📚 Set Notation | 集合表示法
Set notation is the language used to describe collections of objects, known as elements. In A-Level Mathematics, particularly in the Edexcel specification, mastering set notation is essential for expressing solutions to equations and inequalities, defining domains, and understanding probability and statistics. From listing elements within curly braces to using set‑builder notation and interval notation, a solid grasp of these symbols ensures precise mathematical communication.
集合表示法是用来描述对象集合（称为元素）的语言。在 A-Level 数学中，尤其是 Edexcel 考试大纲，掌握集合表示法对于表达方程和不等式的解、定义定义域以及理解概率和统计至关重要。从在花括号内列举元素到使用集合构造式（描述法）和区间表示法，扎实掌握这些符号可以保证数学表达的准确性。
1. What is a Set? Listing and Set‑Builder Notation | 什么是集合？列举法与描述法
A set is a well‑ s A, B, or S. There are two primary ways to describe a set: the listing method and the set‑ . The order of elements does not matter, and repeating an element does not change the set.
集合是明确定义的不同对象的整体。我们通常用大写字母如 A、B 或 S 来表示集合。描述集合主要有两种方法：列举法和集合构造法（描述法）。列举法直接在大括号内写出每一个元素，例如 A = {2, 4, 6, 8}。元素顺序无关紧要，重复元素不改变集合。
Set‑ cifying a property that all its elements satisfy. The general form is {x : condition}, where the colon is read as "such that" al number less than 10} produces the same set as A. In Edexcel, the colon is commonly used, though a vertical bar "|" may also appear.
集合构造法通过规定所有元素都满足的性质来定义集合。一般形式为 {x : 条件}，其中冒号读作“满足”。例如，B = {x : x 是小于 10 的偶数自然数} 得到与 A 相同的集合。在 Edexcel 中，通常使用冒号，有时也会出现竖线 "|"。
2. Element of a Set: ∈ and ∉ | 集合的元素：属于与不属于
We write x ∈ A to mean that x is an element of A, and x ∉ A to mean that x is not an element of A. For instance, if C = {1, 3, 5, 7}, then 3 ∈ C but 4 ∉ C. These symbols are the building blocks for all relational statements about sets.
我们写作 x ∈ A 表示 x 是 A 的元素，x ∉ A 表示 x 不是 A 的元素。例如，若 C = {1, 3, 5, 7}，则 3 ∈ C 但 4 ∉ C。这些符号是所有关于集合的关系陈述的基础构件。
Membership can also be used with infinite sets. If we define D = {x : x ∈ ℝ, x² < 9}, then 0 ∈ D, 2 ∈ D, but 3 ∉ 9. Notice the strict inequality matters; we always respect the condition given in the set‑builder description.
属于关系也可以用于无限集。如果定义 D = {x : x ∈ ℝ, x² < 9}，那么 0 ∈ D，2 ∈ D，但 3 ∉ D，因为 3² = 9 并不严格小于 9。注意严格不等式很重要；我们始终要遵守集合构造描述中给定的条件。
3. Finite, Infinite and Empty Sets | 有限集、无限集和空集
rsday, Friday} has five elements, so it is finite. An infinite set contains countless elements, such as the set of all whole numbers ℤ or the set of all points on a line.
如果一个集合包含可数数量的元素，它就是有限集。例如，E = {星期一, 星期二, 星期三, 星期四, 星期五} 有五个元素，因此是有限集。无限集包含无数个元素，比如所有整数的集合 ℤ 或直线上所有点的集合。
The empty set, denoted by ∅ or sometimes { }, is the unique set that contains no elements. In set notation, ∅ e empty set with the set containing zero; {0} is not empty because it has one element. The empty set is a subset of every set, which we will explore shortly.
空集，记作 ∅ 或有时用 { }，是不包含任何元素的唯一集合。在集合表示法中，∅ = {}。重要的是不要将空集与含有零的集合混淆；{0} 不是空的，因为它有一个元素。空集是任何集合的子集，我们稍后会讨论。
4. Cardinality of a Set | 集合的基数
The cardinality of a finite set A, written as n(A) or |A| , if F = {2, 4, 6, 8, 10}, then n(F) = 5. Repeated elements are ignored: n({a, a, b}) = 2.
有限集合 A 的基数，写作 n(A) 或 |A|，是它包含的不同元素的个数。在 Edexcel 考试中，n(A) 是首选记号。例如，若 F = {2, 4, 6, 8, 10}，则 n(F) = 5。重复元素被忽略：n({a, a, b}) = 2。
ems with Edexcel, you will often combine set notation with Venn diagrams and calculate n(A ∪ B), n(A'), and so on, using the formula n(A ∪ B) = n(A) + n(B) − n(A ∩ B).
无限集具有无限基数。我们不赋予具体数字；只是说集合不是有限的。在 Edexcel 的概率问题中，你经常会结合集合表示法和文氏图，并使用公式 n(A ∪ B) = n(A) + n(B) − n(A ∩ B) 等来计算。
5. Subsets and Proper Subsets | 子集与真子集
A set A is a subset of B, written A ⊆ B, if every element of A is also an element of B. By this definition, every set is a subset of itself, and the empty set is a subset of any set. To indicate that A is a subset of B but A and B are not equal, we use the proper subset symbol A ⊂ B, which means A ⊆ B and A ≠ B.
若集合 A 的每一个元素都是 B 的元素，则称 A 是 B 的子集，写作 A ⊆ B。根据这一定义，每个集合都是自身的子集，且空集是任何集合的子集。为了表示 A 是 B 的子集但 A 与 B 不相等，我们使用真子集符号 A ⊂ B，这意味着 A ⊆ B 且 A ≠ B。
For example, let G = {1, 2, 3}. Then H = {2, 3} is a subset: H ⊆ G and also H ⊂ G. K = {1, 2, 3} satisfies K ⊆ G but K ⊂ G is false. The symbol ⊄ means "is not a subset of": {4, 5} ⊄ . Understanding these relationships is key when working with power sets and Venn diagrams.
例如，设 G = {1, 2, 3}。那么 H = {2, 3} 是一个子集：H ⊆ G 且 H ⊂ G。K = {1, 2, 3} 满足 K ⊆ G 但 K ⊂ G 为假。符号 ⊄ 表示“不是……的子集”：{4, 5} ⊄ G。理解这些关系在处理幂集和文氏图时非常关键。
6. Universal Set and Complement | 全集与补集
set that contains all objects under consideration. The complement of a set A, written as A' in Edexcel, is the set of all elements in ξ that are not in A. In set‑builder notation: A' = {x ∈ ξ : x ∉ A}.
在任何讨论背景下，全集——通常用 ξ（希腊字母 xi）或 U 表示——是包含所有被考虑对象的集合。集合 A 的补集，在 Edexcel 中写作 A'，是 ξ 中所有不在 A 中的元素的集合。用集合构造法表示为：A' = {x ∈ ξ : x ∉ A}。
For example, if ξ = {1, 2, 3, 4, 5} and A = {1, 3, 5}, then A' = {2, 4}. Two important identities arise from complement: A ∪ A' = ξ and A ∩ A' = ∅ are frequently tested, especially when simplifying set expressions using algebra of sets.
例如，若 ξ = {1, 2, 3, 4, 5} 且 A = {1, 3, 5}，则 A' = {2, 4}。由补集产生两个重要恒等式：A ∪ A' = ξ 以及 A ∩ A' = ∅。这些关系经常出现在考题中，尤其是在利用集合代数简化表达式时。
7. Union and Intersection | 并集与交集
The union of two sets A and B, denoted A ∪ the set containing all elements that belong to A, or to B, or to both. The intersection, A ∩ B, is the set of elements that belong to both A and B. In set‑builder form: A ∪ B = {x : x ∈ A or x ∈ B} and A ∩ B = {x : x ∈ A and x ∈ B}.
两个集合 A 和 B 的并集，记作 A ∪ B，是包含所有属于 A、或属于 B、或同时属于两者的元素的集合。交集 A ∩ B 是同时属于 A 和 B 的元素的集合。用集合构造形式可写作：A ∪ B = {x : x ∈ A 或 x ∈ B} 以及 A ∩ B = {x : x ∈ A 且 x ∈ B}。
e overlapping region. If two sets have no common elements, they are called disjoint, and A ∩ B = ∅. Disjoint sets are widely used in probability.
使用文氏图时，并集对应于将两个圆全部涂上阴影，交集对应于只涂重叠区域。如果两个集合没有公共元素，则称它们互斥，且 A ∩ B = ∅。互斥集合在概率中被广泛使用。
8. Set Difference | 差集
The difference of set A and set B, written A \ B or sometimes A − B, is the set of elements that are in A but not in B. In set‑builder notation: A \ B = {x : x ∈ A and x ∉ B}. Note that generally A \ B is not the same as B \ A.
集合 A 与集合 B 的差集，写作 A \ B 或有时用 A − B，是属于 A 但不属于 B 的元素的集合。用集合构造法表示为：A \ B = {x : x ∈ A 且 x ∉ B}。请注意，通常 A \ B 并不等于 B \ A。
For example, if A = {1, 2, 3, 4} and B = {3, 4, 5}, then A \ B = {1, 2} and B \ et that are outside another. It can also be expressed using intersection and complement: A \ B = A ∩ B'.
例如，若 A = {1, 2, 3, 4} 而 B = {3, 4, 5}，则 A \ B = {1, 2}，而 B \ A = {5}。差集运算在分离一个集合中不在另一个集合中的部分时很有用。它也可以用交集和补集表示：A \ B = A ∩ B'。
9. Laws of Set Operations | 集合运算定律
Set algebra obeys several fundamental laws. Commutative laws state that A ∪ B = B ∪ A and A ∩ B = B ∩ A. Associative laws allow us to group operations: (A ∪ B) ∪ C = A ∪ (B ∪ C) and similarly for intersection. The distributive laws link union and intersection: A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C) and A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C).
集合代数遵循若干基本定律。交换律指出 A ∪ B = B ∪ A 且 A ∩ B = B ∩ A。结合律允许我们分组运算：(A ∪ B) ∪ C = A ∪ (B ∪ C)，交集也类似。分配律将并集和交集联系起来：A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C) 以及 A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)。
De Morgan's laws are particularly important for complements of unions and intersections. They are written as:
(A ∪ B)' = A' ∩ B'
(A ∩ B)' = A' ∪ B'
These laws allow you to transform complement expressions into forms with intersections and unions of complements, very useful in simplifying logical statements and in probability calculations.
德摩根定律对于并集和交集的补集尤为重要。它们写作以上居中公式。这些定律使你能够将补集表达式转化为补集的交和并的形式，在简化逻辑陈述和概率计算时非常有用。
10. Common Number Sets | 常见数集
In A‑level mathematics, certain infinite sets are denoted by standard symbols. ℕ represents the set of natural numbers; in Edexcel this typically means positive integers {1, 2, 3, ...}. ℤ denotes integers: {..., −3, −2, −1, 0, 1, 2, 3, ...}. ℚ is the set of rational numbers, numbers that can be written as a fraction of two integers. ℝ is the set of real numbers, including all rational and irrational numbers. Occasionally ℂ denotes complex numbers, though A‑level Pure further covers that.
在 A‑level 数学中，某些无限集用标准符号表示。ℕ 代表自然数集；在 Edexcel 中，通常指正整数 {1, 2, 3, ...}。ℤ 表示整数：{..., −3, −2, −1, 0, 1, 2, 3, ...}。ℚ 是有理数集，即可以写成两个整数之比的数。ℝ 是实数集，包括所有有理数和无理数。偶尔 ℂ 表示复数，不过 A‑level 进阶纯数会涉及。
Knowing these symbols allows us to write conditions compactly, such as {x ∈ ℝ : x² − 4 = 0} = {−2, 2} or {x ∈ ℤ e standard sets form the basis for domain restrictions and solution sets in many problems.
了解这些符号后，我们可以简洁地写出条件，例如 {x ∈ ℝ : x² − 4 = 0} = {−2, 2} 或 {x ∈ ℤ : −3 < x ≤ 2} = {−2, −1, 0, 1, 2}。这些标准集合构成了许多问题中定义域限制和解集的基础。
11. Interval Notation | 区间表示法
l numbers x such that a < x < b. A closed interval [a, b] means all x with a ≤ x ≤ b. Half‑ −∞ are used with open brackets because infinity is not a number and can never be included.
处理实数时，我们常将集合表示为区间。开区间 (a, b) 表示所有满足 a < x < b 的实数 x。闭区间 [a, b] 表示所有满足 a ≤ x ≤ b 的 x。半开区间 (a, b] 或 [a, b) 结合了不等式。无穷符号 ∞ 和 −∞ 总是使用开括号，因为无穷大不是数字，永远无法被包含。
For instance, the set {x ∈ ℝ : x > 4} can be written as (4, ∞). The set {x ∈ ℝ : −2 ≤ x < 5} is [−2, 5). Edexcel questions frequently require you to switch between set‑ ion, especially when giving final answers to inequalities or finding domains of functions.
例如，集合 {x ∈ ℝ : x > 4} 可以写作 (4, ∞)。集合 {x ∈ ℝ : −2 ≤ x < 5} 为 [−2, 5)。Edexcel 题目经常要求你在集合构造表示法和区间表示法之间转换，尤其是在给出不等式最终答案或求函数定义域时。
12. Sets and Inequalities | 集合与不等式
Solving an inequality produces a solution set, which must be written in valid set notation. For example, solving x² − 5x + 6 < 0 gives (x − 2)(x − 3) < 0, so x is between 2 and 3. The solution set can be expressed as {x ∈ ℝ : 2 < x < 3} or as the interval (2, 3). Both forms are acceptable; however, Edexcel mark schemes often expect the set notation for final answers unless specified.
解不等式会产生一个解集，必须用合法的集合表示法写出。例如，解 x² − 5x + 6 < 0 得到 (x − 2)(x − 3) < 0，因此 x 介于 2 和 3 之间。解集可以表示为 {x ∈ ℝ : 2 < x < 3} 或区间 (2, 3)。两种形式都可以接受；不过 Edexcel 评分标准通常要求最终答案使用集合表示法，除非另有说明。
Solve x² − 4 ≥ 0: it gives (x − 2)(x + 2) ≥ 0, so x ≤ −2 or x ≥ 2. The solution set is {x ∈ ℝ : x ≤ −2} ∪ {x ∈ ℝ : x ≥ 2}, or equivalently (−∞, −2] ∪ [2, ∞). The union symbol ∪ is essential for connecting separate parts of the solution.
当二次不等式涉及大于或等于零时，我们会得到区间的并集。解 x² − 4 ≥ 0：得 (x − 2)(x + 2) ≥ 0，所以 x ≤ −2 或 x ≥ 2。解集为 {x ∈ ℝ : x ≤ −2} ∪ {x ∈ ℝ : x ≥ 2}，或写成等价的 (−∞, −2] ∪ [2, ∞)。并集符号 ∪ 对于连接解的不同部分是必不可少的。
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