Set Builder Notation for IGCSE Edexcel Maths — IGCSE数学：集合描述法及其应用

1. 什么是集合描述法：从列举法到描述法 | What Is Set Builder Notation: From Listing to Describing

在 Edexcel IGCSE 数学（4MA1）的集合单元中，我们首先学会用列举法（roster form）表示集合，也就是把集合的所有元素一一写在大括号里。例如，集合 {1, 2, 3, 4} 表示由 1、2、3、4 这四个数字组成的集合。列举法的优点是一目了然，读者可以直接看到集合里有哪些元素。

In the Sets unit of Edexcel IGCSE Mathematics (4MA1), we first learn to represent a set using roster form, which means listing every element of the set inside curly braces. For example, the set {1, 2, 3, 4} represents the set made up of the four numbers 1, 2, 3 and 4. The advantage of roster form is that it is clear at a glance: the reader can see exactly which elements are in the set.

但是列举法有一个严重的局限：当一个集合包含无穷多个元素，或者元素数量多到无法一一写出来时，列举法就失效了。例如，”所有大于 3 的整数”这个集合有无数个元素（4, 5, 6, 7, …），你永远不可能把它们全部写完。这时，我们就需要一种更强大的表示方法 – 集合描述法（set builder notation）。

However, roster form has a serious limitation: when a set contains infinitely many elements, or so many elements that they cannot all be written out one by one, roster form fails. For example, the set of all integers greater than 3 has infinitely many elements (4, 5, 6, 7, …), and you could never write them all down. In this situation, we need a more powerful method of representation: set builder notation.

集合描述法用”元素的共同性质”来定义集合，而不是把元素逐一列出。它回答了这样一个问题：”哪些东西属于这个集合？”答案是：”所有满足某个条件的东西。”这种思路从”罗列”上升到了”描述”，是 IGCSE 集合学习中一个重要的思维跨越，也是后续学习区间、数集和概率论的基础。

Set builder notation defines a set by the common property of its elements rather than by listing them individually. It answers the question: “Which things belong to this set?” The answer is: “Everything that satisfies a certain condition.” This way of thinking moves from listing to describing, and it is an important conceptual step in IGCSE set work, as well as the foundation for later topics such as intervals, number sets and probability.

2. 描述法的核心语法：花括号、变量、竖线与条件 | The Core Syntax: Braces, a Variable, a Vertical Bar and a Condition

集合描述法的标准形式可以写成：{ x : 条件 } 或者 { x | 条件 }。这里的冒号（:）和竖线（|）读作”满足……的条件”（such that），整句话读作”所有满足给定条件的 x 组成的集合”。在 Edexcel IGCSE 试卷中，两种写法都被接受，你只需要保持一致即可。

The standard form of set builder notation can be written as { x : condition } or { x | condition }. Here the colon (:) and the vertical bar (|) are both read as “such that”, and the whole expression is read as “the set of all x such that the given condition holds”. In Edexcel IGCSE exam papers, both notations are accepted, so you simply need to be consistent.

让我们拆解这个结构。第一，花括号 { } 告诉读者这是一个集合；第二，花括号内的字母 x 是变量，它代表集合中的任意一个元素；第三，冒号或竖线相当于”such that”；第四，条件部分（例如 x > 3）规定了元素必须满足的性质。四部分合在一起，就完整地定义了一个集合。

Let us break down this structure. First, the curly braces { } tell the reader that this is a set. Second, the letter x inside the braces is a variable: it stands for any one element of the set. Third, the colon or vertical bar means “such that”. Fourth, the condition part (for example x > 3) states the property that elements must satisfy. Together, the four parts define a set completely.

来看几个具体例子。{ x : x > 3 } 表示所有大于 3 的实数组成的集合；{ x : x 是正整数且 x < 10 } 表示所有小于 10 的正整数，也就是 {1, 2, 3, 4, 5, 6, 7, 8, 9}；{ x : x 是偶数 } 表示所有偶数组成的集合。注意，第三个例子无法用列举法写出，因为偶数有无限多个，这正是描述法不可替代的原因。

Here are some concrete examples. { x : x > 3 } is the set of all real numbers greater than 3; { x : x is a positive integer and x < 10 } is the set of positive integers less than 10, namely {1, 2, 3, 4, 5, 6, 7, 8, 9}; and { x : x is even } is the set of all even numbers. Note that the third example cannot be written in roster form at all, because there are infinitely many even numbers. This is exactly why set builder notation is indispensable.

3. 常用数集符号：自然数、整数、有理数与实数 | Common Number Sets: Natural, Integer, Rational and Real Numbers

在集合描述法中，条件部分经常要用到标准数集符号。Edexcel IGCSE 大纲要求学生认识并正确使用四个基本数集：自然数集 ℕ、整数集 ℤ、有理数集 ℚ 和实数集 ℝ。这些符号来自德语和法语单词的首字母，例如 ℤ 来自德语 “Zahlen”（数字），ℚ 来自英语 “Quotient”（商），因为它们都可以写成两个整数之比。

In set builder notation, the condition part frequently uses standard number set symbols. The Edexcel IGCSE specification requires students to recognise and correctly use four basic number sets: the natural numbers ℕ, the integers ℤ, the rational numbers ℚ and the real numbers ℝ. These symbols come from the initial letters of German and French words: for example, ℤ comes from the German “Zahlen” (numbers), and ℚ comes from the English “Quotient”, because rational numbers can be written as the quotient of two integers.

自然数集 ℕ 包含正整数：ℕ = {1, 2, 3, 4, …}（部分教材把 0 也包含在自然数内，考试时以题目说明为准）。整数集 ℤ 包含所有正整数、负整数和零：ℤ = {…, -2, -1, 0, 1, 2, …}。有理数集 ℚ 包含所有能写成两个整数之比的数，包括有限小数和循环小数。实数集 ℝ 包含所有有理数和无理数，例如 √2、π 和 e 都在 ℝ 中。

The natural numbers ℕ consist of the positive integers: ℕ = {1, 2, 3, 4, …} (some textbooks also include 0; in the exam, follow the wording of the question). The integers ℤ include all positive integers, negative integers and zero: ℤ = {…, -2, -1, 0, 1, 2, …}. The rational numbers ℚ include every number that can be written as the ratio of two integers, including terminating decimals and recurring decimals. The real numbers ℝ include all rational and irrational numbers, for example √2, π and e all belong to ℝ.

这些数集之间存在着包含关系：ℕ ⊂ ℤ ⊂ ℚ ⊂ ℝ。也就是说，每个自然数都是整数，每个整数都是有理数，每个有理数都是实数。理解这条包含链非常重要，因为考试题经常要求你判断某个数属于哪个集合，例如：-3 是整数但不是自然数；1/2 是有理数但不是整数；√2 是实数但不是有理数。

These number sets have an inclusion relationship: ℕ ⊂ ℤ ⊂ ℚ ⊂ ℝ. In other words, every natural number is an integer, every integer is a rational number, and every rational number is a real number. Understanding this chain of inclusion is very important, because exam questions often ask you to decide which set a number belongs to. For example: -3 is an integer but not a natural number; 1/2 is rational but not an integer; and √2 is real but not rational.

4. 区间型集合：用描述法表达不等式 | Interval-Style Sets: Expressing Inequalities in Set Builder Notation

描述法最常见的一类应用是用不等式表示区间。例如，{ x : x ≥ 4 } 表示所有大于或等于 4 的实数，在数轴上表现为从 4 开始向右延伸到无穷的一条射线，其中 4 用实心圆点表示（因为 4 本身属于该集合）。这类集合在解不等式、求函数定义域和值域时反复出现。

The most common application of set builder notation is expressing intervals using inequalities. For example, { x : x ≥ 4 } is the set of all real numbers greater than or equal to 4. On the number line it appears as a ray starting at 4 and extending to the right, with 4 marked by a filled dot (because 4 itself belongs to the set). This type of set appears again and again when solving inequalities and finding the domain and range of functions.

再看一个双端限制的例子。{ x : -2 < x ≤ 3 } 表示所有大于 -2 且小于或等于 3 的实数。在数轴上，-2 用空心圆点表示（-2 不属于集合），3 用实心圆点表示（3 属于集合）。注意，两个条件用”且”（and）连接，意味着元素必须同时满足两个不等式。

Now consider an example with two bounds. { x : -2 < x ≤ 3 } is the set of all real numbers greater than -2 and less than or equal to 3. On the number line, -2 is marked with an open dot (because -2 is not in the set) while 3 is marked with a filled dot (because 3 is in the set). Note that the two conditions are joined by “and”, which means an element must satisfy both inequalities at the same time.

还有一类题目要求你把描述法改写为区间符号或数轴图。区间符号是更简洁的写法：{ x : -2 < x ≤ 3 } 可以写成 (-2, 3]，其中圆括号表示开区间（不含端点），方括号表示闭区间（含端点）。Edexcel 的题目经常同时考察这几种表示法的互译，所以你需要熟练掌握描述法、区间符号和数轴图三者的转换。

There is also a type of question that asks you to rewrite set builder notation as interval notation or as a number line diagram. Interval notation is a more compact way of writing: { x : -2 < x ≤ 3 } can be written as (-2, 3], where a round bracket means an open interval (endpoint excluded) and a square bracket means a closed interval (endpoint included). Edexcel questions often test the translation between these representations, so you need to be fluent in converting among set builder notation, interval notation and number line diagrams.

5. 描述法与维恩图的互译 | Translating Between Set Builder Notation and Venn Diagrams

维恩图（Venn diagram）是集合的图形表示，而描述法是集合的符号表示。在 Edexcel IGCSE 考试中，很多题目会给你一张维恩图，要求你写出某个区域的集合；或者反过来，给你一个描述法集合，要求你在维恩图上涂出对应的区域。掌握两者的互译是拿分的关键。

A Venn diagram is the pictorial representation of a set, while set builder notation is its symbolic representation. In Edexcel IGCSE exams, many questions give you a Venn diagram and ask you to write down the set represented by a region; or conversely, they give you a set in set builder notation and ask you to shade the corresponding region on a Venn diagram. Mastering the translation between the two is the key to scoring.

举例来说，设全集 ξ = { x : x 是 1 到 12 之间的整数 }，集合 A = { x : x 是偶数 }。那么 A 包含 2, 4, 6, 8, 10, 12。如果题目要求你在维恩图上表示 A，你就把代表偶数的元素所在的区域涂满。反过来，如果维恩图上已经涂好了某个区域，你需要观察该区域内的元素有什么共同特征，再用描述法写出来。

For example, let the universal set ξ = { x : x is an integer between 1 and 12 }, and set A = { x : x is even }. Then A contains 2, 4, 6, 8, 10 and 12. If the question asks you to represent A on a Venn diagram, you shade the region containing the even numbers. Conversely, if a region is already shaded on the Venn diagram, you must observe what common property the elements in that region share, and then write it using set builder notation.

互译时最容易出错的地方是边界元素的取舍。例如集合 { x : x < 5 } 是否包含 5？答案是不包含，因为条件是严格小于。而 { x : x ≤ 5 } 包含 5。在维恩图上，这种区别对应着元素是否落在圆圈边界上。做题时养成先判断端点是否属于集合的习惯，可以避免大量低级失误。

The most error-prone part of translation is the treatment of boundary elements. For example, does the set { x : x < 5 } contain 5? The answer is no, because the condition is strictly less than. But { x : x ≤ 5 } does contain 5. On a Venn diagram, this difference corresponds to whether an element falls on the boundary of the circle. If you develop the habit of first deciding whether an endpoint belongs to the set, you will avoid many careless mistakes.

6. 并集与交集：用描述法表示组合运算 | Union and Intersection: Combined Operations in Set Builder Notation

并集（union）和交集（intersection）是集合的两个基本运算，它们都可以用描述法精确定义。A ∪ B（读作 “A union B”）表示属于 A 或属于 B（或同时属于两者）的所有元素组成的集合，即 A ∪ B = { x : x ∈ A 或 x ∈ B }。注意，”或”在这里是包容性的：元素只需要满足其中一个条件。

The union and intersection are the two basic operations on sets, and both can be defined precisely using set builder notation. A ∪ B (read as “A union B”) is the set of all elements that belong to A or belong to B (or both), that is, A ∪ B = { x : x ∈ A or x ∈ B }. Note that “or” here is inclusive: an element only needs to satisfy one of the conditions.

交集 A ∩ B（读作 “A intersection B”）表示同时属于 A 和 B 的所有元素组成的集合，即 A ∩ B = { x : x ∈ A 且 x ∈ B }。两个条件必须同时满足。例如，设 A = {1, 2, 3, 4, 5}，B = {3, 4, 5, 6, 7}，则 A ∪ B = {1, 2, 3, 4, 5, 6, 7}，A ∩ B = {3, 4, 5}。

The intersection A ∩ B (read as “A intersection B”) is the set of all elements that belong to both A and B, that is, A ∩ B = { x : x ∈ A and x ∈ B }. Both conditions must be satisfied simultaneously. For example, let A = {1, 2, 3, 4, 5} and B = {3, 4, 5, 6, 7}. Then A ∪ B = {1, 2, 3, 4, 5, 6, 7} and A ∩ B = {3, 4, 5}.

在维恩图上，A ∪ B 是两个圆圈覆盖的全部区域，A ∩ B 是两个圆圈重叠的中间区域。这两个区域是 Edexcel 图表题的常客。做题时可以用一个小技巧：先分别标出 A 和 B 的元素，再根据”或”和”且”的逻辑合并或取公共部分，这样可以避免数漏元素。

On a Venn diagram, A ∪ B is the whole region covered by the two circles, while A ∩ B is the overlapping middle region. These two regions are regulars in Edexcel diagram questions. Here is a useful trick: first mark the elements of A and B separately, then combine or take the common part according to the logic of “or” and “and”. This prevents you from missing elements.

7. 补集与差集：在全集的框架下描述 | Complements and Differences: Describing Within the Universal Set

补集（complement）运算需要依赖全集的概念。全集 ξ（读作 “xi”）是讨论范围内所有可能元素的集合。集合 A 的补集记作 A′（或 A^c），定义为 A′ = { x : x ∈ ξ 且 x ∉ A }，也就是全集中所有不属于 A 的元素。在维恩图上，A′ 是 A 圆圈外面的所有区域（包括其他集合的圆圈内部）。

The complement operation relies on the concept of the universal set. The universal set ξ (read as “xi”) is the set of all possible elements under discussion. The complement of a set A, written A′ (or A^c), is defined as A′ = { x : x ∈ ξ and x ∉ A }, that is, all elements of the universal set that are not in A. On a Venn diagram, A′ is the whole region outside the circle of A (including the interiors of any other circles).

差集（difference）是另一个常用运算。A − B（或 A B）表示属于 A 但不属于 B 的元素，即 A − B = { x : x ∈ A 且 x ∉ B }。例如，设 A = {1, 2, 3, 4, 5}，B = {3, 4, 6}，则 A − B = {1, 2, 5}，B − A = {6}。注意，差集与补集不同：补集永远相对于全集而言，而差集是相对于另一个集合而言。

The difference is another commonly used operation. A − B (or A B) means the elements that belong to A but not to B, that is, A − B = { x : x ∈ A and x ∉ B }. For example, let A = {1, 2, 3, 4, 5} and B = {3, 4, 6}; then A − B = {1, 2, 5} and B − A = {6}. Note that the difference is not the same as the complement: the complement is always taken relative to the universal set, while the difference is taken relative to another set.

Edexcel 考试喜欢把补集和差集混在一起考，例如要求你写出 (A ∪ B)′ 或者 A′ ∩ B 对应的区域。处理这类复合运算时，最稳妥的方法是一步一步来：先算括号内的部分，再算括号外的运算。例如 (A ∪ B)′ 先求并集 A ∪ B，再对结果取补集，得到的是两个圆圈之外的所有区域。

Edexcel exams like to mix complements and differences, for example asking you to identify the region for (A ∪ B)′ or A′ ∩ B. When dealing with such compound operations, the safest method is to work step by step: first compute the part inside the brackets, then apply the outer operation. For example, for (A ∪ B)′ you first find the union A ∪ B, then take its complement, which gives the whole region outside the two circles.

8. Edexcel IGCSE 真题题型分析 | Edexcel IGCSE Exam Question Patterns

根据近年 Edexcel IGCSE 数学 A（4MA1）真题，集合描述法相关的题目主要有四种题型。第一种是”用描述法写出集合”：题目给出一组数或一个区域，要求你用 { x : … } 的形式表示。这类题考察的是对条件语言的精确把握，例如”大于 5 且小于等于 10 的整数”应写成 { x : x 是整数且 5 < x ≤ 10 }。

Based on recent Edexcel IGCSE Mathematics A (4MA1) papers, questions about set builder notation mainly come in four forms. The first is “write a set using set builder notation”: the question gives a list of numbers or a region, and asks you to express it in the form { x : … }. This type tests your precise command of conditional language. For example, “integers greater than 5 and less than or equal to 10” should be written as { x : x is an integer and 5 < x ≤ 10 }.

第二种题型是”元素判断”：给定一个用描述法定义的集合，判断某个数是否属于它。例如 A = { x : x 是整数且 x² < 20 }，问 5 是否属于 A。因为 5² = 25 > 20，所以 5 ∉ A。这类题要求你既能读懂描述法，又能快速验证条件。第三种题型是”维恩图与描述法互译”，我们已经在第 5 节详细讨论过。

The second type is “element membership”: given a set defined by set builder notation, decide whether a particular number belongs to it. For example, A = { x : x is an integer and x² < 20 }; does 5 belong to A? Since 5² = 25 > 20, we have 5 ∉ A. This type requires you to read set builder notation fluently and verify the condition quickly. The third type is “translation between Venn diagrams and set builder notation”, which we discussed in detail in Section 5.

第四种题型是”集合运算求元素个数”：结合描述法和 n(A) 记号（表示集合 A 的元素个数）出题。例如全集 ξ = {1, 2, 3, …, 20}，A = { x : x 是 3 的倍数 }，B = { x : x 是偶数 }，求 n(A ∩ B)。A ∩ B 中的元素必须既是 3 的倍数又是偶数，即 6 的倍数，在 1 到 20 之间共有 6, 12, 18 三个，所以 n(A ∩ B) = 3。

The fourth type is “counting elements after set operations”: questions combine set builder notation with the n(A) notation (the number of elements in set A). For example, universal set ξ = {1, 2, 3, …, 20}, A = { x : x is a multiple of 3 }, B = { x : x is even }; find n(A ∩ B). Elements of A ∩ B must be multiples of both 3 and 2, that is, multiples of 6. Between 1 and 20 there are exactly three: 6, 12 and 18, so n(A ∩ B) = 3.

9. 常见错误与易混淆点 | Common Mistakes and Confusing Points

第一个高频错误是混淆属于符号 ∈ 和包含符号 ⊆。x ∈ A 表示”x 是 A 的一个元素”，x 是一个元素；A ⊆ B 表示”A 是 B 的子集”，A 是一个集合。两者的对象层次完全不同：元素用小写字母，集合用大写字母。写描述法条件时，若 x 是元素，应该写 x ∈ A，而不是 A ∈ x。

The first high-frequency error is confusing the membership symbol ∈ with the subset symbol ⊆. x ∈ A means “x is an element of A”, where x is an element; A ⊆ B means “A is a subset of B”, where A is a set. The two operate on completely different levels: elements are written in lowercase letters and sets in capital letters. When writing a condition in set builder notation, if x is an element, you should write x ∈ A, never A ∈ x.

第二个常见错误是漏掉全集或选错全集。补集运算必须说明相对于哪个全集，不同的全集会产生不同的补集。例如在全集 ℤ 中，{ x : x > 0 } 的补集是 { x : x ≤ 0 }（包括 0 和所有负整数）；但如果全集是 ℕ，同一个集合的补集就是空集 ∅，因为自然数中没有非正数。

The second common error is forgetting the universal set or choosing the wrong one. A complement operation must specify which universal set it is relative to, because different universal sets give different complements. For example, within the universal set ℤ, the complement of { x : x > 0 } is { x : x ≤ 0 } (including 0 and all negative integers); but if the universal set is ℕ, the complement of the same set is the empty set ∅, because there are no non-positive natural numbers.

第三个错误是不等式方向写反，尤其在”且”和”或”的转换上。{ x : x > 2 且 x < 7 } 是 2 和 7 之间的区间；而 { x : x > 2 或 x < 7 } 却是除了 2 到 7 之外几乎覆盖全部实数（实际是全集 ℝ）。一字之差，集合完全不同。读题时务必圈出”且/and”与”或/or”，养成条件反射。

The third error is writing the inequality direction backwards, especially when converting between “and” and “or”. { x : x > 2 and x < 7 } is the interval between 2 and 7; but { x : x > 2 or x < 7 } covers almost all real numbers (in fact the whole of ℝ). A single word changes the set completely. When reading a question, always circle “and” and “or” so that the distinction becomes a reflex.

第四个错误是混淆空集与含空集的集合。∅ 表示空集，它不含任何元素；而 {∅} 是含有一个元素的集合，这个元素就是空集本身。两者完全不同：n(∅) = 0，而 n({∅}) = 1。此外还要注意，空集是任何集合的子集，即对任意集合 A，都有 ∅ ⊆ A，但空集并不一定是 A 的元素。

The fourth error is confusing the empty set with a set containing the empty set. ∅ is the empty set, which contains no elements; but {∅} is a set with exactly one element, namely the empty set itself. The two are completely different: n(∅) = 0 while n({∅}) = 1. Also note that the empty set is a subset of every set: for any set A, ∅ ⊆ A, but the empty set is not necessarily an element of A.

10. 实战练习与分步解答 | Practice Questions with Step-by-Step Solutions

练习一：用描述法表示集合 {2, 4, 6, 8, 10}。解答：这些元素都是 1 到 10 之间的偶数，因此可以写成 { x : x 是整数且 1 ≤ x ≤ 10 且 x 是偶数 }。更简洁的写法是利用 2 的倍数：{ x : x = 2n，其中 n 是正整数且 n ≤ 5 }。两种写法都正确，考试中任选一种即可。

Practice 1: Express the set {2, 4, 6, 8, 10} using set builder notation. Solution: these elements are all even numbers between 1 and 10, so we can write { x : x is an integer, 1 ≤ x ≤ 10 and x is even }. A more compact form uses multiples of 2: { x : x = 2n, where n is a positive integer and n ≤ 5 }. Both answers are correct; choose either one in the exam.

练习二：设全集 ξ = {1, 2, 3, 4, 5, 6, 7, 8}，A = { x : x 是 2 的倍数 }，求 A′。解答：先在 ξ 中找出 2 的倍数：A = {2, 4, 6, 8}。补集就是全集中不属于 A 的元素：A′ = {1, 3, 5, 7}。用描述法可以写成 A′ = { x : x ∈ ξ 且 x 不是 2 的倍数 }。

Practice 2: Let the universal set ξ = {1, 2, 3, 4, 5, 6, 7, 8} and A = { x : x is a multiple of 2 }. Find A′. Solution: first find the multiples of 2 in ξ: A = {2, 4, 6, 8}. The complement is the set of elements of ξ not in A: A′ = {1, 3, 5, 7}. In set builder notation we can write A′ = { x : x ∈ ξ and x is not a multiple of 2 }.

练习三：A = { x : x 是整数且 -3 < x ≤ 4 }，B = { x : x 是正整数 }。求 A ∩ B 和 A − B。解答：A 的元素为 {-2, -1, 0, 1, 2, 3, 4}，B = {1, 2, 3, …}。交集为 A ∩ B = {1, 2, 3, 4}；差集为 A − B = {-2, -1, 0}。注意 0 不是正整数，所以 0 属于 A 但不属于 B。

Practice 3: A = { x : x is an integer and -3 < x ≤ 4 }, B = { x : x is a positive integer }. Find A ∩ B and A − B. Solution: the elements of A are {-2, -1, 0, 1, 2, 3, 4} and B = {1, 2, 3, …}. The intersection is A ∩ B = {1, 2, 3, 4}; the difference is A − B = {-2, -1, 0}. Note that 0 is not a positive integer, so 0 belongs to A but not to B.

练习四：用维恩图表示三个集合 A、B 和 C，并涂出区域 (A ∩ B) − C。解答：先找出 A 与 B 的重叠部分（同时属于 A 和 B 的区域），再从中去掉同时属于 C 的部分。最终涂出的是 A、B 两圆重叠区域中落在 C 圆之外的那部分。分步作图可以避免把 C 圆内的重叠区域误涂进去。

Practice 4: Draw a Venn diagram with three sets A, B and C, and shade the region (A ∩ B) − C. Solution: first identify the overlap of A and B (the region belonging to both), then remove the part that also belongs to C. The final shading is the part of the A-B overlap that lies outside circle C. Drawing step by step prevents you from accidentally shading the overlap inside circle C.

练习五：已知 n(ξ) = 30，n(A) = 12，n(B) = 15，n(A ∩ B) = 5，求 n(A ∪ B) 和 n(A′ ∩ B)。解答：由容斥原理，n(A ∪ B) = n(A) + n(B) − n(A ∩ B) = 12 + 15 − 5 = 22。A′ ∩ B 是”属于 B 但不属于 A”的元素，即 n(A′ ∩ B) = n(B) − n(A ∩ B) = 15 − 5 = 10。

Practice 5: Given n(ξ) = 30, n(A) = 12, n(B) = 15 and n(A ∩ B) = 5, find n(A ∪ B) and n(A′ ∩ B). Solution: by the inclusion-exclusion principle, n(A ∪ B) = n(A) + n(B) − n(A ∩ B) = 12 + 15 − 5 = 22. The set A′ ∩ B consists of elements in B but not in A, so n(A′ ∩ B) = n(B) − n(A ∩ B) = 15 − 5 = 10.

Summary | 总结

集合描述法是 Edexcel IGCSE 数学中连接”列举”与”抽象”的桥梁。它的核心形式 { x : 条件 } 用元素的共同性质定义集合，特别适合表示无穷集合和区间。本文依次讲解了描述法的语法结构、四大数集符号 ℕ ℤ ℚ ℝ、区间型描述法、与维恩图的互译、并集交集补集差集五种运算，以及 Edexcel 真题的四种题型。

Set builder notation is the bridge between listing and abstraction in Edexcel IGCSE Mathematics. Its core form { x : condition } defines a set by the common property of its elements, and it is especially suitable for infinite sets and intervals. This article has covered the syntax of set builder notation, the four number set symbols ℕ ℤ ℚ ℝ, interval-style sets, translation with Venn diagrams, the five operations (union, intersection, complement and difference), and the four question patterns found in Edexcel papers.

复习时请特别留意四个易错点：区分 ∈ 与 ⊆、明确补集的全集、辨别”且”与”或”、分清 ∅ 与 {∅}。把这四个易错点练熟，再配合足够的真题训练，集合描述法相关的题目就能稳定拿分。希望这篇指南能帮助你在 IGCSE 数学考试中更加从容自信。

When revising, pay special attention to four common pitfalls: distinguishing ∈ from ⊆, specifying the universal set for complements, telling “and” apart from “or”, and separating ∅ from {∅}. Once you have mastered these four pitfalls and practised enough past paper questions, you will score consistently on set builder notation questions. We hope this guide helps you feel more confident and prepared in your IGCSE Mathematics exam.

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