Set-Builder Notation | 集合描述法

📚 Set-Builder Notation | 集合描述法

Set-builder notation is one of the most elegant and compact ways to describe a set in mathematics. Instead of listing every element one by one, we state a rule that determines exactly which objects belong to the set. For the Edexcel IGCSE syllabus, mastering this notation is essential, as it appears in questions on sets, number systems, inequalities, and functions.

集合描述法是数学中描述集合最简洁、最优美的方式之一。我们不必逐一列出每个元素，而是给出一个规则，精确地决定哪些对象属于这个集合。对于 Edexcel IGCSE 考试大纲而言，掌握这种记法至关重要，因为它出现在集合、数系、不等式和函数等各类题目中。

1. What Is a Set? | 什么是集合

A set is a well-defined collection of distinct objects. The objects in a set are called elements or members. We usually name a set with a capital letter such as A, B, or C. In IGCSE mathematics, you will encounter sets of numbers frequently, and you must be able to write them clearly and without ambiguity. For example, the set of positive integers less than 5 is written as A = {1, 2, 3, 4}.

集合是明确定义的互异对象的总称。集合中的对象称为元素或成员。我们通常用大写字母（如 A、B、C）来表示集合。在 IGCSE 数学中，你会频繁遇到各种数集，必须能够清晰无误地将其表达出来。例如，小于 5 的正整数集合写作 A = {1, 2, 3, 4}。

An element belongs to a set is written using the symbol ∈. For instance, 3 ∈ A means “3 is an element of A”, while 5 ∉ A means “5 is not an element of A”. These symbols form the foundation of all set notation.

元素属于集合用符号 ∈ 表示。例如，3 ∈ A 表示”3 是 A 的元素”，而 5 ∉ A 表示”5 不是 A 的元素”。这些符号构成所有集合记法的基础。

2. Two Ways to Describe a Set | 描述集合的两种方式

There are two standard ways to describe a set. The first is roster form, also called listing form, where we write every element inside curly braces. For example, A = {1, 3, 5, 7, 9}. The second is set-builder notation, where we state a rule that every element must satisfy. For example, A = {x : x is an odd integer, 1 ≤ x ≤ 9}.

描述集合有两种标准方式。第一种是列举法（也叫列表法），我们将每个元素写在花括号内。例如 A = {1, 3, 5, 7, 9}。第二种是集合描述法，我们给出每个元素必须满足的规则。例如 A = {x : x 是奇数整数，1 ≤ x ≤ 9}。

Both forms describe exactly the same set. However, when a set contains too many elements to list, or when the elements follow a clear pattern, set-builder notation is far more powerful. For example, the set of all even integers cannot be written in roster form because there are infinitely many of them, but it can be written simply as {x : x ∈ ℤ, x is even}.

两种形式描述的是同一个集合。然而，当集合包含的元素太多无法一一列举，或元素遵循明显的规律时，描述法要强大得多。例如，所有偶整数的集合无法用列举法写出，因为有无穷多个元素，但可以用 {x : x ∈ ℤ, x 是偶数} 简洁地表示。

3. The Anatomy of Set-Builder Notation | 集合描述法的结构

Set-builder notation has three essential parts. Consider the set {x : x ∈ ℤ, x > 0}. The first part is the variable, here x, which represents an arbitrary element. The second part is the “such that” symbol, written as ‘:’ or ‘|’. The third part is the condition, which is a statement that must be true for every element of the set. The whole expression is enclosed in curly braces { }.

集合描述法有三个必要部分。考虑集合 {x : x ∈ ℤ, x > 0}。第一部分是变量（此处为 x），代表任意元素。第二部分是”使得”符号，写作 ‘:’ 或 ‘|’。第三部分是条件，即集合中每个元素都必须满足的陈述。整个表达式用花括号 { } 括起来。

In Edexcel IGCSE, the condition often includes membership of a number set, such as x ∈ ℤ, combined with an inequality or another property. Reading the notation aloud helps to check your understanding: {x : x ∈ ℤ, x > 0} is read as “the set of all x such that x is an integer and x is greater than zero”.

在 Edexcel IGCSE 中，条件通常包含数集归属（如 x ∈ ℤ）以及不等式或其他性质。大声读出这个记法有助于检验理解：{x : x ∈ ℤ, x > 0} 读作”所有使得 x 是整数且 x 大于零的 x 的集合”。

{x : condition} 或 {x | condition}

4. The “Such That” Symbol | “使得”符号

The symbols ‘:’ and ‘|’ inside set-builder notation are both read as “such that”. Many international textbooks prefer the vertical bar |, while some use the colon :. Edexcel accepts both, as long as you are consistent within your answer. For example, {x | x ∈ ℚ, x ≠ 0} is read as “the set of all x such that x is a non-zero rational number”.

描述法中的 ‘:’ 和 ‘|’ 都读作”使得”。许多国际教材偏好竖线 |，而有些则用冒号 :。Edexcel 两种写法都接受，只要你在同一个答案中保持一致。例如，{x | x ∈ ℚ, x ≠ 0} 读作”所有使得 x 是非零有理数的 x 的集合”。

A common question is whether the vertical bar and colon can be used interchangeably. The answer is yes in IGCSE. What matters is that you do not confuse this internal symbol with the ∈ symbol or with the commas that separate different parts of the condition. In China, the colon form {x : …} is more common in textbooks, but both forms are equally valid in the Edexcel examination.

常见问题是竖线和冒号是否可以互换。在 IGCSE 中答案是肯定的。关键在于不要把这个内部符号与 ∈ 符号或用于分隔条件不同部分的逗号混淆。在中国，冒号形式 {x : …} 在教材中更常见，但在 Edexcel 考试中两种形式同样有效。

5. Number Sets You Must Know | 必须掌握的数集

To use set-builder notation effectively, you must be familiar with the standard number-set symbols. The table below summarises the ones that appear regularly in the Edexcel IGCSE syllabus. Remember that ℤ, ℚ, and ℝ are widely used in set-builder conditions.

为了有效使用集合描述法，你必须熟悉标准数集符号。下表总结了 Edexcel IGCSE 大纲中经常出现的数集。请记住 ℤ、ℚ 和 ℝ 在描述法条件中被广泛使用。

Symbol 符号	Name 名称	Example 示例
ℕ	Natural numbers 自然数	{1, 2, 3, …}
ℤ	Integers 整数	{…, −2, −1, 0, 1, 2, …}
ℚ	Rational numbers 有理数	½, −¾, 7
ℝ	Real numbers 实数	√2, π, −3.5
∅	Empty set 空集	{ }

In set-builder notation, the universal set is often stated explicitly before the inequality. For example, {x : x ∈ ℤ, x ≥ 0} means “the set of non-negative integers”, which is {0, 1, 2, 3, …}. If we changed ℤ to ℝ, the same form would describe all non-negative real numbers, an infinite set that includes fractions and irrational numbers.

在描述法中，全集通常明确写在不等式之前。例如 {x : x ∈ ℤ, x ≥ 0} 表示”非负整数集”，即 {0, 1, 2, 3, …}。如果把 ℤ 换成 ℝ，同样的形式描述的则是所有非负实数，这是一个包含分数和无理数的无穷集合。

6. Inequalities Inside Set-Builder Notation | 描述法中的不等式

Set-builder notation is particularly useful for describing sets defined by inequalities. For example, the set of all real numbers greater than −1 and less than or equal to 4 is written as {x : x ∈ ℝ, −1 < x ≤ 4}. Notice that we always state the universal set first, followed by the inequality condition.

描述法尤其适合描述由不等式定义的集合。例如，所有大于 −1 且小于或等于 4 的实数的集合写作 {x : x ∈ ℝ, −1 < x ≤ 4}。注意我们总是先写全集，再写不等式条件。

If the universal set is ℤ, the same inequality gives a very different set. The set {x : x ∈ ℤ, −1 < x ≤ 4} equals {0, 1, 2, 3, 4}, because the only integers satisfying −1 < x ≤ 4 are 0, 1, 2, 3 and 4. This shows why specifying the universal set is so important: the same inequality can produce different sets under different universes.

如果全集是 ℤ，同一个不等式会得到完全不同的集合。集合 {x : x ∈ ℤ, −1 < x ≤ 4} 等于 {0, 1, 2, 3, 4}，因为满足 −1 < x ≤ 4 的整数只有 0、1、2、3 和 4。这说明指定全集非常重要：同一个不等式在不同的全集下会产生不同的集合。

Here is another example. The solution set of the inequality 2x − 5 < 7 in the real numbers is found by solving: 2x < 12, so x < 6. Therefore the solution set is {x : x ∈ ℝ, x < 6}.

再看一个例子。实数范围内不等式 2x − 5 < 7 的解集求解如下：2x < 12，所以 x < 6。因此解集为 {x : x ∈ ℝ, x < 6}。

7. Union and Intersection | 并集与交集

You can combine sets using the union symbol ∪ and the intersection symbol ∩. In set-builder notation, these operations can appear either outside the set or inside the condition. For example, the union of two sets of integers can be written as {x : x ∈ ℤ, x ≥ 3} ∪ {x : x ∈ ℤ, x ≤ −2}, which contains every integer that is either at least 3 or at most −2, that is {…, −4, −3, −2, 3, 4, 5, …}.

你可以用并集符号 ∪ 和交集符号 ∩ 来组合集合。在描述法中，这些运算既可以出现在集合外部，也可以出现在条件内部。例如，两个整数集的并集可以写成 {x : x ∈ ℤ, x ≥ 3} ∪ {x : x ∈ ℤ, x ≤ −2}，它包含所有大于等于 3 或小于等于 −2 的整数，即 {…, −4, −3, −2, 3, 4, 5, …}。

Intersection works in a similar way. The set {x : x ∈ ℝ, x > 1} ∩ {x : x ∈ ℝ, x < 5} equals {x : x ∈ ℝ, 1 < x < 5}. A more compact form is to write the two conditions together inside a single set: {x : x ∈ ℝ, x > 1, x < 5}, which is equivalent. In the Edexcel examination, you may be asked to find the intersection of two solution sets and express it using set-builder notation.

交集以类似方式运作。集合 {x : x ∈ ℝ, x > 1} ∩ {x : x ∈ ℝ, x < 5} 等于 {x : x ∈ ℝ, 1 < x < 5}。更紧凑的写法是把两个条件合并写在同一个集合内：{x : x ∈ ℝ, x > 1, x < 5}，两者等价。在 Edexcel 考试中，你可能会被要求求两个解集的交集并用描述法表示。

8. Common Pitfalls | 常见误区

Students frequently make the following mistakes in set-builder notation. Being aware of them will help you avoid losing easy marks in the exam.

学生在描述法中经常犯以下错误。了解这些错误有助于你在考试中避免不必要的失分。

Forgetting to define the universal set. Writing {x : x > 3} without stating whether x is an integer or a real number is ambiguous. Always specify x ∈ ℤ, x ∈ ℚ, or x ∈ ℝ.

忘记定义全集。写 {x : x > 3} 而不说明 x 是整数还是实数，会造成歧义。务必写明 x ∈ ℤ、x ∈ ℚ 或 x ∈ ℝ。
Confusing ∈ with ⊆. The symbol ∈ means “is an element of”, while ⊆ means “is a subset of”. For example, 5 ∈ ℤ is correct, but 5 ⊆ ℤ is incorrect.

混淆 ∈ 与 ⊆。∈ 表示”是……的元素”，而 ⊆ 表示”是……的子集”。例如 5 ∈ ℤ 是正确的，但 5 ⊆ ℤ 是错误的。
Writing an impossible condition. For example, {x : x ∈ ℝ, x > 5, x < 2} is an empty set, because no real number is simultaneously greater than 5 and less than 2. Recognise that this set equals ∅.

写出不可能满足的条件。例如 {x : x ∈ ℝ, x > 5, x < 2} 是空集，因为没有实数同时大于 5 且小于 2。要能识别出该集合等于 ∅。
Using wrong brackets. Sets must always be written with curly braces { }. Square brackets [ ] and parentheses ( ) denote intervals, not sets, and are not acceptable as set notation in IGCSE.

使用错误的括号。集合必须用花括号 { } 表示。方括号 [ ] 和圆括号 ( ) 表示区间，而非集合，在 IGCSE 中不能用作集合记法。

9. Worked Examples | 例题精解

Let us work through some Edexcel-style questions step by step. These examples reflect the kind of problems you may encounter in the examination.

让我们逐步解答几道 Edexcel 风格的题目。这些例题反映了你在考试中可能遇到的题型。

Example 1: Write the set of even integers between 1 and 11 inclusive in set-builder notation. Solution: The even integers in this range are 2, 4, 6, 8, and 10, so the set is {x : x ∈ ℤ,

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Set-Builder Notation | 集合描述法