IB Mathematics: Equality of Sets, Union and Intersection | IB数学：集合的相等、并集与交集

📚 IB Mathematics: Equality of Sets, Union and Intersection | IB数学：集合的相等、并集与交集

In IB Mathematics, sets are fundamental building blocks that help us organise and compare collections of objects. Understanding the equality of sets, along with the operations of union and intersection, is essential for solving problems in probability, statistics, and pure mathematics.

在 IB 数学中，集合是我们组织和比较对象集合的基础工具。理解集合的相等、并集与交集这些运算，对于解决概率、统计和纯数学中的问题至关重要。

1. Sets and Elements Recap | 集合与元素回顾

A set is a well-defined collection of distinct objects, called elements or members. We usually denote a set by a capital letter, such as A, B or C, and list its elements inside curly braces.

集合是一个定义明确的不同对象的整体，其中的对象称为元素或成员。我们通常用大写字母 A、B、C 表示集合，并把元素写在花括号内。

A = {1, 2, 3}

If a is an element of set A, we write a ∈ A. If a is not an element of A, we write a ∉ A. The order of elements and repetition do not matter.

如果 a 是集合 A 的元素，我们写作 a ∈ A；如果 a 不是 A 的元素，则写作 a ∉ A。元素的顺序和重复不影响集合本身。

Example: Let A = {2, 4, 6}. Then 4 ∈ A and 5 ∉ A.

例如：设 A = {2, 4, 6}，则 4 ∈ A，而 5 ∉ A。

2. Equality of Sets | 集合的相等

Two sets A and B are equal if and only if they contain exactly the same elements. In symbols, A = B means that every element of A is also an element of B, and every element of B is also an element of A.

两个集合 A 和 B 相等，当且仅当它们包含完全相同的元素。用符号表示，A = B 意味着 A 中的每个元素都是 B 的元素，且 B 中的每个元素也都是 A 的元素。

A = B ⇔ A ⊆ B and B ⊆ A

Because order and repetition are ignored, we have, for example, {1, 2, 3} = {3, 2, 1} = {1, 2, 2, 3}.

由于顺序和重复被忽略，例如 {1, 2, 3} = {3, 2, 1} = {1, 2, 2, 3}。

Example: If A = {x | x is an even prime number} and B = {2}, then A = B.

例如：若 A = {x | x 是偶质数}，B = {2}，则 A = B。

3. 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. Every set is a subset of itself.

如果 A 的每一个元素都是 B 的元素，则称 A 是 B 的子集，记作 A ⊆ B。任何集合都是其自身的子集。

If A ⊆ B and A ≠ B, then A is called a proper subset of B, written A ⊂ B. The empty set ∅ is a subset of every set.

如果 A ⊆ B 且 A ≠ B，则称 A 是 B 的真子集，记作 A ⊂ B。空集 ∅ 是任何集合的子集。

∅ ⊆ A for every set A

For example, if A = {1, 2} and B = {1, 2, 3}, then A ⊂ B, because A is contained in B but A ≠ B.

例如，若 A = {1, 2}，B = {1, 2, 3}，则 A ⊂ B，因为 A 包含于 B 但 A ≠ B。

4. Union of Sets | 并集

The union of two sets A and B, written A ∪ B, is the set of all elements that belong to A or B or both. The word ‘or’ in mathematics is inclusive.

两个集合 A 和 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。数学中的“或”是包含性的。

A ∪ B = {x | x ∈ A or x ∈ B}

Example: If A = {1, 2, 3} and B = {3, 4}, then A ∪ B = {1, 2, 3, 4}.

例如：若 A = {1, 2, 3}，B = {3, 4}，则 A ∪ B = {1, 2, 3, 4}。

The union of sets combines the elements of both sets and removes duplicates. It answers the question: what is in at least one of the sets?

并集将两个集合中的元素合并并去除重复。它回答的问题是：哪些元素至少属于其中一个集合？

5. Intersection of Sets | 交集

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

两个集合 A 和 B 的交集，记作 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 and x ∈ B}

Example: If A = {1, 2, 3} and B = {3, 4}, then A ∩ B = {3}.

例如：若 A = {1, 2, 3}，B = {3, 4}，则 A ∩ B = {3}。

The intersection represents the common elements shared by the sets. It answers the question: what is in both sets at the same time?

交集表示两个集合共有的元素。它回答的问题是：哪些元素同时属于两个集合？

6. Disjoint Sets and the Empty Set | 不相交集合与空集

If two sets A and B have no elements in common, then A ∩ B = ∅. We say that A and B are disjoint sets.

如果两个集合 A 和 B 没有公共元素，则 A ∩ B = ∅。此时称 A 和 B 是不相交集合。

Example: A = {1, 2} and B = {3, 4} are disjoint, so A ∩ B = ∅.

例如：A = {1, 2}，B = {3, 4} 不相交，因此 A ∩ B = ∅。

The empty set ∅ is the set with no elements. It is a subset of every set, and it plays the role of a neutral element in the sense that A ∪ ∅ = A and A ∩ ∅ = ∅.

空集 ∅ 是不含任何元素的集合。它是任何集合的子集，并且在运算中起到特殊作用：A ∪ ∅ = A，A ∩ ∅ = ∅。

Be careful not to confuse the empty set ∅ with the number 0, or with the set {0} which contains one element.

注意不要把空集 ∅ 与数字 0 混淆，也不要与含有元素 0 的集合 {0} 混淆。

7. Universal Set and Complement | 全集与补集

In many problems, all sets are considered within a fixed universal set U, which contains all objects under discussion.

在许多问题中，所有集合都处于一个固定的全集 U 之内，U 包含了讨论中的所有对象。

The complement of set A, written A′, is the set of all elements in U that are not in A.

集合 A 的补集，记作 A′，是 U 中所有不属于 A 的元素组成的集合。

A′ = {x | x ∈ U and x ∉ A}

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

例如，若 U = {1, 2, 3, 4, 5}，A = {1, 2}，则 A′ = {3, 4, 5}。

We also have the basic facts: (A′)′ = A, A ∪ A′ = U and A ∩ A′ = ∅.

我们还有基本事实：(A′)′ = A，A ∪ A′ = U，A ∩ A′ = ∅。

8. Venn Diagrams for Union and Intersection | 并集与交集的维恩图

A Venn diagram represents sets as circles inside a rectangle that represents the universal set U. The union A ∪ B is the region covered by all parts of both circles. The intersection A ∩ B is the overlapping region of the two circles.

维恩图用矩形表示全集 U，用圆表示集合。并集 A ∪ B 是两个圆覆盖的所有区域；交集 A ∩ B 是两个圆重叠的区域。

For three sets, A ∪ B ∪ C is the union of all three shaded regions, while A ∩ B ∩ C is the small central region common to all three circles.

对于三个集合，A ∪ B ∪ C 是三个圆全部覆盖的区域，而 A ∩ B ∩ C 是三个圆共同重叠的中央小区域。

When drawing Venn diagrams, always label the universal set and each set clearly. Shade the required region and check whether elements at boundaries are included.

画维恩图时，要标出全集和每个集合。按要求涂色，并检查边界元素是否包含在内。

9. Cardinality: n(A ∪ B) | 基数：n(A ∪ B)

The cardinality of a finite set A, written n(A), is the number of elements in A. For example, if A = {2, 4, 6}, then n(A) = 3.

有限集合 A 的基数，记作 n(A)，是 A 中元素的个数。例如，若 A = {2, 4, 6}，则 n(A) = 3。

For two sets A and B, the number of elements in the union is given by the inclusion-exclusion formula.

对于两个集合 A 和 B，并集中元素的个数由包含排除公式给出。

n(A ∪ B) = n(A) + n(B) − n(A ∩ B)

We subtract n(A ∩ B) because elements in the intersection are counted twice when we add n(A) and n(B).

我们减去 n(A ∩ B)，是因为 n(A) 和 n(B) 相加时，交集部分的元素被重复计算了一次。

Example: If n(A) = 10, n(B) = 8 and n(A ∩ B) = 3, then n(A ∪ B) = 10 + 8 − 3 = 15.

例如：若 n(A) = 10，n(B) = 8，n(A ∩ B) = 3，则 n(A ∪ B) = 10 + 8 − 3 = 15。

If A and B are disjoint, then n(A ∩ B) = 0 and n(A ∪ B) = n(A) + n(B).

如果 A 和 B 不相交，则 n(A ∩ B) = 0，于是 n(A ∪ B) = n(A) + n(B)。

10. Properties and Distributive Laws | 性质与分配律

Union and intersection satisfy several important algebraic properties.

并集和交集满足许多重要的代数性质。

Commutative: A ∪ B = B ∪ A, A ∩ B = B ∩ A

交换律：A ∪ B = B ∪ A，A ∩ B = B ∩ A。

Associative: A ∪ (B ∪ C) = (A ∪ B) ∪ C, A ∩ (B ∩ C) = (A ∩ B) ∩ C

结合律：A ∪ (B ∪ C) = (A ∪ B) ∪ C，A ∩ (B ∩ C) = (A ∩ B) ∩ C。

Idempotent: A ∪ A = A, A ∩ A = A

幂等律：A ∪ A = A，A ∩ A = A。

The distributive laws connect union and intersection.

分配律将并集与交集联系起来。

A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)

A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)

For example, let A = {1, 2}, B = {2, 3} and C = {1, 3}. Then B ∩ C = {3}, so A ∪ (B ∩ C) = {1, 2} ∪ {3} = {1, 2, 3}. Also A ∪ B = {1, 2, 3} and A ∪ C = {1, 2, 3}, so (A ∪ B) ∩ (A ∪ C) = {1, 2, 3}. The law holds.

例如，设 A = {1, 2}，B = {2, 3}，C = {1, 3}。则 B ∩ C = {3}，所以 A ∪ (B ∩ C) = {1, 2} ∪ {3} = {1, 2, 3}。同时 A ∪ B = {1, 2, 3}，A ∪ C = {1, 2, 3}，因此 (A ∪ B) ∩ (A ∪ C) = {1, 2, 3}。分配律成立。

11. De Morgan’s Laws | 德摩根定律

De Morgan’s laws describe how complements interact with union and intersection.

德摩根定律描述了补集与并集、交集之间的相互作用。

(A ∪ B)′ = A′ ∩ B′

The complement of a union is the intersection of the complements.

并集的补集等于各补集的交集。

(A ∩ B)′ = A′ ∪ B′

The complement of an intersection is the union of the complements.

交集的补集等于各补集的并集。

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}. The two results agree.

例如：设 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}。两者结果一致。

These laws are especially useful when simplifying expressions involving complements in IB exam questions.

在 IB 考试中，这些定律特别有助于化简包含补集的表达式。

12. Common Mistakes and Exam Tips | 常见错误与应试技巧

One common mistake is confusing the symbols ∈ and ⊆. The symbol ∈ relates an element to a set, while ⊆ relates a set to another set.

一个常见错误是混淆 ∈ 与 ⊆。符号 ∈ 用于元素与集合之间，而 ⊆ 用于集合与集合之间。

Another mistake is writing A ∩ B = ∅ when the sets actually share no common elements; this is correct only if they really have no intersection. Also, do not write {∅} when you mean ∅.

另一个错误是在两个集合没有公共元素时随意写 A ∩ B = ∅；只有当真无交集时才是正确的。同时，不要用 {∅} 代替 ∅。

When solving word problems, identify the universal set first, then translate the words ‘and’ into intersection and ‘or’ into union.

解决应用题时，先确定全集，然后把“且”翻译为交集，把“或”翻译为并集。

Always check whether a set operation changes the set. For example, A ∪ ∅ = A and A ∪ U = U, while A ∩ ∅ = ∅ and A ∩ U = A.

始终检查集合运算是否改变集合。例如：A ∪ ∅ = A，A ∪ U = U，而 A ∩ ∅ = ∅，A ∩ U = A。

In IB examinations, draw a Venn diagram whenever possible. It helps you avoid double-counting elements in n(A ∪ B).

在 IB 考试中，只要可能就画维恩图。它可以帮助你避免在计算 n(A ∪ B) 时重复计数。

Finally, remember that set notation is precise: write {x | x ∈ A and x ∈ B} rather than a vague sentence, and simplify using the properties and laws introduced above.

最后，请记住集合记号是精确的：应写 {x | x ∈ A 且 x ∈ B}，而不是含糊的句子，并利用上面介绍的性质和定律进行化简。

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IB Mathematics: Equality of Sets, Union and Intersection | IB数学：集合的相等、并集与交集