📚 Equality, Union and Intersection of Sets | 集合的相等、并集与交集
Set theory provides the language of modern mathematics. Three fundamental concepts — equality, union, and intersection — allow us to compare sets and build new ones. This guide explains these ideas with precise definitions, properties, and practical examples, following the IB Mathematics syllabus. Understanding these operations is essential for topics such as probability, logic, and functions.
集合论为现代数学提供了语言基础。相等、并集和交集这三个基本概念能让我们比较集合并构造新集合。本指南按照IB数学课程大纲，用精确的定义、性质和实例讲解这些思想。掌握这些运算对概率、逻辑和函数等主题至关重要。
1. Sets and Elements | 集合与元素
sets by capital letters such as A, B, C, and elements by lowercase letters. The notation a ∈ A means 'a is an element of A', while a ∉ A means 'a is not an element of A' of A'. Sets can be described by listing all elements inside curly braces, e.g., A = {2, 4, 6, 8}, or by a characteristic property, e.g., B = {x | x is an even integer between 1 and 9}.
集合是一些明确定义的、不同对象的总体，这些对象称为元素或成员。通常用大写字母如A、B、C表示集合，用小写字母表示元素。记号a ∈ A表示“a是A的元素”，而a ∉ A表示“a不是A的元素”。集合可以通过在花括号内列举所有元素来描述，例如A = {2, 4, 6, 8}，也可以通过特征性质来描述，例如B = {x | x是1到9之间的偶数}。
Two sets are considered the same if they contain exactly the same elements, regardless of order or repetition. The empty set, denoted ∅ or { }, is the set with no elements. A universal set U is the set of all possible elements under consideration in a particular context.
如果两个集合包含完全相同的元素，无论顺序或重复如何，它们就被视为相同。空集，记作∅或{ }，是不含任何元素的集合。全集U是在特定背景下所考虑的所有可能元素组成的集合。
2. 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 called a proper subset of B, denoted A ⊂ B. Every set is a subset of itself, so A ⊆ A is always true. The empty set is a subset of every set: ∅ ⊆ A for any A.
如果A的每个元素都是B的元素，则称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。每个集合都是自身的子集，因此A ⊆ A始终成立。空集是任何集合的子集：对任意A有∅ ⊆ A。
To prove A ⊆ B, we take an arbitrary element x ∈ A and show that x ∈ ets, we can simply check membership. For instance, if A = {1, 2} and B = {1, 2, 3}, then A ⊆ B and indeed A ⊂ B.
要证明A ⊆ B，我们任取一个元素x ∈ A并证明x ∈ B。对于有限集，我们只需检查成员资格。例如，若A = {1, 2} 而B = {1, 2, 3}，则A ⊆ B，且实际上A ⊂ B。
3. Equality of Sets | 集合的相等
f and only if they have precisely the same elements. This is equivalent to the condition A ⊆ B and B ⊆ Order and repeated elements do not affect equality: {a, b, c} = {c, b, a} = {a, a, b, c}.
两个集合A和B相等，记作A = B，当且仅当它们包含完全相同的元素。这等价于条件A ⊆ B 且 B ⊆ A。因此，证明集合相等往往需要证明两个子集关系。顺序和重复元素不影响相等性：{a, b, c} = {c, b, a} = {a, a, b, c}。
If A = {x | equality may be used to identify unknown set members when two set descriptions are given.
若A = {x | x是小于3的正整数}，而B = {1, 2}，则A = B。空集是唯一的；任意两个空集都相等。在IB题目中，当给出两个集合的描述时，可以利用相等性来识别未知集合成员。
4. Union of Sets | 并集
The union of two sets A and B, denoted A ∪ B, is the set containing all elements that belong to A, to B, or to both. Formally, A ∪ B = {x | x ∈ A or x ∈ B}. The 'or' here is inclusive, meaning elements present in both sets are included once.
两个集合A和B的并集，记作A ∪ B，是包含所有属于A、属于B或同时属于两者的元素的集合。形式定义为A ∪ B = {x | x ∈ A 或 x ∈ B}。这里的“或”是包含性的，即同时出现在两个集合中的元素只计入一次。
For example, if A = {2, 4, 6} and B = {4, 5, 6, 7}, then A ∪ B = {2, 4, 5, 6, 7}. The union of a set with itself is itself: A ∪ A = A. The union of A with the empty set is A: A ∪ ∅ = A. Union can be extended to more than two sets, e.g., A ∪ B ∪ C.
例如，若A = {2, 4, 6}，B = {4, 5, 6, 7}，则A ∪ B = {2, 4, 5, 6, 7}。集合与自身的并集仍是其本身：A ∪ A = A。A与空集的并集是A：A ∪ ∅ = A。并集可以推广到多个集合，如A ∪ B ∪ C。
5. Intersection of Sets | 交集
The intersection of A and B, written A ∩ B, is the set of elements that are common to both A and B. In set-builder notation: A ∩ B = {x | x ∈ A and x ∈ no elements in common, their intersection is the empty set, and they are called disjoint.
A与B的交集，记作A ∩ B，是同时属于A和B的公共元素组成的集合。用集合构建式表示为：A ∩ B = {x | x ∈ A 且 x ∈ B}。如果两个集合没有公共元素，它们的交集是空集，则称它们不相交。
Using the same example, A ∩ B = {4, 6}. The intersection of a set with itself is itself: A ∩ A = A. The intersection with the empty set yields the empty set: A ∩ ∅ = ∅ ection is also defined for multiple sets, and it corresponds to the overlap common to all.
沿用上例，A ∩ B = {4, 6}。集合与自身的交集是自身：A ∩ A = A。与空集的交集得到空集：A ∩ ∅ = ∅。交集也可对多个集合定义，它对应于所有集合的共同部分。
6. Venn Diagrams for Union and Intersection | 并集和交集的维恩图
Venn diagrams represent sets as circles (or other shapes) inside a rectangle denoting the universal set U. The union A ∪ B is the total region covered by both circles. The intersection A ∩ B is the overlapping region of the circles. Disjoint sets appear as non-overlapping circles.
维恩图用矩形（表示全集U）内的圆（或其他形状）来表示集合。并集A ∪ B是两个圆所覆盖的全部区域。交集A ∩ B是两圆重叠的区域。不相交的集合表现为两个不相交的圆。
Shading the appropriate region helps visualise complex set expressions. For example, (A ∪ B) ∩ C is the part of C that lies inside the union of A and B. Venn diagrams also illustrate relationships like A ⊆ B, where circle A lies entirely inside circle B.
给相应区域填色有助于可视化复杂的集合表达式。例如，(A ∪ B) ∩ C是C中位于A和B的并集内部的部分。维恩图也能展示A ⊆ B这样的关系，此时圆A完全位于圆B内部。
7. Algebraic Properties of Union and Intersection | 并集和交集的性质
Union and intersection satisfy several algebraic laws that mirror arithmetic operations to some extent. Key properties are commutativity, associativity, and distributivity.
并集和交集满足若干代数定律，这些定律在某种程度上与算术运算相似。主要性质包括交换律、结合律和分配律。
A ∪ B = B ∪ A (Commutative law for union)
A ∪ B = B ∪ A （并集的交换律）
A ∩ B = B ∩ A (Commutative law for intersection)
A ∩ B = B ∩ A （交集的交换律）
(A ∪ B) ∪ C = A ∪ (B ∪ C) and (A ∩ B) ∩ C = A ∩ (B ∩ ive laws. They allow us to omit parentheses when writing multiple unions or intersections.
(A ∪ B) ∪ C = A ∪ (B ∪ C) 以及 (A ∩ B) ∩ C = A ∩ (B ∩ C) 是结合律。这使得我们可以省略多个并集或交集之间的括号。
The distributive laws connect the two operations: A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C) and A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C). These are extremely useful for simplifying set expressions or proving equalities.
分配律将两种运算联系起来：A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C) 以及 A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)。这些法则对于简化集合表达式或证明相等式非常有用。
Other important properties include idempotent laws (A ∪ A = A, A ∩ A = A) and absorption laws: A ∪ (A ∩ B) = A, A ∩ (A ∪ B) = A.
其他重要性质还包括幂等律（A ∪ A = A，A ∩ A = A）和吸收律：A ∪ (A ∩ B) = A，A ∩ (A ∪ B) = A。
8. Cardinality and the Inclusion-Exclusion Principle | 基数与容斥原理
The cardinality of a finite set A, denoted |A| or n(A), is the number of distinct elements it contains. For two finite sets, the size of their union is not simply the sum of sizes; we must subtract the intersection to avoid double-counting. The inclusion-exclusion principle states:
有限集A的基数，记作|A|或n(A)，是它所含不同元素的个数。对于两个有限集合，它们并集的大小不是简单地相加，而必须减去交集以避免重复计数。容斥原理表述如下：
|A ∪ B| = |A| + |B| − |A ∩ B|
If A and B are disjoint, then |A ∩ B| = 0, so |A ∪ B| = |A| + |B|. This principle extends to three sets: |A ∪ B ∪ C| = |A| + |B| + |C| − |A ∩ B| − |B ∩ C| − |C ∩ A| + |A ∩ B ∩ C|.
如果A和B不相交，则|A ∩ B| = 0，故|A ∪ B| = |A| + |B|。该原理可推广到三个集合：|A ∪ B ∪ C| = |A| + |B| + |C| − |A ∩ B| − |B ∩ C| − |C ∩ A| + |A ∩ B ∩ C|。
In IB problems, cardinality arguments often appear in probability and counting questions. Knowing the formula allows you to find an unknown cardinality when other values are given.
在IB问题中，基数讨论常出现在概率与计数题目里。掌握该公式可在已知其他数值时求出未知基数。
9. Worked Examples | 例题解析
Example 1: Let A = {1, 2, 3, 4, 5} and B = {4, 5, 6, 7}. Find A ∪ B, A ∩ B, and determine whether A = B.
例题1：令A = {1, 2, 3, 4, 5}，B = {4, 5, 6, 7}。求A ∪ B、A ∩ B，并判断A是否等于B。
A ∪ B = {1, 2, 3, 4, 5, 6, 7}. A ∩ 4, 5}. Since the two sets do not contain exactly the same elements, A ≠ B. For instance, 1 ∈ A but 1 ∉ B, which immediately disproves equality.
A ∪ B = {1, 2, 3, 4, 5, 6, 7}。A ∩ B = {4, 5}。由于两个集合不完全包含相同元素，A ≠ B。例如，1 ∈ A 但 1 ∉ B，这直接推翻了相等性。
Example 2: In a class of 30 students, 18 study Mathematics, 15 study Physics, and 8 study both. How many study neither subject?
例题2：一个班级有30名学生，其中18人学习数学，15人学习物理，8人同时学习两门科目。有多少人两门都不学？
Let M be the set of Mathematics students, P the set of Physics students. |M| = 18, |P| = 15, |M ∩ P| = 8. The number studying at least one subject is |M ∪ P| = 18 + 15 − 8 = 25. Therefore, those studying neither = total − |M ∪ P| = 30 − 25 = 5.
设M为学习数学的学生集合，P为学习物理的学生集合。|M| = 18, |P| = 15, |M ∩ P| = 8。至少学习一门科目的人数为|M ∪ P| = 18 + 15 − 8 = 25。因此两门都不学的人数为总数 − |M ∪ P| = 30 − 25 = 5。
Example 3: Prove that A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C) using element arguments.
例题3：用元素论证法证明 A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)。
Take any x ∈ A ∩ (B ∪ C). Then x ∈ A and x ∈ B ∪ C. Hence x ∈ A and (x ∈ B or x ∈ C). If x ∈ B, then x ∈ A ∩ B; if x ∈ C, then x ∈ A ∩ C. So in all cases x ∈ (A ∩ B) ∪ (A ∩ C). Thus A ∩ (B ∪ C) ⊆ (A ∩ B) ∪ (A ∩ C). The reverse inclusion is proved similarly, establishing equality.
任取x ∈ A ∩ (B ∪ C)，则x ∈ A且x ∈ B ∪ C。因此x ∈ A并且(x ∈ B 或 x ∈ C)。若x ∈ B，则x ∈ A ∩ B；若x ∈ C，则x ∈ A ∩ C。因此在所有情形下x ∈ (A ∩ B) ∪ (A ∩ C)。故A ∩ (B ∪ C) ⊆ (A ∩ B) ∪ (A ∩ C)。反向包含可类似证明，从而等式成立。
10. Summary and Key Takeaways | 总结与核心要点
Equality of sets is determined by the subset condition in both directions: A = B iff A ⊆ B and B ⊆ A. The union A ∪ B gathers all elements from both sets, while the intersection A ∩ t, and algebraic laws help manipulate set expressions efficiently. The cardinality formula |A ∪ B| = |A| + |B| − |A ∩ B| or further study in probability, logic, and discrete mathematics within the IB curriculum.
集合的相等由双向子集条件决定：A = B 当且仅当 A ⊆ B 且 B ⊆ A。并集A ∪ B收集了两个集合中的所有元素，而交集A ∩ B只保留公共元素。维恩图提供了直观的理解方式，代数定律有助于高效地操作集合表达式。基数公式|A ∪ B| = |A| + |B| − |A ∩ B| 是解决计数问题的重要工具。掌握这些概念能为IB课程中概率、逻辑和离散数学的进一步学习打下坚实的基础。
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