Set Notation and Operations | 集合的符号表示与运算规则

📚 Set Notation and Operations | 集合的符号表示与运算规则

In A-Level Mathematics, sets are used to describe collections of objects that share a common property. From probability to number theory, set notation provides a precise language for expressing mathematical ideas efficiently. This article covers the standard symbols, definitions, and rules for set operations that you will need in your examinations.

在 A-Level 数学中，集合用于描述具有共同属性的对象集合。从概率论到数论，集合符号提供了一种精确的语言，能够高效地表达数学思想。本文将涵盖考试中所需的标准符号、定义和集合运算规则。

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1. The Definition of a Set | 集合的定义

A set is a well-defined collection of distinct objects, called elements or members. The objects in a set can be anything: numbers, letters, points, functions, or even other sets. The defining requirement is that for any given object, we can determine unambiguously whether it belongs to the set.

集合是一个定义明确的不同对象组成的整体，这些对象称为元素或成员。集合中的对象可以是任何事物：数字、字母、点、函数，甚至是其他集合。定义性要求是：对于任意给定的对象，我们都能明确地判断它是否属于该集合。

For example, the set of natural numbers less than 6 is written as:

例如，小于6的自然数集合写作：

A = {1, 2, 3, 4, 5}

The symbol ∈ means “is an element of”, and ∉ means “is not an element of”. Thus, 3 ∈ A and 7 ∉ A.

符号 ∈ 表示”属于”，∉ 表示”不属于”。因此，3 ∈ A 且 7 ∉ A。

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2. Set Notation and Standard Sets | 集合符号与标准数集

Certain sets of numbers appear so frequently that they have standard names and symbols. You must be able to recognize and use these symbols fluently.

某些数集出现频率极高，因此拥有标准名称和符号。你必须能够熟练识别和使用这些符号。

Symbol | 符号	Meaning | 含义
ℕ	Natural numbers | 自然数集 {1, 2, 3, …}
ℤ	Integers | 整数集 {…, −2, −1, 0, 1, 2, …}
ℚ	Rational numbers | 有理数集
ℝ	Real numbers | 实数集
ℂ	Complex numbers | 复数集

The empty set, denoted ∅ or { }, contains no elements. It is a subset of every set. A set with exactly one element, such as {5}, is called a singleton set.

空集，记作 ∅ 或 { }，不包含任何元素。它是任何集合的子集。只含一个元素的集合，如 {5}，称为单元素集。

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3. Listing and Rule Methods | 列举法与描述法

There are two main ways to describe a set. The first is the roster method (also called tabular form), which lists all elements inside braces. For example, A = {2, 4, 6, 8}. The second is the set-builder notation, which describes the property that elements must satisfy: A = {x : x is an even positive integer less than 10}.

描述集合主要有两种方式。第一种是列举法（也称列举形式），即将所有元素列在大括号内。例如 A = {2, 4, 6, 8}。第二种是描述法，即描述元素必须满足的性质：A = {x : x 是小于10的正偶数}。

Set-builder notation is particularly useful for infinite sets. For instance, the set of all real numbers greater than 3 is written as:

描述法对无限集合尤为有用。例如，所有大于3的实数构成的集合写作：

{x ∈ ℝ : x > 3}

The colon “:” is read as “such that”. Some authors use a vertical bar “|” instead of a colon.

冒号 “:” 读作”使得”。有些教材用竖线 “|” 代替冒号。

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

Set A is a subset of set B, written A ⊆ B, if every element of A is also an element of B. If A ⊆ B and A ≠ B, then A is a proper subset of B, written A ⊂ B. Equivalently, B contains at least one element not in A.

若集合 A 的每个元素都是集合 B 的元素，则称 A 是 B 的子集，记作 A ⊆ B。若 A ⊆ B 且 A ≠ B，则称 A 是 B 的真子集，记作 A ⊂ B。等价地说，B 中至少含有一个 A 中没有的元素。

For any set A, the empty set and A itself are always subsets of A. When not allowing proper subsets, it is helpful to remember: every set is a subset of itself; no set is a proper subset of itself.

对任意集合 A，空集和 A 本身都是 A 的子集。需要记住：每个集合都是自身的子集；但没有任何集合是自身的真子集。

If a finite set has n elements, then it has exactly 2ⁿ subsets in total, of which 2ⁿ − 1 are non-empty and 2ⁿ − 1 are proper subsets (excluding the set itself).

若一个有限集合含有 n 个元素，则它共有 2ⁿ 个子集，其中 2ⁿ − 1 个是非空子集，2ⁿ − 1 个是真子集（不包括自身）。

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5. Union and Intersection | 并集与交集

The union of sets A and B, written A ∪ B, is the set of all elements that belong to A or to B or to both. The intersection of A and B, written A ∩ B, is the set of all elements that belong to both A and B simultaneously.

集合 A 与 B 的并集，记作 A ∪ B，是由属于 A 或属于 B 或同时属于两者的所有元素组成的集合。交集 A ∩ B 是由同时属于 A 和 B 的所有元素组成的集合。

A ∪ B = {x : x ∈ A 或 x ∈ B}

A ∩ B = {x : x ∈ A 且 x ∈ B}

If A ∩ B = ∅, the sets are said to be disjoint — they share no common elements.

若 A ∩ B = ∅，则称这两个集合不相交——它们没有任何公共元素。

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6. Difference and Complement | 差集与补集

The difference of A and B, written A − B (or A \ B), is the set of elements that are in A but not in B:

集合 A 与 B 的差集，记作 A − B（或 A \ B），是由属于 A 但不属于 B 的元素组成的集合：

A − B = {x : x ∈ A 且 x ∉ B}

When a universal set U is fixed, the complement of A, written A′ or Aᶜ, is the set of all elements in U that are not in A:

当确定一个全集 U 时，A 的补集，记作 A′ 或 Aᶜ，是 U 中所有不属于 A 的元素组成的集合：

A′ = U − A = {x : x ∈ U 且 x ∉ A}

Note that A − B = A ∩ B′. This identity is often useful when simplifying expressions.

注意 A − B = A ∩ B′。这个恒等式在化简表达式时非常有用。

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7. Laws of Set Operations | 集合运算律

The following laws hold for all sets A, B, and C. They mirror the algebraic laws of arithmetic but operate on sets.

以下运算律对任意集合 A、B、C 均成立。它们与算术中的代数律相似，但作用于集合。

Law | 名称	Formula | 公式
Commutative | 交换律	A ∪ B = B ∪ A; A ∩ B = B ∩ A
Associative | 结合律	(A ∪ B) ∪ C = A ∪ (B ∪ C); (A ∩ B) ∩ C = A ∩ (B ∩ C)
Distributive | 分配律	A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C); A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)
Identity | 单位元	A ∪ ∅ = A; A ∩ U = A
Idempotent | 幂等律	A ∪ A = A; A ∩ A = A
Absorption | 吸收律	A ∪ (A ∩ B) = A; A ∩ (A ∪ B) = A

These laws are useful not only for set algebra but also in logic and Boolean algebra, where ∪ corresponds to OR and ∩ corresponds to AND.

这些运算律不仅在集合代数中有用，在逻辑学和布尔代数中同样重要，其中 ∪ 对应”或”，∩ 对应”与”。

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8. De Morgan’s Laws | 德摩根定律

De Morgan’s Laws describe how complementation interacts with union and intersection. They are essential for transforming between different forms of set expressions.

德摩根定律描述了补运算与并集、交集之间的相互关系。它们对于在集合表达式的不同形式之间进行转换至关重要。

(A ∪ B)′ = A′ ∩ B′

(A ∩ B)′ = A′ ∪ B′

In words: the complement of a union is the intersection of the complements, and the complement of an intersection is the union of the complements.

用语言描述：并集的补集等于补集的交集；交集的补集等于补集的并集。

For three sets, the laws extend naturally:

对三个集合，定律自然推广为：

(A ∪ B ∪ C)′ = A′ ∩ B′ ∩ C′

(A ∩ B ∩ C)′ = A′ ∪ B′ ∪ C′

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9. Cardinality and Counting | 基数与计数

The cardinality of a set A, written n(A) or |A|, is the number of elements in A. For finite sets, the following counting formula relates the cardinalities of unions and intersections:

集合 A 的基数，记作 n(A) 或 |A|，是 A 中元素的个数。对有限集合，以下计数公式将并集与交集的基数联系起来：

n(A ∪ B) = n(A) + n(B) − n(A ∩ B)

This formula avoids double-counting the elements that lie in both sets. For three sets, the general formula is:

该公式避免了重复计算同时属于两个集合的元素。对三个集合，推广公式为：

n(A ∪ B ∪ C) = n(A) + n(B) + n(C) − n(A ∩ B) − n(A ∩ C) − n(B ∩ C) + n(A ∩ B ∩ C)

These formulas are fundamental when solving problems involving Venn diagrams with overlapping regions.

这些公式在解决涉及韦恩图重叠区域的问题时是基础工具。

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10. Venn Diagrams | 韦恩图

A Venn diagram represents sets as enclosed regions within a universal set U, usually drawn as a rectangle. Each set is represented by a circle, and overlapping regions represent intersections.

韦恩图将集合表示为全集 U（通常画成矩形）内的封闭区域。每个集合用一个圆圈表示，重叠区域表示交集。

Venn diagrams are not just illustrations — they serve as powerful tools for visualizing relationships between sets, verifying identities, and solving counting problems. When drawing a Venn diagram for three sets, ensure that all eight regions (corresponding to the 2³ possible membership combinations) are present.

韦恩图不仅是插图——它们是可视化集合关系、验证恒等式和解决计数问题的强大工具。在绘制三个集合的韦恩图时，要确保所有八个区域（对应 2³ 种可能的归属组合）都存在。

When using a Venn diagram to solve problems, always start by filling in the cardinality of the innermost region (the triple intersection) and work outward.

使用韦恩图解题时，务必先从最内层区域（三重交集）的基数开始填写，然后向外推算。

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11. Sets and Probability | 集合与概率

Set notation provides the natural language for probability. An event is a subset of the sample space S (the universal set of all possible outcomes). The probability of event A is the sum of probabilities of all outcomes in A.

集合符号为概率提供了自然的语言。事件是样本空间 S（所有可能结果的集合）的子集。事件 A 的概率是 A 中所有结果的概率之和。

Key correspondences between set operations and probability:

集合运算与概率之间的关键对应关系：

• P(A ∪ B) = P(A) + P(B) − P(A ∩ B) — the addition rule for probabilities.
• P(A′ ) = 1 − P(A) — the complement rule.
• If A and B are mutually exclusive (disjoint), then P(A ∩ B) = 0 and P(A ∪ B) = P(A) + P(B).

• P(A ∪ B) = P(A) + P(B) − P(A ∩ B) —— 概率的加法公式。
• P(A′ ) = 1 − P(A) —— 补事件公式。
• 若 A 与 B 互斥（不相交），则 P(A ∩ B) = 0，且 P(A ∪ B) = P(A) + P(B)。

This correspondence allows many probability problems to be modelled and solved accurately using set algebra.

这种对应关系使得许多概率问题可以通过集合代数来精确建模和求解。

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12. Set Identities: Proof Strategies | 集合恒等式的证明策略

To prove that two sets are equal, you may use either the element method or algebraic manipulation. The element method involves showing that every element of the first set is in the second, and vice versa. The algebraic method uses established laws — such as commutative, associative, distributive, and De Morgan’s laws — to transform one expression into the other.

要证明两个集合相等，可以使用元素法或代数法。元素法是指证明第一个集合的每个元素都在第二个集合中，反之亦然。代数法运用已建立的运算律——如交换律、结合律、分配律和德摩根定律——将其中一个表达式变形为另一个。

For the element method, take an arbitrary x ∈ A and deduce x ∈ B; then take an arbitrary x ∈ B and deduce x ∈ A. Because x was chosen arbitrarily, the result holds for all elements.

对于元素法，先取任意 x ∈ A，推得 x ∈ B；再取任意 x ∈ B，推得 x ∈ A。由于 x 是任意选取的，结论对所有元素都成立。

Always state which law you are using in each algebraic step — this demonstrates rigour and earns full marks in examinations.

在代数法中，每一步都应标明所用的运算律——这体现了严谨性，也是考试中获取满分的保证。

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