“传递关系”的版本间的差异

2020年11月1日 (日) 22:45的版本

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In mathematics, a homogeneous relation $R$ over a set $X$ is transitive if for all elements $a$ , $b$ , $c$ in $X$ , whenever $R$ relates $a$ to $b$ and $b$ to $c$ , then $R$ also relates $a$ to $c$ . Each partial order as well as each equivalence relation needs to be transitive.

In mathematics, a homogeneous relation over a set is transitive if for all elements , , in , whenever relates to and to , then also relates to . Each partial order as well as each equivalence relation needs to be transitive.

在数学中，一个集合上的齐次关系是可传递的，如果对于所有元素，In，whenever releved to 和 to，then 也 releved to。每个偏序和每个等价关系都需要是可传递的。

Definition

模板:Stack

A homogeneous relation $R$ on the set $X$ is a transitive relation if,^[1]

A homogeneous relation on the set is a transitive relation if,

集合上的齐次关系是传递关系，如果,

for all

a, b, c \in X

, if

a R b

and

b R c

, then

a R c

.

for all , if and , then .

对所有人来说，如果，那么。

Or in terms of first-order logic:

或者就一阶逻辑而言:

[math]\displaystyle{ \forall a,b,c \in X: (aRb \wedge bRc) \Rightarrow aRc, }[/math]

对于 x 中的所有 a，b，c: (aRb 楔形 bRc)右塔罗弧

where $a R b$ is the infix notation for $(a, b) \in R$ .

where is the infix notation for .

中缀符号在哪里。

Examples

As a nonmathematical example, the relation "is an ancestor of" is transitive. For example, if Amy is an ancestor of Becky, and Becky is an ancestor of Carrie, then Amy, too, is an ancestor of Carrie.

作为一个非数学的例子，关系“ is an ancestor of”是可传递的。例如，如果艾米是贝基的祖先，而贝基是嘉莉的祖先，那么艾米也是嘉莉的祖先。

On the other hand, "is the birth parent of" is not a transitive relation, because if Alice is the birth parent of Brenda, and Brenda is the birth parent of Claire, then Alice is not the birth parent of Claire. What is more, it is antitransitive: Alice can never be the birth parent of Claire.

On the other hand, "is the birth parent of" is not a transitive relation, because if Alice is the birth parent of Brenda, and Brenda is the birth parent of Claire, then Alice is not the birth parent of Claire. What is more, it is antitransitive: Alice can never be the birth parent of Claire.

另一方面，is the birth parent of 不是一个传递关系，因为如果 Alice 是 Brenda 的生身父母，而 Brenda 是 Claire 的生身父母，那么 Alice 不是 Claire 的生身父母。更重要的是，它是反及物动词: Alice 永远不可能是 Claire 的生身父母。

"Is greater than", "is at least as great as", and "is equal to" (equality) are transitive relations on various sets, for instance, the set of real numbers or the set of natural numbers:

“ Is greater than”、“ Is at least as great as”和“ Is equal to”(相等)是各种集合上的传递关系，例如实数集合或自然数集合:

whenever x > y and y > z, then also x > z

whenever x > y and y > z, then also x > z

只要 x > y 和 y > z，那么 x > z

whenever x ≥ y and y ≥ z, then also x ≥ z

whenever x ≥ y and y ≥ z, then also x ≥ z

每当 x & ge; y 和 y & ge; z，那么 x & ge; z

whenever x = y and y = z, then also x = z.

whenever x = y and y = z, then also x = z.

当 x = y，y = z，那么 x = z。

More examples of transitive relations:

更多传递关系的例子:

"is a subset of" (set inclusion, a relation on sets)

"divides" (divisibility, a relation on natural numbers)

"implies" (implication, symbolized by "⇒", a relation on propositions)

Examples of non-transitive relations:

不传递关系的例子:

"is the successor of" (a relation on natural numbers)

"is a member of the set" (symbolized as "∈")^[2]

"is perpendicular to" (a relation on lines in Euclidean geometry)

The empty relation on any set [math]\displaystyle{ X }[/math] is transitive^[3]^[4] because there are no elements [math]\displaystyle{ a,b,c \in X }[/math] such that [math]\displaystyle{ aRb }[/math] and [math]\displaystyle{ bRc }[/math], and hence the transitivity condition is vacuously true. A relation $R$ containing only one ordered pair is also transitive: if the ordered pair is of the form [math]\displaystyle{ (x, x) }[/math] for some [math]\displaystyle{ x \in X }[/math] the only such elements [math]\displaystyle{ a,b,c \in X }[/math] are [math]\displaystyle{ a=b=c=x }[/math], and indeed in this case [math]\displaystyle{ aRc }[/math], while if the ordered pair is not of the form [math]\displaystyle{ (x, x) }[/math] then there are no such elements [math]\displaystyle{ a,b,c \in X }[/math] and hence [math]\displaystyle{ R }[/math] is vacuously transitive.

The empty relation on any set [math]\displaystyle{ X }[/math] is transitive because there are no elements [math]\displaystyle{ a,b,c \in X }[/math] such that [math]\displaystyle{ aRb }[/math] and [math]\displaystyle{ bRc }[/math], and hence the transitivity condition is vacuously true. A relation containing only one ordered pair is also transitive: if the ordered pair is of the form [math]\displaystyle{ (x, x) }[/math] for some [math]\displaystyle{ x \in X }[/math] the only such elements [math]\displaystyle{ a,b,c \in X }[/math] are [math]\displaystyle{ a=b=c=x }[/math], and indeed in this case [math]\displaystyle{ aRc }[/math], while if the ordered pair is not of the form [math]\displaystyle{ (x, x) }[/math] then there are no such elements [math]\displaystyle{ a,b,c \in X }[/math] and hence [math]\displaystyle{ R }[/math] is vacuously transitive.

任何集合上的空关系是可传递的，因为 x </math > 中没有元素 < math > a，b，c，以至于 < math > aRb </math > 和 < math > bRc </math > ，因此传递条件是空的。一个只包含一个有序对的关系也是可传递的: 如果有序对是 < math > (x，x) </math > 对于 x </math > 中的某些 < math > x，那么 x </math > 中仅有的这样的元素 a，b，c 是 < math > a = b = c = x </math > ，在这种情况下确实是 < math > aRc </math > ，而如果有序对不是 < math > (x，x) </math > ，那么在 x </math > 中就不存在这样的数学元素 a，b，c，因此数学是虚的。

Properties

Closure properties

The inverse (converse) of a transitive relation is always transitive. For instance, knowing that "is a subset of" is transitive and "is a superset of" is its inverse, one can conclude that the latter is transitive as well.

The intersection of two transitive relations is always transitive. For instance, knowing that "was born before" and "has the same first name as" are transitive, one can conclude that "was born before and also has the same first name as" is also transitive.

The union of two transitive relations need not be transitive. For instance, "was born before or has the same first name as" is not a transitive relation, since e.g. Herbert Hoover is related to Franklin D. Roosevelt, which is in turn related to Franklin Pierce, while Hoover is not related to Franklin Pierce.

The complement of a transitive relation need not be transitive. For instance, while "equal to" is transitive, "not equal to" is only transitive on sets with at most one element.

Other properties

A transitive relation is asymmetric if and only if it is irreflexive.^[5]

A transitive relation is asymmetric if and only if it is irreflexive.

传递关系是不对称的当且仅当它是非反射性的。

A transitive relation need not be reflexive. When it is, it is called a preorder. For example, on set X = {1,2,3}:

传递关系不一定是反射性的。当它是时，它被称为一个预序。例如，在集合 x = {1,2,3}上:

R = {模板:Hair space(1,1), (2,2), (3,3), (1,3), (3,2)模板:Hair space} is reflexive, but not transitive, as the pair (1,2) is absent,

R = {模板:Hair space(1,1), (2,2), (3,3), (1,3)模板:Hair space} is reflexive as well as transitive, so it is a preorder,

R = {模板:Hair space(1,1), (2,2), (3,3)模板:Hair space} is reflexive as well as transitive, another preorder.

Transitive extensions and transitive closure

Let $R$ be a binary relation on set $X$ . The transitive extension of $R$ , denoted $R 1$ , is the smallest binary relation on $X$ such that $R 1$ contains $R$ , and if $(a, b) \in R$ and $(b, c) \in R$ then $(a, c) \in R 1$ .^[6] For example, suppose $X$ is a set of towns, some of which are connected by roads. Let $R$ be the relation on towns where $(A, B) \in R$ if there is a road directly linking town $A$ and town $B$ . This relation need not be transitive. The transitive extension of this relation can be defined by $(A, C) \in R 1$ if you can travel between towns $A$ and $C$ by using at most two roads.

Let be a binary relation on set . The transitive extension of , denoted , is the smallest binary relation on such that contains , and if and then . For example, suppose is a set of towns, some of which are connected by roads. Let be the relation on towns where if there is a road directly linking town and town . This relation need not be transitive. The transitive extension of this relation can be defined by if you can travel between towns and by using at most two roads.

设是集合上的二元关系。表示的传递扩展是包含和 if 及 then 的最小二进制关系。例如，假设是一组城镇，其中一些由公路连接。如果有一条直接连接城镇的公路，就让它成为城镇之间的关系吧。这个关系不一定是可传递的。这种关系的传递性扩展可以定义为，如果你可以在城镇之间旅行，最多使用两条道路。

If a relation is transitive then its transitive extension is itself, that is, if $R$ is a transitive relation then $R 1 = R$ .

If a relation is transitive then its transitive extension is itself, that is, if is a transitive relation then .

如果一个关系是可传递的，那么它的可传递扩展就是它自己，也就是说，如果它是一个传递关系。

The transitive extension of $R 1$ would be denoted by $R 2$ , and continuing in this way, in general, the transitive extension of $R i$ would be $R i + 1$ . The transitive closure of $R$ , denoted by $R *$ or $R \infty$ is the set union of $R$ , $R 1$ , $R 2$ , ... .^[7]

The transitive extension of would be denoted by , and continuing in this way, in general, the transitive extension of would be . The transitive closure of , denoted by or is the set union of , , , ... .

的传递性扩展可以用来表示，并以这种方式继续下去，一般来说，的传递性扩展可以是。传递闭包的集合，表示或者是，，，... 的集合。

The transitive closure of a relation is a transitive relation.^[7]

The transitive closure of a relation is a transitive relation. However, there is a formula for finding the number of relations that are simultaneously reflexive, symmetric, and transitive – in other words, equivalence relations – , those that are symmetric and transitive, those that are symmetric, transitive, and antisymmetric, and those that are total, transitive, and antisymmetric. Pfeiffer has made some progress in this direction, expressing relations with combinations of these properties in terms of each other, but still calculating any one is difficult. See also.

关系的传递闭包就是传递关系。然而，有一个公式可以计算同时具有自反性、对称性和传递性的关系的数量——换句话说，就是等价关系——这些关系具有对称性和传递性，具有对称性、传递性和反对称性，以及具有完全性、传递性和反对称性。Pfeiffer 已经在这个方向上取得了一些进展，将这些性质的组合表示为彼此的关系，但是仍然计算任何一个是困难的。参见。

The relation "is the birth parent of" on a set of people is not a transitive relation. However, in biology the need often arises to consider birth parenthood over an arbitrary number of generations: the relation "is a birth ancestor of" is a transitive relation and it is the transitive closure of the relation "is the birth parent of".

For the example of towns and roads above, $(A, C) \in R *$ provided you can travel between towns $A$ and $C$ using any number of roads.

The Rock–paper–scissors game is based on an intransitive and antitransitive relation "x beats y".

[石头-剪刀-布游戏是基于一个不及物和反传递的关系“ x 拍 y”]

Relation properties that require transitivity

A relation R is called intransitive if it is not transitive, that is, if xRy and yRz, but not xRz, for some x, y, z.

如果关系 r 不是传递的，则称之为不传递的，也就是说，对于某些 x，y，z，如果 xRy 和 yRz 是不传递的，则称之为不传递的。

Preorder – a reflexive transitive relation

In contrast, a relation R is called antitransitive if xRy and yRz always implies that xRz does not hold.

相反，如果 xRy 和 yRz 总是暗示 xRz 不成立，则关系 r 被称为反传递关系。

Partial order – an antisymmetric preorder

For example, the relation defined by xRy if xy is an even number is intransitive, but not antitransitive. The relation defined by xRy if x is even and y is odd is both transitive and antitransitive.

例如，如果 xy 是偶数，由 xRy 定义的关系是不传递的，而不是反传递的。如果 x 是偶数，y 是奇数，则 xRy 定义的关系既是传递的又是反传递的。

Total preorder – a total preorder

The relation defined by xRy if x is the successor number of y is both intransitive and antitransitive. Unexpected examples of intransitivity arise in situations such as political questions or group preferences.

如果 x 是 y 的继承数，则 xRy 定义的关系既是不及物关系，又是反及物关系。不及物性的意外例子出现在政治问题或群体偏好等情况下。

Equivalence relation – a symmetric preorder

Strict weak ordering – a strict partial order in which incomparability is an equivalence relation

Generalized to stochastic versions (stochastic transitivity), the study of transitivity finds applications of in decision theory, psychometrics and utility models.

将及物性推广到随机变量(随机及物性) ，及物性的研究在决策理论、心理测量学和效用模型中都有应用。

Total ordering – a total, antisymmetric transitive relation

A quasitransitive relation is another generalization; it is required to be transitive only on its non-symmetric part. Such relations are used in social choice theory or microeconomics.

拟顺序关系是另一种推广，它只要求在它的非对称部分是可传递的。这种关系被用于社会选择理论或微观经济学。

Counting transitive relations

No general formula that counts the number of transitive relations on a finite set 模板:OEIS is known.^[8] However, there is a formula for finding the number of relations that are simultaneously reflexive, symmetric, and transitive – in other words, equivalence relations – 模板:OEIS, those that are symmetric and transitive, those that are symmetric, transitive, and antisymmetric, and those that are total, transitive, and antisymmetric. Pfeiffer^[9] has made some progress in this direction, expressing relations with combinations of these properties in terms of each other, but still calculating any one is difficult. See also.^[10]

模板:Number of relations

Related properties

文件:Rock-paper-scissors.svg

The Rock–paper–scissors game is based on an intransitive and antitransitive relation "x beats y".

A relation R is called intransitive if it is not transitive, that is, if xRy and yRz, but not xRz, for some x, y, z.

In contrast, a relation R is called antitransitive if xRy and yRz always implies that xRz does not hold.

For example, the relation defined by xRy if xy is an even number is intransitive,^[11] but not antitransitive.^[12] The relation defined by xRy if x is even and y is odd is both transitive and antitransitive.^[13]

The relation defined by xRy if x is the successor number of y is both intransitive^[14] and antitransitive.^[15] Unexpected examples of intransitivity arise in situations such as political questions or group preferences.^[16]

Generalized to stochastic versions (stochastic transitivity), the study of transitivity finds applications of in decision theory, psychometrics and utility models.^[17]

A quasitransitive relation is another generalization; it is required to be transitive only on its non-symmetric part. Such relations are used in social choice theory or microeconomics.^[18]

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 此词条暂由彩云小译翻译，翻译字数共800，未经人工整理和审校，带来阅读不便，请见谅。
+  --[[用户:小趣木木|小趣木木]]（[[用户讨论:小趣木木|讨论]]）英文译文缺失 需要补充
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