Mastering Set Notation for IGCSE Mathematics | IGCSE数学集合表示法精讲

📚 Mastering Set Notation for IGCSE Mathematics | IGCSE数学集合表示法精讲

Set notation is a compact language used to describe collections of objects. In IGCSE Mathematics, questions on sets appear in both Paper 2 and Paper 4, often alongside probability and data handling. Knowing how to read and write set notation correctly can save time and prevent confusion.

集合表示法是一种简洁的数学语言，用来描述对象的集合。在 IGCSE 数学考试中，集合题目常见于 Paper 2 和 Paper 4，并常与概率、数据处理结合出现。准确掌握集合符号，能帮助你在考试中节省时间、减少失误。

1. What Is a Set? | 什么是集合？

A set is an unordered collection of distinct objects, called elements or members. In IGCSE, sets are usually written with curly braces, for example A = {2, 4, 6}. The order inside the braces does not matter, and repeated elements are not listed twice.

集合是按照某种规则确定的一群不同对象，这些对象称为元素。IGCSE 中通常用花括号表示集合，例如 A = {2, 4, 6}。大括号内的顺序不影响集合本身，重复元素不需要写两次。

Roster notation: A = {1, 2, 3}
枚举法：把集合中的所有元素逐一写出来，例如 A = {1, 2, 3}。
Set-builder notation: B = {x : x is an integer and 1 ≤ x ≤ 4}
描述法：用条件描述集合中元素的共同特征，例如 B = {x : x 是整数且 1 ≤ x ≤ 4}，即 B = {1, 2, 3, 4}。

2. The Empty Set and the Universal Set | 空集与全集

The empty set, written as ∅ or { }, is the set with no elements. It is a subset of every set. For example, if A = {1, 2}, then ∅ ⊆ A is always true.

空集是不含任何元素的集合，记作 ∅ 或 { }。空集是任何集合的子集。例如，若 A = {1, 2}，则一定有 ∅ ⊆ A。

The universal set, often written as ξ or U, contains all the objects under consideration in a particular problem. For IGCSE questions, the universal set is usually given at the start of the question. If ξ = {1, 2, 3, 4, 5} and A = {1, 2}, then the remaining elements 3, 4 and 5 belong to the complement of A.

全集是某个问题中所有可能对象的集合，通常用 ξ 或 U 表示。在 IGCSE 题目中，全集一般会在题干开头给出。若 ξ = {1, 2, 3, 4, 5}，且 A = {1, 2}，则剩下的元素 3、4、5 都属于 A 的补集。

3. Important Set Symbols | 常用集合符号

The table below lists the most important symbols you may see in an IGCSE paper.

下表整理了 IGCSE 考卷中常见的集合符号，请务必熟记。

Symbol	Meaning 含义	Example 示例
{ }	the set of … 集合；花括号	{1, 2} is the set containing 1 and 2. {1, 2} 表示含有 1 和 2 的集合。
∅ or { }	the empty set 空集	∅ has no elements. 空集不含任何元素。
∈	is an element of 属于；是……的元素	2 ∈ {1, 2, 3} 2 属于集合 {1, 2, 3}。
∉	is not an element of 不属于；不是……的元素	4 ∉ {1, 2, 3} 4 不属于集合 {1, 2, 3}。
⊆	is a subset of 是……的子集	{1} ⊆ {1, 2} {1} 是 {1, 2} 的子集。
⊂	is a proper subset of 是……的真子集	{1} ⊂ {1, 2} {1} 是 {1, 2} 的真子集。
∪	union; the set of elements in either set 并集；两个集合中的所有元素	{1, 2} ∪ {2, 3} = {1, 2, 3} 两个集合的并集是 {1, 2, 3}。
∩	intersection; the set of elements in both sets 交集；两个集合的公共元素	{1, 2} ∩ {2, 3} = {2} 两个集合的交集是 {2}。
A′	complement of A A 的补集	If ξ = {1, 2, 3, 4} and A = {1, 2}, then A′ = {3, 4}. 若 ξ = {1, 2, 3, 4}，A = {1, 2}，则 A′ = {3, 4}。
ξ or U	universal set 全集	ξ = {1, 2, 3, 4, 5} 全集包含问题中所有元素。
n(A)	the number of elements in set A 集合 A 的元素个数	n({x, y, z}) = 3 集合 {x, y, z} 有 3 个元素。
: or \|	such that 使得；满足条件	{x : x is a positive integer, x < 4} 所有满足 x < 4 的正整数组成的集合。

4. ∈ and ∉ | 属于与不属于

The symbol ∈ means belongs to, and ∉ means does not belong to. These symbols connect an element to a set, not one set to another set.

符号 ∈ 表示“属于”，∉ 表示“不属于”。这两个符号用于表示元素与集合之间的关系，而不是集合与集合之间的关系。

If A = {2, 4, 6}, then 2 ∈ A is true and 3 ∈ A is false.
如果 A = {2, 4, 6}，那么 2 ∈ A 成立，3 ∈ A 不成立。
3 ∉ A means 3 is not an element of A.
3 ∉ A 表示 3 不是 A 的元素。

5. Subsets and Proper Subsets | 子集与真子集

If every element of set A is also an element of set B, then A is a subset of B, written as A ⊆ B. If A ⊆ B and A ≠ B, then A is a proper subset of B, written as A ⊂ B.

如果集合 A 的所有元素都是集合 B 的元素，那么 A 是 B 的子集，记作 A ⊆ B。如果 A ⊆ B 且 A ≠ B，那么 A 是 B 的真子集，记作 A ⊂ B。

A ⊆ B means every element of A is also an element of B.

A ⊆ B 表示 A 中的每一个元素都属于 B。

Be careful: the empty set is a subset of every set, and every set is a subset of itself. For example, if B = {1, 2}, then ∅ ⊆ B, {1} ⊆ B, {2} ⊆ B and B ⊆ B.

注意：空集是任何集合的子集，任何集合都是它自身的子集。例如，若 B = {1, 2}，则 ∅ ⊆ B、{1} ⊆ B、{2} ⊆ B、B ⊆ B 都成立。

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Mastering Set Notation for IGCSE Mathematics | IGCSE数学集合表示法精讲