In mathematics, a binary relation over sets X and Y is a subset of the Cartesian product X × Y; that is, it is a set of ordered pairs (x, y) consisting of elements x in X and y in Y. It encodes the common concept of relation: an element x is related to an element y, if and only if the pair (x, y) belongs to the set of ordered pairs that defines the binary relation. A binary relation is the most studied special case n = 2 of an n-ary relation over sets X1, ..., Xn, which is a subset of the Cartesian product X1 × ... × Xn.An example of a binary relation is the "divides" relation over the set of prime numbers
P
{\displaystyle \mathbb {P} }
and the set of integers
Z
{\displaystyle \mathbb {Z} }
, in which each prime p is related to each integer z that is a multiple of p, but not to an integer that is not a multiple of p. In this relation, for instance, the prime number 2 is related to numbers such as −4, 0, 6, 10, but not to 1 or 9, just as the prime number 3 is related to 0, 6, and 9, but not to 4 or 13.
Binary relations are used in many branches of mathematics to model a wide variety of concepts. These include, among others:
the "is greater than", "is equal to", and "divides" relations in arithmetic;
the "is congruent to" relation in geometry;
the "is adjacent to" relation in graph theory;
the "is orthogonal to" relation in linear algebra.A function may be defined as a special kind of binary relation. Binary relations are also heavily used in computer science.
A binary relation over sets X and Y is an element of the power set of X × Y. Since the latter set is ordered by inclusion (⊆), each relation has a place in the lattice of subsets of X × Y. A binary relation is either a homogeneous relation or a heterogeneous relation depending on whether X = Y or not.
Since relations are sets, they can be manipulated using set operations, including union, intersection, and complementation, and satisfying the laws of an algebra of sets. Beyond that, operations like the converse of a relation and the composition of relations are available, satisfying the laws of a calculus of relations, for which there are textbooks by Ernst Schröder, Clarence Lewis, and Gunther Schmidt. A deeper analysis of relations involves decomposing them into subsets called concepts, and placing them in a complete lattice.
In some systems of axiomatic set theory, relations are extended to classes, which are generalizations of sets. This extension is needed for, among other things, modeling the concepts of "is an element of" or "is a subset of" in set theory, without running into logical inconsistencies such as Russell's paradox.
The terms correspondence, dyadic relation and two-place relation are synonyms for binary relation, though some authors use the term "binary relation" for any subset of a Cartesian product X × Y without reference to X and Y, and reserve the term "correspondence" for a binary relation with reference to X and Y.
Hello, I need to solve the commutator relations above. I found the equation above for the last one, but I am not sure, if something similar applys to the first one. I am a little bit confused, because I know there has to be a trick and you don't solve it like other commutator.
Thanks for your help!
Here is the equation I obtain after simplification, I don't know if it is correct:
gmc * V1 + s * C2 * Vout = [{s * (C1 + C2) * ro2 + 1} * Vout - s * C1 * ro2 * V1] * (s * rb * C2 + 1) / {ro2 * rb * (s * C2 - gm2)}
I need to eliminate V1 to find the relation between Vin and Vout.
Hi
Would you explain to me what is the q^ and how they are related to completeness.How can i solve this exercise?It is from "Quarks and leptons An Introductory course in Modern Particle Physics" of Halzen and Alan D.Martin.Also, can you point me to a useful bibliography?
For example what is ##\frac {169}{13} = ?##
This says “When ##169## is divided into ##13## groups how many there are in each group?”
This can be converted into a multiplication problem like this “##13## groups of how many in each group makes ##169##?”
This is ##13 * ? = 169##. It can be solved...
As you can see in this picture: This explanation "relation between the normal and the slope of a curve" is formulated here:
$$\frac{1}{\rho} \frac{d\rho }{d\psi }=\tan\left(\frac{\theta+\psi}{2}\right)$$
I got confused because I don't have the curve equation(regarding the slope of the curve...
I just learned that if two linear operators do not commute, this means when we use operators to characterize observables in quantum mechanics, the corresponding observables cannot both be definite at the same time. This seems hard to believe to me since I have a strong intuition, perhaps...
The below is the diagram i want to find the phase and line-line voltage relation
I am finding difficulty in identifying the loop and applying the KVL.
One attempt is
U_{VA} - U_R - U_L -U_O - U_L - U_R = U_VB
U_{VA} - 2(U_R+U_L) = U_VB -> eq1
Is my attempt correct? i am not confident please help.
These are the FDs:
AB=>CD
C=>A
D=>B
My method of finding candidate keys is:
1) Look at RHS
2) Whatever isn't there could be a candidate key. (Find its closure).
But here everything is in RHS. So, I'm confused.
Can you share a better method to find candidate keys without getting too...
So, a friend of mine has attempted a solution. Unfortunately, he's having numbers spawn out of nowhere and a lot of stuff is going on there which I can't make sense of. I'm going to write down the entire attempt.
$$
0 \in X \; \text{otherwise no subgroup since neutral element isn't included}...
I drive car only at country roads allways at speeds 100-120km/h, no city and no idle time-heating engine etc.
Why computer allways show very low average speed , 40-48km/h?I allways have feeling that this speed is too low because I allways drive way faster then this. Indeed all my friends have...
SO2(g)+1/2O2(g)⇌SO3(g);ΔHo=-98.32KJ/mole,ΔSo=-95J/(mole-K).
find Kp at 298 Kelvin?
In given question at first Δ G will be calculated using formula ΔG = Δ H – T x ΔS, by putting the given values in formula we get ΔG = -70.01 kJ/mol.
Then Keq will be calculated using equation = Δ G = -RT ln Keq...
Hello,
When we consider a block sitting on a surface, the gravitational force ##W## and the normal force ##F_N## are applied to the block. Both equal i magnitude and opposite in direction. We call the normal force the reaction force exerted by the surface on the block.
Now we consider the...
Hi
I have just been looking at the derivation of the uncertainty relationship for non-commutating operators. I have come across the following quote in Quantum Mechanics by Mandl regarding the time-energy relationship. "Time is not an operator ; it is an ordinary parameter which commutes with...
Here, it's shown how white light, after passing from air to another medium, gets broken down into its constituent coloured rays. Each has its own refractive index in the medium, but it's only shown here red, blue and yellow. The auther comments on this image and says that, for small angles of...
I am going through my class notes and trying to prove the middle commutator relation,
I am ending up with a negative sign in my work. It comes from [a†,a] being invoked during the commutation. I obviously need [a,a†] to appear instead.
Why am I getting [a†,a] instead of [a,a†]?
Hello,
Given a set of numbers, we can calculate their average and there are different types of averages (arithmetic, weighted, harmonic, geometric, etc.) The choice of the average depends on the situation. The average, also called mean or expectation value, is a number that can replace all the...
Find text (question and working to solution here ...this is very clear to me...on the use of implicit differentiation and quotient rule to solution). I am seeking an alternative approach.
Now from my study we can also have; using partial derivatives...
I don't know how to do a search for information on a specific equation. It's f(n + 1) = 2 - \dfrac{d(n)}{f(n)}, where d(n) is more or less arbitrary. It came up in some work I've been doing and I can't seem to get anywhere with it. Being non-linear it may not even have a closed form solution...
So IPV4 header has a topic called "total header length" and it is of 16 bits. That means it can count from 0-65535. Book says it means IP datagram is limited to 65535 bytes. how do we get to idea of 65535 bytes? is it 1 memory location=1 byte idea?
It doesn't make any sense to me(I have studied...
We mainly have to prove that this quantity## \bra{\varphi} A^{\otimes n } \ket{\varphi} \pm \bra{\varphi} B^{\otimes n } \ket{\varphi} ##
is greater or equal than zero for all ##\ket{\varphi}##.
Being ##\ket{\varphi}## a product state it is straightforward to demonstrate such inequality. I am...
hello,
1. according to Robert Wald, General Relativity, equation (4.2.22)
the magnetic field as measured by an observer with 4-velocity ## v^b ## is given by
## B_a = - \frac {1}{2} {ϵ_{ab}}^{cd} F_{cd} v^b ##
where ## {ϵ_{ab}}^{cd}##, the author says, is the totally antisymmetric tensor (for...
Hello,
this is my first thread.
Robert Wald, in General Relativity, equation (4.2.8) says :
E = – pa va
where E is the energy of a particle, pa the energy-momentum 4-vector and va the 4-velocity of the particle. How can I see this is compatible with the common energy-momentum-relation E2 – p2 =...
I'm studying orbital angular momentum in the quantum domain, and I've come up with the Robertson uncertainty relation for the components of orbital angular momentum. Therefore, I read that it is necessary to pay attention to the triviality problem, because in the case where the commutator is...
How does electrical potential and electric fields change from dipole to parallel dipoles? What does this math demonstrate? What does the equations mean?
There's a famous functional equation that was asked in the 2019 IMO. It looks like this: find all f: Z -> Z where f(2a)+2f(b)=f(f(a+b)).
I thought of solving it using a recurrence relation where a_n=f(nx). But when I substituted values in the functional equation (after setting a and b equal...
Hi. A electromagnetic wave consists of an electric and a magnetic component. I believe that the electric field strength is measured in volts per meter. The magnetic field I think is measured in Tesla. Let's imagine that I measure the electic field strength of two different radio stations and...
Hello,
I am running experiments where materials are heated at high temperatures during tens of hours under high-vacuum conditions. Since what I am investigating lies in the first hundreds of nanometers of the materials, I must take into account (and anticipate) surface sublimation.
Therefore I...
In a scenario of a free-falling object in a vacuum on earth, the object will be acceleration towards the earth. According to the theorem of Work-Kinetic and Work-Potential:
* Since the object is accelerating towards the earth, we know that the object's Kinetic energy is increasing because the...
It is a theorem that: two propositions implying each other, in the sense that the set of outcomes making one true is the same as the one making the other true) have the same probability. this comes from the fact that if p --> q, the P(p&q) = P(p), we have that if p <-> q, then P(p&q) = P(p)=...
my_age = 10
if my_age >= 100:
print("One hundred years old! Very impressive.")
elif my_age <= 3:
print("Awwww. Just a baby.")
else:
print("Ah - a very fine age indeed")
https://www.fullstackpython.com/blog/python-basic-data-types-booleans.html
Article says-:
my_age = 10
if my_age >= 100:
print("One hundred years old! Very impressive.")
elif my_age <= 3:
print("Awwww. Just a baby.")
else:
print("Ah - a very fine age indeed")
https://www.fullstackpython.com/blog/python-basic-data-types-booleans.html
Article says-:
Hey! :giggle:
Give for the following expressions of relation algebra the equivalent expression in relational calculus.
1. $\sigma_{B=A}(R(A,B,C))$
2. $\pi_{B,C}(R(A,B,C))$
3. $R(A,B,C)\cup S(A,B,C)$
4. $R(A,B,C)\cap S(A,B,C)$
5. $R(A,B,C)\setminus S(A,B,C)$
6. $R(A,B,C)\times S(D,C,E)$
7...
In post #30 of a now closed thread, vanhees71 wrote:
The quote in question is from his 1930 book "The Physical Principles of the Quantum Theory" on p.20 of the English translation of the German original. There Heisenberg writes:
Hi all,
I've a doubt about the following formula for the number of sidereal vs solar days for a generic planet orbit (e.g. the Earth's orbit around the Sun):
$$N_{sid} = N_{sol} + 1$$
Section 1.5 of the book "Foundation of Astrophysics" - B. Ryden shows how to calculate the above equation...
First, since A and B are articulated, the moments due to A and B are zero. Now, we may call reaction forces in A, ##V_A## and ##H_A## and in the same way, call the reactions in B as ##V_B## and ##H_B##. With that and Newton's third law, I managed to find three equations (equilibrium of...
I am guessing time-energy uncertainty relation is the way to solve this. I solved the Schrodinger equation for both the regions and used to continuity at ##x=-a, 0,a## and got ##\psi(-a<x<0) = A\sin(\kappa(x+a))## and ##\psi(0<x<a) = -A\sin(\kappa(x-a))## where ##\kappa^2 = 2mE/\hbar^2##...
Suppose we have specific amount of hydrogen gas enclosed inside a metallic chamber and that is connected to a very high positive voltage source. As the voltage is positive and that's so high that all the molecules inside the chamber lost their electrons and there is nothing but a nuclei gas is...
Are there any papers or articles that reference black holes being the creations of the big bang or being considered in creating universes in alternate dimensions? Thanks for the help.
Hello,
I understand that webpages are created using HTML which defines the structure of the document using a finite number of specified tags, like <p>, <head> etc.
In XML, the user can create customized tags (as long as certain rules are respected). The XML file recipient must then be able to...
Hello,
If i have this relation:
## M_p^2 = M^3_s V ##
where ##M_p ## and ##M_s ## are masses in GeV and V is a length. Let ## M_p = 10^{18} ~ ## GeV and ##M_s = 10^3 ## GeV , what is V in meters ?
My solution :
The equation becomes
## V = 10^{30}## GeV , but ## 1 m \sim 10^{15} ~...
The given lagrangian doesn't seem to correspond to any of the basic systems (like simple/ coupled harmonic oscillators, etc). So I calculated the momentum ##p## which is the partial derivative of ##L## with respect to generalized velocity ##\dot{q}##. Doing so I obtain
$$p =...
I have a simple question sort of about exact differentials and deciding which variables matter and when.
I know we can write entropy ##S## as ##S(P,T)## and ##S(V,T)## to derive different relations between heat capacities ##C_V## and ##C_P##. I was wondering if it is technically correct to...
From a previous post about the Relationship between the angular and 3D power spectra , I have got a demonstration making the link between the Angular power spectrum ##C_{\ell}## and the 3D Matter power spectrum ##P(k)## :
1) For example, I have the following demonstration,
##
C_{\ell}\left(z...
Given the commutation relation
$$\left[\phi\left(t,\vec{x}\right),\pi\left(t,\vec{x}'\right)\right]=i\delta^{n-1}\left(\vec{x}-\vec{x}'\right)$$
and define the Fourier transform as...
Hi,
On slide 9 of this presentation: http://www.globalcommhost.com/rogers/acs/techsupporthub/en/docs/MWJ_webinar_June20_2017_JC_microstrip_coplanar_stripline_final.pdf it states the signal wavelength can be changed with dielectric constant Dk.
As far as I understand the wavelength and the...
For a steady, non-viscous and incompressible flow, one can apply both Bernoulli's principle (no potentials) as
$$p+\frac{\rho v^2}{2} = p_t$$
where ##p##, ##\rho,##, ##v##, and ##p_t## are static pressure, density, flow velocity, and total pressure, respectively,
and continuitiy principle as...
In AdS/CFT, we have the GKP-witten relation
$$\left< \exp \left( i \int \phi^{(0)} O \right) \right> = e^{-S[\phi^{(0)}]}$$
why is it natural to formulate string theory on an AdS space? is it a natural background for some particular definite reasons? Is holography naturally formulated in an...