“relation”的概念、定义、翻译、参考文献-科学参考

relation

Mathematics

A relation on a set S is usually a binary relation on S, though the notion can be extended to involve more than two elements. An example of a ternary relation, involving three elements, is ‘a lies between b and c’, where a, b, and c are real numbers. See also equivalence relation.
See presentation.

Computer

(defined on sets S₁, S₂,… S_n) A subset R of the Cartesian product
$S_{1} \times S_{2} \times \dots \times S_{n}$
of the n sets S₁, …, S_n. This is called an n-ary relation. When a relation R is defined on a single set S the implication is that R is a subset of
$S \times S \times \dots \times S (n terms)$
The most common situation occurs when n= 2, i.e. R is a subset of S₁ × S₂. Then R is called a binary relation on S₁ to S₂ or between S₁ and S₂. S₁ is the domain of R and S₂ the codomain of R. If the ordered pair (s₁,s₂) belongs to the subset R, a notation such as
$s_{1} R s_{2} or s_{1} ρ s_{2}$
is usually adopted and it is then possible to talk about the relation R or ρ‎ and to say that s₁ and s₂ are related.
An example of a binary relation is the usual ‘is less than’ relation defined on integers, where the subset R consists of ordered pairs such as (4,5); it is however more natural to write 4 < 5. Other examples include: ‘is equal to’ defined on strings, say; ‘is the square root of’ defined on the nonnegative reals; ‘is defined in terms of’ defined on the set of subroutines within a particular program; ‘is before in the queue’ defined on the set of jobs awaiting execution at a particular time.
The function is a special kind of relation. Graphs are often used to provide a convenient pictorial representation of a relation.
Relations play an important part in theoretical aspects of many areas of computing, including the mathematical foundations of the subject, databases, compiling techniques, and operating systems. See also equivalence relation, partial ordering.

Philosophy

Philosophically relations are interesting because of the historic prejudice, given its most forceful expression by Leibniz, that they are somehow ‘unreal’ compared to the intrinsic, monadic properties of things. A way of putting the idea is that if all the monadic properties of the objects of a domain are fixed, then the relational properties are fixed as well (relations supervene on monadic properties). But in modern logic and science there is no justification for this claim.
The central notions in the logical treatment of relations are as follows. The domain of a relation is the set of things that bear the relation to something. The range is the set of things that have something bear the relation to them. The field of a relation is the set of things that belong either to its domain, or to its range. A binary relation relates one element from its domain to one of its range: Rxy. Relations may be defined over greater numbers of things: for example, we can define Rxyz to be the relation holding between three numbers when x+y=z, and so up to relations defined over n-tuples of any size. For formal and mathematical purposes a relation may be identified with the class of ordered pairs (or in general ordered n-tuples) that satisfy it. So ‘father of’ becomes the set of all pairs x,y, such that x is the father of y, ‘is greater than’ becomes the set of all pairs x,y, such that x is greater than y, and so on. The main properties to be noticed in the theory of relations are indexed under their own headings: see antisymmetric, asymmetric, equivalence relation, irreflexive, ordering relation, reflexive, symmetric, transitive.

单词	relation
释义	relation Mathematics A relation on a set S is usually a binary relation on S, though the notion can be extended to involve more than two elements. An example of a ternary relation, involving three elements, is ‘a lies between b and c’, where a, b, and c are real numbers. See also equivalence relation. See presentation. Computer (defined on sets S₁, S₂,… S_n) A subset R of the Cartesian product $S_{1} \times S_{2} \times \dots \times S_{n}$ of the n sets S₁, …, S_n. This is called an n-ary relation. When a relation R is defined on a single set S the implication is that R is a subset of $S \times S \times \dots \times S (n terms)$ The most common situation occurs when n= 2, i.e. R is a subset of S₁ × S₂. Then R is called a binary relation on S₁ to S₂ or between S₁ and S₂. S₁ is the domain of R and S₂ the codomain of R. If the ordered pair (s₁,s₂) belongs to the subset R, a notation such as $s_{1} R s_{2} or s_{1} ρ s_{2}$ is usually adopted and it is then possible to talk about the relation R or ρ‎ and to say that s₁ and s₂ are related. An example of a binary relation is the usual ‘is less than’ relation defined on integers, where the subset R consists of ordered pairs such as (4,5); it is however more natural to write 4 < 5. Other examples include: ‘is equal to’ defined on strings, say; ‘is the square root of’ defined on the nonnegative reals; ‘is defined in terms of’ defined on the set of subroutines within a particular program; ‘is before in the queue’ defined on the set of jobs awaiting execution at a particular time. The function is a special kind of relation. Graphs are often used to provide a convenient pictorial representation of a relation. Relations play an important part in theoretical aspects of many areas of computing, including the mathematical foundations of the subject, databases, compiling techniques, and operating systems. See also equivalence relation, partial ordering. Philosophy Philosophically relations are interesting because of the historic prejudice, given its most forceful expression by Leibniz, that they are somehow ‘unreal’ compared to the intrinsic, monadic properties of things. A way of putting the idea is that if all the monadic properties of the objects of a domain are fixed, then the relational properties are fixed as well (relations supervene on monadic properties). But in modern logic and science there is no justification for this claim. The central notions in the logical treatment of relations are as follows. The domain of a relation is the set of things that bear the relation to something. The range is the set of things that have something bear the relation to them. The field of a relation is the set of things that belong either to its domain, or to its range. A binary relation relates one element from its domain to one of its range: Rxy. Relations may be defined over greater numbers of things: for example, we can define Rxyz to be the relation holding between three numbers when x+y=z, and so up to relations defined over n-tuples of any size. For formal and mathematical purposes a relation may be identified with the class of ordered pairs (or in general ordered n-tuples) that satisfy it. So ‘father of’ becomes the set of all pairs x,y, such that x is the father of y, ‘is greater than’ becomes the set of all pairs x,y, such that x is greater than y, and so on. The main properties to be noticed in the theory of relations are indexed under their own headings: see antisymmetric, asymmetric, equivalence relation, irreflexive, ordering relation, reflexive, symmetric, transitive.
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