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




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.

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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

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  4. M

    B Relation between Division and multiplication

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  5. baby_1

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  6. richard_andy

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  7. E

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

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  8. PhysicsTest

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  9. shivajikobardan

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  10. P

    I Deriving the Planck relation

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  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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  14. F

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  15. D

    I Time-energy uncertainty relation

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  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

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  18. F

    I Averages and average speed/instantaneous speed relation

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  19. F

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  20. chwala

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  21. topsquark

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  22. shivajikobardan

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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

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  26. Yan Campo

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  27. coulthardb

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  28. MevsEinstein

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  29. R

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  30. D

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  31. aljan9559

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  32. gregthenovelist

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  33. shivajikobardan

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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

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  36. A. Neumaier

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  37. cianfa72

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  38. Moara

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  39. Wannabe Physicist

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  40. T

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  41. E

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  42. F

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  43. Safinaz

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  44. Wannabe Physicist

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  45. cwill53

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  46. F

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  47. U

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  48. S

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  49. lahanadar

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  50. D

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