Relations Between Institutions | 机构间的关系

📚 Relations Between Institutions | 机构间的关系

In Edexcel A-level Mathematics, a relation is not just a social link: it is a precise way of pairing elements from one set with elements of another. The phrase ‘Relations Between Institutions’ reminds us that many real-world connections, such as regulatory oversight, partnership, or membership, can be described using the formal language of sets and relations. This article develops the core ideas of relations and applies them to institutional networks, building the skills that underpin functions, graphs, and further mathematics.

在 Edexcel A-level 数学中，关系不仅仅是一种社会联系：它是将一个集合的元素与另一个集合的元素进行配对的精确方式。“机构间的关系”这一短语提醒我们，许多现实世界中的联系，例如监管、合作或成员关系，都可以用集合和关系的形式语言来描述。本文发展关系的核心思想，并将其应用于机构网络，培养支撑函数、图像和进阶数学的技能。

1. Sets, Elements and Institutions | 集合、元素与机构

A set is a well-defined collection of distinct objects, called elements or members. When we model relations between institutions, we first identify the relevant sets. For example, let A be the set of regulators and B be the set of schools or banks under supervision.

集合是一个定义明确的、由不同对象组成的整体，这些对象称为元素或成员。当我们对机构之间的关系进行建模时，首先要确定相关的集合。例如，设 A 为监管机构集合，B 为受监管的学校或银行集合。

We write A = {FCA, PRA, CMA} and B = {School X, School Y, School Z}. The order of elements in a set does not matter, and repeated elements are ignored.

写作 A = {FCA, PRA, CMA} 以及 B = {School X, School Y, School Z}。集合中元素的顺序无关紧要，重复元素会被忽略。

The size of a set is its cardinality, denoted n(A). If A has three institutions, then n(A) = 3. This basic counting will be important when we count possible relations.

集合的大小称为势，记作 n(A)。如果 A 有三个机构，那么 n(A) = 3。这个基本计数在我们计算可能的关系数量时会很重要。

2. Cartesian Products and Ordered Pairs | 笛卡尔积与有序对

An ordered pair (a, b) records a first element a and a second element b in a fixed order. The Cartesian product A × B is the set of all possible ordered pairs from A to B.

有序对 (a, b) 按固定顺序记录第一个元素 a 和第二个元素 b。笛卡尔积 A × B 是从 A 到 B 的所有可能有序对组成的集合。

A × B = {(a, b) : a ∈ A and b ∈ B}

If A = {FCA, PRA} and B = {School X, School Y}, then A × B = {(FCA, School X), (FCA, School Y), (PRA, School X), (PRA, School Y)}.

如果 A = {FCA, PRA}，B = {School X, School Y}，那么 A × B = {(FCA, School X), (FCA, School Y), (PRA, School X), (PRA, School Y)}。

The number of ordered pairs is n(A) × n(B), because each element of A can be paired with each element of B. This product forms the universe from which any relation is selected.

有序对的数量为 n(A) × n(B)，因为 A 中的每个元素都可以与 B 中的每个元素配对。这个笛卡尔积构成了所有关系从中选取的“全集”。

3. Defining a Relation | 定义二元关系

A binary relation R from set A to set B is any subset of the Cartesian product A × B. This means R is a collection of ordered pairs (a, b) that satisfy a particular condition, such as ‘a regulates b’ or ‘a partners with b’.

从集合 A 到集合 B 的二元关系 R 是笛卡尔积 A × B 的任意子集。这意味着 R 是满足特定条件的有序对 (a, b) 的集合，例如“a 监管 b”或“a 与 b 合作”。

For example, if FCA regulates School X and School Y, while PRA regulates only School Z, we write:

例如，如果 FCA 监管 School X 和 School Y，而 PRA 仅监管 School Z，我们写作：

R = {(FCA, School X), (FCA, School Y), (PRA, School Z)}

This set of ordered pairs is a subset of A × B. The statement (FCA, School X) ∈ R means ‘FCA is related to School X’ under the relation R.

这个有序对集合是 A × B 的子集。语句 (FCA, School X) ∈ R 表示在关系 R 下“FCA 与 School X 相关”。

A relation can be empty, total, or somewhere in between. The empty relation ∅ contains no ordered pairs, while the universal relation A × B contains every possible pair.

关系可以是空关系、全关系或介于两者之间。空关系 ∅ 不包含任何有序对，而全关系 A × B 包含所有可能的有序对。

4. Domain and Range of a Relation | 关系的定义域与值域

The domain of a relation R is the set of all first elements that appear in the ordered pairs of R. The range is the set of all second elements that appear. These are subsets of A and B respectively.

关系 R 的定义域是在 R 的有序对中出现的所有第一个元素组成的集合。值域是出现的所有第二个元素组成的集合。它们分别是 A 和 B 的子集。

For R = {(FCA, School X), (FCA, School Y), (PRA, School Z)}, the domain is {FCA, PRA} and the range is {School X, School Y, School Z}.

对于 R = {(FCA, School X), (FCA, School Y), (PRA, School Z)}，定义域为 {FCA, PRA}，值域为 {School X, School Y, School Z}。

Notice that FCA appears twice in the list of ordered pairs, but it is only listed once in the domain. Sets do not contain duplicates.

注意 FCA 在有序对列表中出现了两次，但在定义域中只列出一次。集合不包含重复元素。

If an element of A is not related to any element of B, it is not in the domain of R. This distinction is essential when a relation is a partial, rather than total, correspondence.

如果 A 中的某个元素与 B 中的任何元素都不相关，它就不在 R 的定义域中。当关系是部分对应而非全对应时，这一区别至关重要。

5. Representing Relations with Diagrams | 用关系图表示关系

A relation between two institutional sets can be shown using an arrow diagram. Place the elements of A on the left and the elements of B on the right, then draw an arrow from a to b whenever (a, b) belongs to R.

两个机构集合之间的关系可以用箭头图表示。将 A 的元素放在左侧，B 的元素放在右侧，每当 (a, b) 属于 R 时，就从 a 向 b 画一个箭头。

For our example, arrows would point from FCA to School X and School Y, and from PRA to School Z. Elements with no arrows are simply left unconnected.

在我们的例子中，箭头从 FCA 指向 School X 和 School Y，从 PRA 指向 School Z。没有箭头的元素保持不连接。

Arrow diagrams make it easy to see whether a relation is one-to-many, many-to-one, or many-to-many. Institutional relations are often many-to-many because one regulator may oversee several institutions, and one institution may answer to several regulators.

箭头图可以直观地显示关系是一对多、多对一还是多对多。机构间关系通常是多对多的，因为一个监管机构可能监管多个机构，而一个机构可能对多个监管机构负责。

In an exam sketch, always label the sets clearly and use a ruler for neatness. A clear diagram can support your reasoning even if the relation is written in set notation.

在考试草图中，务必清晰地标注集合，并使用直尺保持整洁。即使关系用集合符号书写，清晰的图也能支持你的推理。

6. Relation Matrices | 关系矩阵

A relation from set A to set B can also be represented by a matrix. If A has m elements and B has n elements, the relation matrix has m rows and n columns. Put 1 in position (i, j) if the ith element of A is related to the jth element of B, and 0 otherwise.

从集合 A 到集合 B 的关系也可以用矩阵表示。如果 A 有 m 个元素，B 有 n 个元素，关系矩阵就有 m 行 n 列。如果 A 的第 i 个元素与 B 的第 j 个元素相关，则在位置 (i, j) 填 1，否则填 0。

Using A = {FCA, PRA} and B = {School X, School Y, School Z}, the relation R = {(FCA, School X), (FCA, School Y), (PRA, School Z)} gives the matrix:

使用 A = {FCA, PRA} 和 B = {School X, School Y, School Z}，关系 R = {(FCA, School X), (FCA, School Y), (PRA, School Z)} 对应的矩阵为：

1	1	0
0	0	1

The first row corresponds to FCA and the second row to PRA. The columns correspond to School X, School Y, and School Z in that order.

第一行对应 FCA，第二行对应 PRA。各列依次对应 School X、School Y 和 School Z。

Relation matrices are useful when institutions have many members, because the matrix can be stored, added, or multiplied in a systematic way.

当机构有很多成员时，关系矩阵非常有用，因为矩阵可以系统地存储、相加或相乘。

7. Inverse Relations | 逆关系

The inverse relation, written R⁻¹, reverses the direction of every ordered pair in R. Formally, R⁻¹ = {(b, a) : (a, b) ∈ R}. If R means ‘a regulates b’, then R⁻¹ means ‘b is regulated by a’.

逆关系写作 R⁻¹，它反转 R 中每个有序对的方向。形式化地，R⁻¹ = {(b, a) : (a, b) ∈ R}。如果 R 表示“a 监管 b”，那么 R⁻¹ 表示“b 受 a 监管”。

For R = {(FCA, School X), (FCA, School Y), (PRA, School Z)}, the inverse is R⁻¹ = {(School X, FCA), (School Y, FCA), (School Z, PRA)}.

对于 R = {(FCA, School X), (FCA, School Y), (PRA, School Z)}，其逆关系为 R⁻¹ = {(School X, FCA), (School Y, FCA), (School Z, PRA)}。

The domain of R⁻¹ is the range of R, and the range of R⁻¹ is the domain of R. This symmetry is useful when switching between ‘regulator to institution’ and ‘institution to regulator’ perspectives.

R⁻¹ 的定义域是 R 的值域，R⁻¹ 的值域是 R 的定义域。这种对称性在“监管机构到被监管机构”与“被监管机构到监管机构”的视角之间切换时非常有用。

To find the matrix of R⁻¹, simply transpose the matrix of R: swap rows and columns.

求 R⁻¹ 的矩阵时，只需对 R 的矩阵进行转置：交换行和列。

8. Composite Relations | 复合关系

If R is a relation from A to B and S is a relation from B to C, the composite relation S ∘ R is defined from A to C. An element a is related to c in S ∘ R if there exists some b in B such that (a, b) ∈ R and (b, c) ∈ S.

如果 R 是从 A 到 B 的关系，S 是从 B 到 C 的关系，那么复合关系 S ∘ R 定义在 A 到 C 上。如果存在 B 中的某个元素 b，使得 (a, b) ∈ R 且 (b, c) ∈ S，则 a 与 c 在 S ∘ R 中相关。

This is like a chain: a regulator R relates to an institution, and that institution S relates to a regional body. The composite relation links the regulator directly to the regional body through the institution.

这就像一条链条：一个监管机构 R 与某个机构相关，而该机构 S 与某个地区机构相关。复合关系通过该机构将监管机构直接与地区机构联系起来。

For example, if R = {(FCA, School X)} and S = {(School X, Region 1)}, then S ∘ R = {(FCA, Region 1)}. The middle element School X is not included in the final ordered pair.

例如，如果 R = {(FCA, School X)}，S = {(School X, Region 1)}，那么 S ∘ R = {(FCA, Region 1)}。中间元素 School X 不出现在最终的有序对中。

Note that composition is not commutative: R ∘ S is not generally equal to S ∘ R, and in many cases one of them is not even defined.

注意复合运算不满足交换律：R ∘ S 通常不等于 S ∘ R，而且在很多情况下其中一个甚至没有定义。

9. Functions as Special Relations | 作为特殊关系的函数

A function is a special type of relation in which every element of the domain is paired with exactly one element of the codomain. This is the key condition that distinguishes a function from a general relation between institutions.

函数是一种特殊类型的关系，其中定义域中的每个元素都与值域中的恰好一个元素配对。这是区分函数与机构之间一般关系的关键条件。

If FCA regulates School X and School Y, the relation R = {(FCA, School X), (FCA, School Y)} is not a function from A to B, because FCA maps to two different schools.

如果 FCA 监管 School X 和 School Y，关系 R = {(FCA, School X), (FCA, School Y)} 就不是从 A 到 B 的函数，因为 FCA 映射到了两所不同的学校。

However, the relation H = {(FCA, School X), (PRA, School Y)} is a function, because each regulator maps to exactly one institution. Functions are one-to-one or many-to-one, but never one-to-many.

然而，关系 H = {(FCA, School X), (PRA, School Y)} 是一个函数，因为每个监管机构恰好映射到一个机构。函数是一对一或多对一的，但绝不一对多。

In institutional modelling, functions often arise when each institution has exactly one chief regulator or one unique registration number.

在机构建模中，当每个机构恰好有一个主要监管机构或一个唯一注册号时，就会产生函数。

10. Properties of Relations: Reflexive, Symmetric, Transitive | 关系的性质：自反、对称、传递

When a relation is defined from a set A to itself, we can test three important properties. A relation R on A is reflexive if every element is related to itself: (a, a) ∈ R for all a ∈ A.

当关系定义在集合 A 到其自身时，我们可以检验三个重要性质。如果每个元素都与自身相关，即对所有 a ∈ A 都有 (a, a) ∈ R，则称 A 上的关系 R 是自反的。

A relation is symmetric if whenever (a, b) ∈ R, then (b, a) ∈ R. A relation is transitive if whenever (a, b) ∈ R and (b, c) ∈ R, then (a, c) ∈ R.

如果只要 (a, b) ∈ R，就有 (b, a) ∈ R，则关系是对称的。如果只要 (a, b) ∈ R 且 (b, c) ∈ R，就有 (a, c) ∈ R，则关系是传递的。

For institutions, a ‘shares data with’ relation might be symmetric, while a ‘supervises’ relation is rarely symmetric. A ‘has the same regulator as’ relation is reflexive, symmetric, and transitive, so it is an equivalence relation.

对于机构而言，“与……共享数据”关系可能是对称的，而“监管”关系很少是对称的。“与……有相同监管机构”关系是自反的、对称的和传递的，因此它是一个等价关系。

Reflexive: each institution has the same regulator as itself.
自反：每个机构与自身有相同的监管机构。
Symmetric: if X has the same regulator as Y, then Y has the same regulator as X.
对称：如果 X 与 Y 有相同的监管机构，那么 Y 与 X 也有相同的监管机构。
Transitive: if X has the same regulator as Y and Y has the same regulator as Z, then X has the same regulator as Z.
传递：如果 X 与 Y 有相同的监管机构且 Y 与 Z 有相同的监管机构，那么 X 与 Z 有相同的监管机构。

11. Equivalence Relations in Institutional Networks | 机构网络中的等价关系

An equivalence relation is a relation that is reflexive, symmetric, and transitive. It partitions a set into disjoint subsets called equivalence classes, where all elements within a class are related to each other.

等价关系是自反、对称且传递的关系。它将集合划分为互不相交的子集，称为等价类，同一类中的所有元素都彼此相关。

In an institutional setting, the relation ‘is in the same regulatory tier as’ can be an equivalence relation. All Tier 1 institutions form one equivalence class, all Tier 2 institutions form another, and so on.

在机构环境中，“与……处于同一监管层级”的关系可以是等价关系。所有第一层机构形成一个等价类，所有第二层机构形成另一个等价类，依此类推。

Equivalence classes are powerful because they reduce a large set of individual institutions into a small number of categories that share a common property.

等价类之所以强大，是因为它们将大量个体机构缩减为少数具有共同属性的类别。

To verify an equivalence relation in an exam, always check the three properties separately and state your conclusion with a clear partition diagram or set notation.

在考试中验证等价关系时，务必分别检验这三个性质，并用清晰的划分图或集合符号陈述结论。

12. Worked Example and Exam Tips | 例题与考试技巧

Worked example: Let A = {Bank P, Bank Q, Bank R} and let R be the relation on A defined by ‘has the same credit rating as’. Suppose Bank P and Bank Q are rated A, and Bank R is rated BBB.

例题：设 A = {Bank P, Bank Q, Bank R}，定义 A 上的关系 R 为“信用评级相同”。假设 Bank P 和 Bank Q 评级为 A，Bank R 评级为 BBB。

Then R = {(P, P), (Q, Q), (R, R), (P, Q), (Q, P)}. The equivalence classes are {P, Q} and {R}. The relation is reflexive, symmetric, and transitive.

那么 R = {(P, P), (Q, Q), (R, R), (P, Q), (Q, P)}。等价类为 {P, Q} 和 {R}。该关系是自反、对称且传递的。

Exam tip: when writing a relation, give it in set notation as a list of ordered pairs or as a rule. Never leave a relation as a vague description unless the question asks for a description only.

考试技巧：书写关系时，请用集合符号将其表示为有序对列表或规则。除非题目只要求描述，否则不要把关系写成模糊的叙述。

Always count ordered pairs carefully. If n(A) = 3 and n(B) = 4, the universal relation A × B has 12 ordered pairs, but a specific relation can have any number from 0 to 12.

务必仔细计算有序对数量。如果 n(A) = 3 且 n(B) = 4，全关系 A × B 有 12 个有序对，但特定关系可以包含 0 到 12 之间的任意数量。

Draw a quick arrow diagram or matrix if you need to find an inverse or composite. These visual methods reduce errors and make your answer easier to check under timed conditions.

如果需要求逆关系或复合关系，请快速画出箭头图或矩阵。这些可视化方法能减少错误，并使答案在限时条件下更易于检查。

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Relations Between Institutions | 机构间的关系