What is Relation: Definition and 1000 Discussions

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


{\displaystyle \mathbb {P} }
and the set of integers


{\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.

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  1. J

    Solve the commutator relations

    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!
  2. D

    Find the relation between 2 variables

    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.
  3. D

    Question related to completeness relation for photons

    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?
  4. M

    B Relation between Division and multiplication

    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...
  5. baby_1

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  6. richard_andy

    A Relation between the density matrix and the annihilation operator

    This question is related to equation (1),(3), and (4) in the [paper][1] [1]: https://arxiv.org/abs/2002.12252
  7. E

    I Non-Commutation Property and its Relation to the Real World

    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...
  8. PhysicsTest

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    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.
  9. shivajikobardan

    Comp Sci How to find candidate keys in this relation?

    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...
  10. P

    I Deriving the Planck relation

    How do one derive the relation E=hf?
  11. PhysicsRock

    Prove relation between the group of integers and a subgroup

    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}...
  12. J

    I Car low average speed vs aero drag relation

    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...
  13. tbn032

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    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...
  14. F

    I Reaction force of and its relation to normal force and friction

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  15. D

    I Time-energy uncertainty relation

    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...
  16. Ahmed1029

    I How can I derive this relation from Snell's law?

    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...
  17. guyvsdcsniper

    Proving commutator relation between H and raising operator

    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†]?
  18. F

    I Averages and average speed/instantaneous speed relation

    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...
  19. F

    A Relation between a_{\ell m} noise and Poisson noise with C_{\ell}

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  20. chwala

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  21. topsquark

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  22. shivajikobardan

    Comp Sci Total length IP header and 65535 bytes relation-:

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  23. J

    A Tensor product matrices order relation

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  24. C

    A Relation of Electromagnetic Field & Field Tensor

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  25. C

    B Robert Wald's General Relativity: Energy-Momentum Relation

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  26. Yan Campo

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  27. coulthardb

    Please explain the relation behind this?

    How does electrical potential and electric fields change from dipole to parallel dipoles? What does this math demonstrate? What does the equations mean?
  28. MevsEinstein

    B Recurrence relation in a recurrence relation?

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  29. R

    I Relation between electric & magnetic fields in terms of field strength

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  30. D

    Relation between enthelpy of sublimation and rate of sublimation?

    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...
  31. aljan9559

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    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...
  32. gregthenovelist

    I Equivalence relation and different sample spaces

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  33. shivajikobardan

    Python What does this code signify in relation to boolean logic?

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  34. shivajikobardan

    Python-What does this code signifies in relation to boolean logic?

    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-:
  35. M

    MHB Relation Algebra - Relational Calculus

    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...
  36. A. Neumaier

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  37. cianfa72

    I About the relation between number of sidereal and solar days

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  38. Moara

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  39. Wannabe Physicist

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  40. T

    A Relation of ionisation voltage vs. pressure for hydrogen

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  41. E

    I Black holes and relation to creation

    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.
  42. F

    Where does XML fit in relation to website and webpages?

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  43. Safinaz

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  44. Wannabe Physicist

    Lagrangian Problem (Find Relation between Amplitude and Momentum)

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  45. cwill53

    I Entropy and Heat Capacity Relation

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  46. F

    A Relation between Matter Power spectrum and Angular power spectrum

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  47. U

    Question on discrete commutation relation in QFT

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  48. S

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  49. lahanadar

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  50. D

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