📚 1.10 Set Language and Notation | 集合语言与符号
Sets and set notation form the language we use to describe collections of objects in a precise way. In Edexcel IGCSE Mathematics, you need to be confident working with universal sets, subsets, complements, intersections and unions. Venn diagrams bring these ideas to life and help you solve problems involving overlapping categories or counting elements that satisfy given conditions.
集合与集合符号为我们提供了一种精确描述一组对象的语言。在爱德思 IGCCSE 数学中，你需要熟练运用全集、子集、补集、交集与并集。文氏图将这些概念变得直观可见，并帮助你解决涉及重叠类别或计算满足特定条件元素个数的问题。
1. Sets, Elements and Membership | 集合、元素与属于关系
A set is a well-defined collection of distinct objects, called elements curly brackets. For example, A = {2, 4, 6, 8} is the set of even numbers between 1 and 9.
集合是一组确定且互不相同的对象的总体，这些对象称为元素。我们通常用大写字母如 A、B、C 命名集合，并把元素写在一对花括号内。例如 A = {2, 4, 6, 8} 表示 1 到 9 之间的偶数集合。
When an object x belongs to set A, we write x ∈ A. If x is not in A, we write x ∉ A. The order of elements inside the brackets does not matter, and repeated elements are listed only once.
当某个对象 x 属于集合 A 时，我们记作 x ∈ A；若 x 不属于 A，则记作 x ∉ A。花括号内元素的排列顺序无关紧要，重复的元素只写一次。
2. Universal Set and Empty Set | 全集与空集
The universal set, often denoted by U or ξ, contains all the elements we are considering in a particular context. Every other set in the problem is a subset of the universal set. The empty set (or null set) contains no elements and is written as ∅ or { }.
全集通常用 U 或 ξ 表示，它包含了在特定情境下我们所考虑的所有元素。问题中出现的每一个其他集合都是全集的子集。空集（也称零集）不含任何元素，记作 ∅ 或 { }。
9, we could take U = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}. The set of digits greater than 10 is ∅ because no digit satisfies that condition within this universe.
举例来说，如果我们考虑 0 到 9 的数字，可取 U = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}。大于 10 的数字集合就是 ∅，因为在这个全集中没有任何数字满足该条件。
3. Subsets and Proper Subsets | 子集与真子集
Set A is a subset of set B, written A ⊆ B, if every element of A is also an element of B. If A is a subset of B but A is not equal to B, we say A is a proper subset of B, sometimes written as A ⊂ B.
如果集合 A 中的每一个元素都属于集合 B，那么 A 是 B 的子集，记作 A ⊆ B。如果 A 是 B 的子集且 A 不等于 B，则称 A 是 B 的真子集，有时写为 A ⊂ B。
et of every set, and every set is a subset of itself. For instance, if B = {1, 2, 3}, then ∅ ⊆ B, {1} ⊆ B and B ⊆ B are all true, but {1} ⊂ B is also true because {1} ≠ B.
一定要记住：空集是任何集合的子集，而每个集合也是自身的子集。例如，若 B = {1, 2, 3}，则 ∅ ⊆ B、{1} ⊆ B 以及 B ⊆ B 均成立，而 {1} ⊂ B 也成立，因为 {1} ≠ B。
4. Complement of a Set | 补集
The complement of a set A, written as A' (or Aᶜ), is the set of all elements in the universal set that are not in A. In symbols, A' = {x ∈ U : x ∉ A}.
集合 A 的补集记作 A'（或 Aᶜ），它是全集中所有不属于 A 的元素构成的集合。用符号表示为 A' = {x ∈ U : x ∉ A}。
For example, if U = {1, 2, 3, 4, 5} and A = {2, 4}, then A' = {1, 3, 5}. Notice that A ∪ A' = U and A ∩ A' = ∅ – two very useful facts when checking your answers.
例如，若 U = {1, 2, 3, 4, 5}，A = {2, 4}，那么 A' = {1, 3, 5}。请注意 A ∪ A' = U 且 A ∩ A' = ∅——这两个事实在检验答案时非常有用。
5. Intersection of Sets | 交集
The intersection of two sets A and B, written A ∩ B, is the set containing all elements that belong to both A and B. In formal notation, A ∩ B = {x : x ∈ A and x ∈ B}.
两个集合 A 与 B 的交集记作 A ∩ B，它由同时属于 A 和 B 的所有元素组成。形式化表示为 A ∩ B = {x : x ∈ A 且 x ∈ B}。
If A = {2, 3, 5, 7} and B = {1, 2, 3, 4, 5}, then A ∩ 2, 3, 5}. When two sets have no elements in common, their intersection is the empty set: A ∩ B = ∅, and the sets are said to be disjoint.
若 A = {2, 3, 5, 7}，B = {1, 2, 3, 4, 5}，则 A ∩ B = {2, 3, 5}。当两个集合没有公共元素时，它们的交集为空集：A ∩ B = ∅，此时称这两个集合是不交的。
6. Union of Sets | 并集
The union of A and B, written A ∪ B ll elements that are in A, in B, or in both. Duplicate elements are listed only once. So A ∪ B = {x : x ∈ A or x ∈ B}.
A 与 B 的并集记作 A ∪ B，它是所有属于 A、属于 B 或同属两者的元素构成的集合，重复的元素只列出一次。因此 A ∪ B = {x : x ∈ A 或 x ∈ B}。
Using the previous example, A ∪ B = {1, 2, 3, 4, 5, 7}. You can think of union as 'combining' the two sets, while intersection identifies their overlap.
沿用上例，A ∪ B = {1, 2, 3, 4, 5, 7}。你可以把并集想象成两个集合的“合并”，而交集则找出它们重叠的部分。
7. Number of Elements in a Set | 集合中的元素个数
The notation n(A) elements in set A. For finite sets, n(A) is a non-negative integer. For the empty set, n(∅) = 0.
符号 n(A) 表示集合 A 中不同元素的个数。对于有限集，n(A) 是一个非负整数。对空集而言，n(∅) = 0。
A key formula connects the cardinalities of unions and intersections: n(A ∪ B) = n(A) + n(B) − n(A ∩ B). This adjustment avoids double-counting the elements that lie in the intersection.
一条关键公式将并集与交集的元素个数联系在一起：n(A ∪ B) = n(A) + n(B) − n(A ∩ B)。减去交集确保了重叠部分的元素不会被重复计算。
For three sets, the formula extends to: n(A ∪ B ∪ C) = n(A) + n(B) + n(C) − n(A ∩ B) − n(A ∩ C) − n(B ∩ C) + n(A ∩ B ∩ C).
对于三个集合的情况，公式扩展为：n(A ∪ B ∪ C) = n(A) + n(B) + n(C) − n(A ∩ B) − n(A ∩ C) − n(B ∩ C) + n(A ∩ B ∩ C)。
8. Venn Diagrams: Representing Two Sets | 文氏图：表示两个集合
A Venn diagram is a visual tool where the universal set U is shown as a rectangle, and sets inside U are drawn as circles (or ovals). Overlapping regions represent intersections, while the area outside all circles but inside the rectangle shows elements not in the drawn sets.
文氏图是一种可视化工具：全集 U 通常用一个矩形表示，而 U 内的各个集合用圆形（或椭圆形）画出。重叠区域表示交集，而位于所有圆圈之外、矩形之内的区域则表示不属于任何一个所画集合的元素。
When shading or labelling regions, you can clearly see A ∩ B, A ∪ B, A' ∩ B, etc. IGCSE questions often ask you to shade a specific region such as (A ∪ B)' or A ∩ B'.
通过给不同区域涂色或标注，你可以清晰地看到 A ∩ B、A ∪ B、A' ∩ B 等。IGCSE 题目经常要求你给某个特定区域涂色，例如 (A ∪ B)' 或 A ∩ B'。
To shade (A ∪ B)', shade everything outside both circles; this is the complement of the union. To shade A ∩ B', shade the part of circle A that does not intersect B.
要涂色 (A ∪ B)'，你需要给两个圆以外的整个矩形区域涂色；这就是并集的补集。要涂色 A ∩ B'，则要给圆 A 中不与 B 相交的部分涂色。
9. Venn Diagrams with Three Sets | 三集合文氏图
When three sets A, B and C are involved, the Venn diagram shows eight distinct regions formed by all possible combinations of belonging (or not) to each set. The central region where all three circles intersect represents A ∩ B ∩ C.
当涉及三个集合 A、B、C 时，文氏图会显示出由各个集合“属于/不属于”的所有可能组合所形成的八个互不相交的区域。三个圆共同相交的中心区域代表 A ∩ B ∩ C。
Problems often provide information such as n(A) = 20, n(B) = 25, n(C) = 30, n(A ∩ B) = 8, n(A ∩ C) = 10, n(B ∩ C) = 12, n(A ∩ B ∩ and then ask you to complete the diagram or find how many elements are in exactly one set.
题目通常会给出类似这样的信息：n(A) = 20，n(B) = 25，n(C) = 30，n(A ∩ B) = 8，n(A ∩ C) = 10，n(B ∩ C) = 12，n(A ∩ B ∩ C) = 4，然后要求你完成示意图，或找出恰好只属于一个集合的元素有多少个。
Working systematically from the centre outwards is the safest method: place the central value first, then fill in the 'exclusive' pairwise intersections by subtracting the central overlap, and finally complete the remaining parts of each circle.
从中心向外逐步计算是最稳妥的方法：先填入中心值，然后通过减去中心重叠部分获得各个“纯两两”交集的数值，最后补全每个圆的剩余部分。
10. Problem Solving with Venn Diagrams | 用文氏图解决问题
Many word problems in IGCSE ask you to classify a group of people or objects according to two or three attributes. Drawing a Venn diagram and placing the given numbers in the correct regions often makes the solution straightforward.
IGCSE 中的许多文字题要求你根据两个或三个属性对一组人或物进行分类。画出文氏图并将已知数据放入正确区域，往往能让解答变得清晰直接。
For example: ' 15 study Physics, and 8 study both. Find the number of students who study neither subject.' Let M = Mathematics, P = Physics. n(M ∩ nts studying at least one subject = 10 + 8 + 7 = 25. Those studying neither = 30 − 25 = 5.
例如：“班上有 30 名学生，18 人选修数学，15 人选修物理，8 人两科都选。求两科都不选的学生人数。”设 M = 数学，P = 物理。n(M ∩ P) = 8，所以只选数学的人数为 18 − 8 = 10，只选物理的人数为 15 − 8 = 7。于是至少选修一科的人数为 10 + 8 + 7 = 25，两科都不选的人数为 30 − 25 = 5。
Always check that all the placed numbers add up to the total given. Writing a small equation can also help when the universal set contains elements that lie outside all considered sets.
每次都应检查所有填入的数字之和是否等于给定的总数。当全集包含不属于任何被考虑集合的元素时，写出一个小方程也会有所帮助。
11. Key Notation Summary | 关键符号速查
The table below collects the essential set notation you will encounter in Edexcel IGCSE exam papers. Familiarity with these symbols is crucial for reading questions accurately.
下表汇总了你在爱德思 IGCSE 试卷中会遇到的基本集合符号。熟悉这些符号对于准确理解题意至关重要。
| Symbol | Meaning | 中文含义 |
|---|---|---|
| { } | Set delimiters | 集合括号 |
| ∈ | Is an element of | 属于 |
| ∉ | Is not an element of | 不属于 |
| ∅ or { } | Empty set | 空集 |
| U (or ξ) | Universal set | 全集 |
| A' | Complement of A | A 的补集 |
| ⊆ | Is a subset of | 是……的子集 |
| ⊂ | Is a proper subset of | 是……的真子集 |
| ∩ | Intersection | 交集 |
| ∪ | Union | 并集 |
| n(A) | Number of elements in A | 集合 A 的元素个数 |
12. Exam Tips and Common Mistakes | 应试技巧与常见错误
When working with set notation questions, always begin by identifying the universal set and listing its elements if possible. Read the question carefully: 'A and B' usually means intersection, while 'A or B' means union. Do not confuse the symbols ∩ and ∪.
做集合符号题时，总是先确定全集，如有可能就列出它的所有元素。仔细审题：“A 与 B”通常指交集，“A 或 B”则指并集。不要混淆符号 ∩ 和 ∪。
A common mistake is forgetting that n(A ∪ B) = n(A) + n(B) − n(A ∩ tion. Always subtract it once. For three sets, remember the inclusion-exclusion principle.
一个常见错误是忘记 n(A ∪ B) = n(A) + n(B) − n(A ∩ B)。如果直接把总数相加，你会把交集部分重复计算。必须减去一次交集。对三个集合，务必记住容斥原理。
When shading Venn diagrams, it is helpful to use a pencil and shade lightly so that you can check your region against the required expression. Practice shading complicated combinations such as (A ∩ B') ∪ C or A' ∩ (B ∪ C).
涂色文氏图时，建议用铅笔轻轻涂画，以便对照所需表达式检查涂色的区域是否正确。多练习一些较复杂的组合涂色，例如 (A ∩ B') ∪ C 或 A' ∩ (B ∪ C)。
Finally, always label your diagrams clearly and write intermediate steps on your paper. Even if your final answer is incorrect, clear working often earns method marks in the Edexcel IGCSE examination.
最后，一定要在图上清晰地标注，并在试卷上写出中间步骤。即便最终答案有误，清晰的解题过程在爱德思 IGCSE 考试中也常常能获得方法分。
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