A-Level Mathematics: Set Notation | A-Level 数学：集合符号

📚 A-Level Mathematics: Set Notation | A-Level 数学：集合符号

In A-Level mathematics, set notation provides a universal language for describing collections of objects, from simple lists of numbers to complex solution sets of equations and inequalities. Mastering set notation is essential for topics such as algebra, sequences, and probability.

在 A-Level 数学中，集合符号为描述对象的集合提供了一种通用语言，从简单的数字列表到方程和不等式的复杂解集。掌握集合符号对于代数、数列和概率等主题至关重要。

1. What is a Set? | 什么是集合？

A set is a well-defined collection of distinct objects, called elements or members. The order in which elements are listed is irrelevant, and repeated elements are usually written only once. For example, the set of vowels in the English alphabet can be written as {a, e, i, o, u}.

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。元素列出的顺序无关紧要，重复元素通常只写一次。例如，英文字母表中元音字母的集合可以写成 {a, e, i, o, u}。

Sets can be finite or infinite. Some common examples are:

集合可以是有限的，也可以是无限的。一些常见例子如下：

{1, 2, 3} is the set of the first three positive integers.
{1, 2, 3} 是前三个正整数的集合。
{x, y} is a set with two elements.
{x, y} 是包含两个元素的集合。
∅ is the empty set, the set with no elements.
∅ 是空集，即不含任何元素的集合。

2. Roster and Set-Builder Notation | 列举法与描述法

There are two common ways to describe a set: roster notation and set-builder notation. Roster notation lists all elements between curly braces. Set-builder notation describes a property that determines membership, using a vertical bar “|” read as “such that”.

描述集合有两种常见方法：列举法和描述法。列举法在大括号中列出所有元素。描述法用竖线 “|”（读作“使得”）描述决定成员资格的性质。

{2, 4, 6, 8} = {x : x is an even positive integer less than 10}

{2, 4, 6, 8} = {x : x 是小于 10 的正偶数}

Similarly, the solution set of an equation can be written in set-builder notation. The equation x² = 4 has two solutions:

类似地，方程的解集可以用描述法表示。方程 x² = 4 有两个解：

{x : x² = 4} = {−2, 2}

3. Important Number Sets | 重要数集

A-Level mathematics uses standard symbols for important number sets. You must be able to recognise and use these symbols fluently.

A-Level 数学对重要数集使用标准符号。你必须能够熟练地识别和使用这些符号。

Symbol	English Name	中文名称	Description
ℕ	Natural numbers	自然数	{1, 2, 3, …} (in some courses, {0, 1, 2, …})
ℤ	Integers	整数	{…, −2, −1, 0, 1, 2, …}
ℚ	Rational numbers	有理数	Numbers expressible as p/q where p, q ∈ ℤ, q ≠ 0
ℝ	Real numbers	实数	All points on the number line
ℂ	Complex numbers	复数	Numbers of the form a + bi where a, b ∈ ℝ

Notations such as ℤ⁺ and ℝ⁺ are sometimes used for positive integers and positive real numbers respectively.

有时使用 ℤ⁺ 和 ℝ⁺ 分别表示正整数和正实数。

4. Membership and Cardinality | 属于关系与基数

The symbol ∈ means “is an element of”, and ∉ means “is not an element of”. The cardinality of a finite set A, written |A|, is the number of elements in A.

符号 ∈ 表示“属于”，∉ 表示“不属于”。有限集合 A 的基数，记作 |A|，是 A 中元素的个数。

2 ∈ {1, 2, 3}
5 ∉ {1, 2, 3}
|{a, b, c}| = 3
|∅| = 0

For any finite set, the cardinality is a non-negative integer. The empty set has cardinality 0.

任何有限集合的基数都是非负整数。空集的基数为 0。

5. Subsets and the Power Set | 子集与幂集

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

如果 A 的每个元素都是 B 的元素，则 A 是 B 的子集，记作 A ⊆ B。如果 A 是 B 的子集但 A ≠ B，则 A 是 B 的真子集，记作 A ⊂ B。

The power set of A, denoted P(A), is the set of all subsets of A. Its cardinality follows a simple rule:

A 的幂集，记作 P(A)，是 A 的所有子集组成的集合。它的基数遵循一个简单的规则：

|P(A)| = 2ⁿ, where n = |A|

For example, if A = {1, 2}, then P(A) = {∅, {1}, {2}, {1, 2}}, so |P(A)| = 4 = 2².

例如，如果 A = {1, 2}，则 P(A) = {∅, {1}, {2}, {1, 2}}，所以 |P(A)| = 4 = 2²。

6. Union and Intersection | 并集与交集

The union of two sets A and B, written A ∪ B, contains every element that belongs to A, to B, or to both. The intersection, written A ∩ B, contains every element that belongs to both A and B.

两个集合 A 和 B 的并集，记作 A ∪ B，包含属于 A、属于 B 或同时属于两者的所有元素。交集，记作 A ∩ B，包含同时属于 A 和 B 的所有元素。

A ∪ B = {x : x ∈ A or x ∈ B}

A ∩ B = {x : x ∈ A and x ∈ B}

If A ∩ B = ∅, the sets A and B are said to be disjoint.

如果 A ∩ B = ∅，则称集合 A 和 B 不相交。

7. Complement and Difference | 补集与差集

The complement of a set A relative to a universal set U, written A′ (or Aᶜ), is the set of elements in U that are not in A. The difference of A and B, written A \ B, is the set of elements in A that are not in B.

集合 A 相对于全集 U 的补集，记作 A′（或 Aᶜ），是 U 中不属于 A 的元素组成的集合。A 与 B 的差集，记作 A \ B，是 A 中不属于 B 的元素组成的集合。

A′ = {x : x ∈ U and x ∉ A}

A \ B = {x : x ∈ A and x ∉ B}

For example, if U = {1, 2, 3, 4, 5}, A = {1, 2}, and B = {2, 3}, then A′ = {3, 4, 5} and A \ B = {1}.

例如，如果 U = {1, 2, 3, 4, 5}，A = {1, 2}，B = {2, 3}，则 A′ = {3, 4, 5}，A \ B = {1}。

8. Interval Notation and Inequalities | 区间表示与不等式

Interval notation is a shorthand for describing subsets of ℝ that are continuous ranges. It is frequently used when writing solution sets of inequalities.

区间表示法是描述 ℝ 上连续范围的简写形式，常用于书写不等式的解集。

(a, b) = {x : a < x < b} — open interval / 开区间
[a, b] = {x : a ≤ x ≤ b} — closed interval / 闭区间
[a, b) = {x : a ≤ x < b} — half-open interval / 半开半闭区间
(a, ∞) = {x : x > a} — unbounded above / 上方无界
(−∞, b] = {x : x ≤ b} — unbounded below / 下方无界

When a solution set consists of two or more disjoint intervals, the union symbol ∪ is used. For example, the solution set of x < 1 or x ≥ 3 is written as (−∞, 1) ∪ [3, ∞).

当解集由两个或多个不相交的区间组成时，使用并集符号 ∪。例如，x < 1 或 x ≥ 3 的解集写作 (−∞, 1) ∪ [3, ∞)。

9. Cartesian Product | 笛卡尔积

The Cartesian product of sets A and B, written A × B, is the set of all ordered pairs (a, b) where a ∈ A and b ∈ B.

集合 A 与 B 的笛卡尔积，记作 A × B，是所有有序对 (a, b) 组成的集合，其中 a ∈ A 且 b ∈ B。

A × B = {(a, b) : a ∈ A, b ∈ B}

For example, if A = {1, 2} and B = {x, y}, then A × B = {(1, x), (1, y), (2, x), (2, y)}. In general, |A × B| = |A| · |B|.

例如，如果 A = {1, 2}，B = {x, y}，则 A × B = {(1, x), (1, y), (2, x), (2, y)}。一般地，|A × B| = |A| · |B|。

10. De Morgan’s Laws | 德摩根定律

De Morgan’s laws relate unions, intersections, and complements. They are useful for simplifying expressions involving multiple set operations.

德摩根定律揭示了并集、交集和补集之间的联系，在化简涉及多种集合运算的表达式时非常有用。

(A ∪ B)′ = A′ ∩ B′

(A ∩ B)′ = A′ ∪ B′

For example, let U = {1, 2, 3, 4}, A = {1, 2}, and B = {2, 3}. Then A ∪ B = {1, 2, 3}, so (A ∪ B)′ = {4}. Also A′ = {3, 4} and B′ = {1, 4}, so A′ ∩ B′ = {4}. Both sides agree.

例如，令 U = {1, 2, 3, 4}，A = {1, 2}，B = {2, 3}。则 A ∪ B = {1, 2, 3}，所以 (A ∪ B)′ = {4}。又 A′ = {3, 4}，B′ = {1, 4}，因此 A′ ∩ B′ = {4}。两边结果一致。

11. Venn Diagrams and Practical Problems | 韦恩图与实际问题

Venn diagrams provide a visual representation of sets and their relationships. They are especially useful for solving problems involving overlapping sets, such as counting problems with three categories.

韦恩图为集合及其关系提供了直观的图形表示，在解决涉及重叠集合的问题时特别有用，例如涉及三个类别的计数问题。

Consider a class of 30 students. Let M be the set of students who study Mathematics and P be the set of students who study Physics. Suppose 18 study Mathematics, 15 study Physics, and 8 study both. The number who study at least one subject is found using the formula:

考虑一个 30 名学生的班级。设 M 为学习数学的学生集合，P 为学习物理的学生集合。假设 18 人学习数学，15 人学习物理，8 人两科都学。至少学习一科的人数由以下公式得出：

|M ∪ P| = |M| + |P| − |M ∩ P| = 18 + 15 − 8 = 25

Therefore the number who study neither is 30 − 25 = 5.

因此两科都不学的人数是 30 − 25 = 5。

12. Set Notation in Probability | 概率中的集合符号

In probability, the sample space S plays the role of the universal set. Events are subsets of S, and set operations correspond to combinations of events.

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A-Level Mathematics: Set Notation | A-Level 数学：集合符号