operators satisfies the commutation relation, $${\varphi}_{x'}{\varphi}_{x} - {\varphi}_{x}{\varphi}_{x'} = 0$$ at equal times. While the fermions satisfies, $${\psi}_{x'}{\psi}_{x} + {\psi}_{x}{\psi}_{x'} = 0$$ at equal times. I interpret ##{\varphi}_{x}## and...

I'm studying Thermodynamics and I'm a little stuck at this problem. 1. Homework Statement Starting with the first Maxwell relation, derive the remaining three by using only the relations: $$\left(\frac{\partial x}{\partial y}\right) _{z} \left(\frac{\partial y}{\partial z}\right) _{x}...

I have this question:2. Let R be a relation on Z with $$R = {(a,b) : |a−b| < 3}.$$ (1) Is R reflexive? (If yes, prove it; if no, give a counterexample) (2) Is R symmetric? (If yes, prove it; if no, give a counterexample) (3) Is R antisymmetric? (If yes, prove it; if no, give a counterexample) (4)...

Hi, I have a question about the bending of spacetime and its proportion and relation to the mass which causes the bending; and also how the bent space would interact with other objects as they come closer. I'm going to ask a more detailed question with some possible problems I would like to...

Homework Statement Given that ##t_1 = 1## and ##\displaystyle t_{n+1} = \frac{t_n^2 + 2}{2t_n}## for ##n \ge 1##. Show that ##t_n > 0## for all ##n##. Homework EquationsThe Attempt at a Solution Intuitively this is obvious. Since ##t_1## is positive, so is ##t_2##, and so on. But I am having...

Hello - I am having a tough time understanding the problems in the attached picture (Problem 13). My issue is understanding how I plug in the proposed solutions, specifically those that include n. I am able to solve A and B but unable to solve the rest. For instance, how do I plug in C or...

Homework Statement Derive, using the canonical commutation relation of the position space representation of the fields φ(x) and π(y), the corresponding commutation relation in momentum space.Homework Equations [φ(x), π(y)] = iδ3(x-y) My Fourier transforms are defined by: $$ φ^*(\vec p)=\int...

<Moderator's note: This is a spin-off from another thread.> I will find out axioms to find out an answer to a question - axioms guarantees that my solution to a mathematical problem is correct. I have another question: A function 'y' in 'x' yields a single value as output on an input. Is there...

New to set and graph theory and need help on how to approach these exercise questions: For each of the following relations, state whether the relation is: i) reflexive ii) irreflexive iii) symmetric iv) anti-symmetric v) transitive Also state whether the relation is an equivalence or partial...

Hey! :o I want to prove the following relations of condition number: $\operatorname{cond}(\alpha A)=\operatorname{cond}(A)$. The matrixnorm is submultiplicativ. $\operatorname{cond}_2(U)=1$ if $U$ is an orthogonal matrix. $\operatorname{cond}_2(UA)=\operatorname{cond}_2(A)$, $U$ is...

I am reading "Introduction to Set Theory" (Third Edition, Revised and Expanded) by Karel Hrbacek and Thomas Jech (H&J) ... ... I am currently focused on Chapter 2: Relations, Functions and Orderings; and, in particular on Section 5: Orderings I need some help with H&J's depiction of...

I am reading the book: "Discovering Modern Set Theory. I The Basics" (AMS) by Winfried Just and Martin Weese. I am currently focused on Chapter 2 Partial Order Relations ... I need some help with Exercise 1(a) ... indeed, I have been unable to make a meaningful start on the exercise ... :(...

Given a right ABC trigon with the right angle at A and two points D,E on BC such that: (BD)=(DE)=(EC) Prove: $$(AD)^2+(DE)^2+(AE)^2= \frac{2}{3}(BC)^2$$

(This is not a homework problem). I'm an undergrad physics student taking my second course in quantum. My question is about operator methods. Most of the proofs for different commutation relations for qm operators involve referring to specific forms of the operators given some basis. For...

What math is useful for distinguishing and classifying things based only on relations they satisfy? For example the relation ##R_1 = \{(a,b), (b,a)\}## isn't useful for distinguishing "a" from "b" while the relation ##R_2 = \{(a,b), (c,b) \}## let's us distinguish "b" by the description "The...

Homework Statement Prove or disprove: Every translation is a product of two non-involutory rotations. Homework EquationsThe Attempt at a Solution :[/B] I am not sure if I got the right proof for the special situation: A translation is the product of two reflections with parallel reflections...

I have an exercise in my set theory book that states the following: Show that a set ##A## is a relation iff ##A \subseteq \operatorname{dom} A\times \operatorname{ran} B##. This is an easy exercise, so I am not asking how to prove it. However, I am confused about one thing. The forward...

I use the word «field» in purely algebraic sense here. Sometimes, when reading textbooks I encounter sentences like «Although the formulae in this section derived for the field of real numbers, they remain valid for complex numbers field as well». Or even more general variant of it: «...remain...

I'm trying to derive the commutation relations of the raising and lowering operators for a complex scalar field and I had a question. Let's start with the commutation relations: $$[\varphi(\mathbf{x},t),\varphi(\mathbf{x}',t)]=0$$ $$[\Pi(\mathbf{x},t),\Pi(\mathbf{x}',t)]=0$$...

Hello! I am reading Peskin's book on QFT and at a point he wants to show that the Dirac field can't be quantified using this commutation relations: ##[\psi_a(x),\psi_b^\dagger(x)]=\delta^3(x-y)\delta_{ab}## (where ##\psi## is the solution to Dirac equation). I am not sure I understand the math...

Homework Statement Show that ##|l, m\rangle## for ##l=1## vanishes for the commutator ##[l_i^2, l_j^2]##. Homework Equations ##L^2 = l_1^2 + l_2^2 + l_3^2## and ##[l_i^2,L^2]=0## The Attempt at a Solution I managed to so far prove that ##[l_1^2, l_2^2] = [l_2^2, l_3^2] = [l_3^2, l_1^2]##. I...

Hello all, I have a few questions related to the different number of equivalence classes under some constraint. I don't know how to approach them, if you could guide me to it, maybe if I do a few I can do the others. Thank you. Given the set A={1,2,3,4,5}, 1) How many different equivalence...

Homework Statement Consider a two-dimensional harmonic oscillator, described by the Hamiltonian ##\hat H_0 = \hbar \omega (\hat a_x \hat a_x ^{\dagger} + \hat a_y \hat a_y^{\dagger} + 1)## Calculate ##\hat H_0 \hat L | n_1, n_2 \rangle## and ##\hat L \hat H_0 |n_1, n_2 \rangle##. What does...

Homework Statement Im stuck on a old exam in plasma physics. It is about how to determine dispersion relations for high frequency waves in cold plasma's. I'm not sure how they do in the solution manual. Homework Equations B = B_0z^ E = E_0exp(i(kx-wt))z^ The Attempt at a Solution The...

Homework Statement [/B] In the equation between (3) and (2), why does the author says that ? Isn't the trigonometric identity actually ? 2. Homework Equations The Attempt at a Solution

Lagrangian is defined by ##L=L(q_i,\dot{q}_i,t)## and hamiltonian is defined by ##H=H(q_i,p_i,t)##. Why there is relation H=\sum_i p_i\dot{q}_i-L end no H=L-\sum_i p_i\dot{q}_i or why ##H## is Legendre transform of ##-L##?

Thermal coefficient of pressure is defined by \gamma=\frac{1}{P}(\frac{\partial P}{\partial T})_V . Why in books authors uses ##\gamma=\gamma(P,T)## and no ##\gamma=\gamma(P,T,V)##. Could you explain me this. I am sometimes confused with this dependences in thermodynamics.

Simultaneous tracking of spin angle and amplitude beyond classical limits Nature 543, 525–528 (23 March 2017) www.nature.com/nature/journal/v543/n7646/abs/nature21434.html The authors remark that "because spins obey non-Heisenberg uncertainty relations, enables simultaneous precise knowledge...

Suppose that the relation R is defined on the set Z where aRb means a = ±b. Establish whether R is an equivalence relation giving your justifications. Find the set of solutions of each of the linear congruence: a) x ≡ 3 (mod 5). b) 2x ≡ 5 (mod 9).(please write the full solutions thanks)

Homework Statement "Determine the max number of elements in a three-element set that is not reflexive, symmetric, or transitive?" Homework Equations ##a R b⇔(a,b)∈R## The Attempt at a Solution Basically, my professor has stated that there are a total number of seven possible elements in a...

Homework Statement A set ##P=\left\{ \ p1,p2,p3,p4 \right\}## is given. Determine the number of antisimmetric relations of this set so that ##p1## is in relation with ##p3##, ##p2## is in relation with ##p4## but ##p2## is not in relation with ##p1##. Homework Equations 3. The Attempt at a...

I'm having issues with the first four questions and have uploaded them. My attempts are shown below. 1. a) True, all elements of E are even b) False, 0 is not a multiple of 3 c) True, 8 is even and 9 is a multiple of 3 d) No idea e) False, 6 is an element of E and T f) No idea 2. a) You can...

Homework Statement I am trying to calculate the expectation value of ##\hat{P}^3## for the harmonic oscillator in energy eigenstate ##|n\rangle## Homework EquationsThe Attempt at a Solution [/B] ##\hat{P}^3 = (i \sqrt{\frac{\hbar \omega m}{2}} (\hat{a}^\dagger - \hat{a}))^3 = -i(\frac{\hbar...

Homework Statement In an attempt to find a transfer function of the system, I need to come up with equations that I can use to solve for unknowns. See the attached image to see the diagram of the pulley system. J is the moment of inertia, r is the radius. The smaller radius on pulley 2 is r1...

Homework Statement I need to prove the following equation: Homework Equations The 4 maxwell relations and their derivations: https://en.wikipedia.org/wiki/Maxwell_relations The Attempt at a Solution I started out with the fundamental equations of dU=TdS - PdV and as dS=0, and Cv=(dU/dT)v; I...

Given a set, there are subsets and possible relations between those arbitrary subsets. For a given example set, the possible relation between the subsets of the example set will narrow down to the "true" possible relations between those subsets. a) {1} Number of Subsets: ##2^1 = 2## (∅, {1})...

When it comes to analytic geometry, I am little confused about the use of vectors. For example, throughout high school, one works in ##\mathbb{R}^2##, and geometric objects such as lines are described using equations relating two variables, the x and y coordinate, such as y = 2x + 1. However...

Photon energy E_p= hv=hc/lamda taking lamda= h/mc which is the electron Compton wavelength and substituting in above E_p=mc^2 L(angular momentum)=r X P=(Lamda/2)*(E_p/c)=h/2 are these results coincident or have any physical meaning, they relate a photon wavelength equal to an electron...

I'm a little stuck here. I should determine the following determinant. I first tried to simplify it a little by using elemntary transformations. And then I did Laplace expansion on the last row. $\begin{vmatrix}2 & 2 & \cdots & 2 & 2 & 1 \\ 2 & 2 & \cdots & 2 & 2 & 2 \\ 2 & 2 & \cdots & 3 & 2 &...

Hello! (Wave) We set $L_k \equiv u_{tt}+\frac{k}{t}-\Delta u=0, k \in \mathbb{N}_0$. I am looking at the proof of the following proposition: We suppose that $g(x)$ is twice differentiable in $\mathbb{R}^n$. Then the mean value of $g(x)$ at the sphere with radius $t$ and center at $x$, which...

How many different relations are possible? Is the question. Is the answer the power set of AxA? 2^36.

Homework Statement Given the entropy of a system : $$ S = AU^αV^βN^{1-α-β} $$ The problem requires me to write $$ (\frac{∂T}{∂U})_{V,N} > 0, 　(\frac{∂P}{∂V})_{U,N} < 0, (\frac{∂μ}{∂N})_{U,V} > 0$$ to find the mathematical constraint of α and β Homework Equations dU = TdS - PdV + μdN The...

Homework Statement This is question 2.18 from Bowley and Sanchez, "Introductory Statistical Mechanics" . Show with the help of Maxwell's Relations that $$T dS = C_v dT + T (\frac{\partial P}{\partial T})_V dV$$ and $$TdS = C_p dT - T( \frac{\partial V}{\partial T})_P dP.$$ Then, prove that...

Mentor note: moved to homework section y = sin(x) y = cos(x) y = tan(x) y = csc(x) y = sec(x) y = cot(x) (a) 0 (b) 4 (c) 6 (d) 2 I thought it was (c) because i graphed all the trig functions and they passed the vertical line test but the answer sheet is saying (d) 2

Homework Statement Homework Equations I don't think there are any in this case The Attempt at a Solution I know that in order to prove R is an equivalence relation, I'd have to show that it is Reflexive, Symmetric, and Transitive. I'm not sure why, but I'm finding this a bit difficult in...