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关系(relation)

在逻辑上，关系R被认为是被指定的两者、三者或四者等间的关联。所指定的两者被称为两地(或双值)关系，所指定的三者则称为三地（或三值）关系，依此类推。一般说来，关系是一种被指定的数项物件之间的关联。关系的重要特质包括了对称性（symmetry）、传递性（transitivity）和反射性（reflexivity）。两地（或双值）关系被称为关系R，R若存在於x和y值之间则可视为对称，可以(@8(LogicAll.jpgx) (@8(LogicAll.jpgy) [RxyRyx]表示之，「x与y并行」则为对称关系之一例。R在一个物件和第二个物件、或第二个物件与第三个物件之间则为传递性，可以(@8(LogicAll.jpgx) (@8(LogicAll.jpgy) (@8(LogicAll.jpgz ) [(Rxy@8(logicAnd.jpgRyz)Rxz]表示之，「x大於y」则为其例。R在任何物件与其自身皆具反射性，可以(@8(LogicAll.jpgx) Rxx表示之，「x至少跟y一样高」为其例，因为x总是「至少与自己一般高」。

English version：

relation

In logic, a relation R is defined as a set of ordered pairs, triples, quadruples, and so on. A set of ordered pairs is called a two-place (or dyadic) relation; a set of ordered triples is a three-place (or triadic) relation; and so on. In general, a relation is any set of ordered n-tuples of objects. Important properties of relations include symmetry, transitivity, and reflexivity. Consider a two-place (or dyadic) relation R. R can be said to be symmetrical if, whenever R holds between x and y, it also holds between y and x (symbolically, (∀x) (∀y) [Rxy ⊃ Ryx]); an example of a symmetrical relation is "x is parallel to y". R is transitive if, whenever it holds between one object and a second and also between that second object and a third, it holds between the first and the third (symbolically, (∀x) (∀y) (∀z ) [(Rxy ∧ Ryz) ⊃ Rxz]); an example is "x is greater than y". R is reflexive if it always holds between any object and itself (symbolically, (∀x) Rxx); an example is "x is at least as tall as y" since x is always also "at least as tall" as itself.

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