📚 Essential Set Notation in IB Mathematics | IB数学：常用集合符号精讲
Sets form the foundation of many areas of IB Mathematics, from probability to functions and calculus. Mastering the standard set symbols is essential for clear communication, rigorous proofs, and correctly interpreting exam questions.
集合是 IB 数学中从概率到函数、微积分等多个领域的基石。掌握标准集合符号对清晰表达、严谨证明以及准确理解考试题目至关重要。
1. What Is a Set? | 集合的定义与基本表示
A set is a well-defined collection of distinct objects, called elements or members. In mathematics, we usually denote a set by a capital letter, such as A, B or U, and list its elements inside curly braces.
集合是一个定义明确的对象总体，这些对象互不相同，称为元素或成员。在数学中，我们通常用大写字母如 A、B 或 U 表示集合，并把元素写在大括号内。
For example, the set of natural numbers less than 5 can be written as {1, 2, 3, 4}. The order of elements does not matter, and each element appears only once.
例如，小于 5 的自然数集合可以写成 {1, 2, 3, 4}。元素的顺序无关紧要，每个元素只出现一次。
2. Element of: ∈ and ∉ | 属于与不属于
The symbol ∈ means “is an element of” or “belongs to”. When an object belongs to a set, we use ∈ between the object and the set. If it does not belong, we use ∉.
符号 ∈ 表示“是……的元素”或“属于”。当一个对象属于某个集合时，我们在对象与集合之间使用 ∈；如果它不属于，则使用 ∉。
For instance, 2 ∈ ℕ means 2 is a natural number, while -1 ∉ ℕ for typical IB notation where natural numbers are non-negative. Always check the definition used in your course.
例如，2 ∈ ℕ 表示 2 是自然数，而按 IB 通常的定义（自然数为非负整数），-1 ∉ ℕ。务必注意你所在课程中的具体定义。
a ∈ S : a is an element of S | a ∈ S：a 是 S 的元素
3. Describing a Set: List Notation and Set-Builder Notation | 列举法与描述法
There are two common ways to describe a set: listing all elements explicitly, and giving a rule. Listing is called roster notation; the rule method is called set-builder notation.
描述集合有两种常用方式：逐一列出所有元素，以及给出规则。前者称为列举法，后者称为描述法。
In set-builder notation, we write {x : condition} or {x | condition}, where the colon or vertical bar is read as “such that”. For example, A = {x : x is an even integer greater than 0}.
在描述法中，我们写作 {x : 条件} 或 {x | 条件}，其中冒号或竖线读作“使得”。例如，A = {x : x 是大于 0 的偶数}。
This notation is especially useful for infinite sets, such as B = {x : x > 3} when the universal set is ℚ. The variable x ranges over the universal set unless stated otherwise.
这种表示法对无限集合特别有用，例如当全集为 ℚ 时，B = {x : x > 3}。除非另有说明，变量 x 在全集范围内取值。
4. Standard Number Sets: ℕ, ℤ, ℚ, ℝ, ℂ | 常用数集符号
IB Mathematics uses special double-struck capital letters to denote important number sets. These symbols are standard in all international exams.
IB 数学使用特殊的空心大写字母表示重要的数集。这些符号在所有国际考试中都是标准的。
| Symbol | Meaning | 含义 |
| ℕ | Natural numbers | 自然数 |
| ℤ | Integers | 整数 |
| ℚ | Rational numbers | 有理数 |
| ℝ | Real numbers | 实数 |
| ℂ | Complex numbers | 复数 |
Note that some textbooks define ℕ as starting from 1, while others include 0. Always follow the convention stated in your IB course. For example, in the IB Analysis and Approaches formula booklet, ℕ often includes 0.
注意，有些教材把 ℕ 定义为从 1 开始，而另一些则包含 0。务必遵循 IB 课程中规定的约定。例如，在 IB 数学分析与方法公式手册中，ℕ 通常包含 0。
5. Subsets and Proper Subsets: ⊆, ⊂, ⊄ | 子集与真子集
If every element of A is also an element of B, we say A is a subset of B and write A ⊆ B. If A is a subset of B but not equal to 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，就称 A 是 B 的子集，记作 A ⊆ B。如果 A 是 B 的子集但不等于 B，则称 A 是 B 的真子集，记作 A ⊂ B。
The symbol ⊄ means “not a subset of”. For example, {1, 2} ⊄ {2, 3, 4}. Be careful: A ⊆ A is always true, and ∅ ⊆ A is true for every set A.
符号 ⊄ 表示“不是……的子集”。例如，{1, 2} ⊄ {2, 3, 4}。注意：A ⊆ A 总是成立，且对任意集合 A，∅ ⊆ A 也总是成立。
Some books use ⊂ to mean “subset” instead of “proper subset”. In IB, the notation is generally consistent, but you should read the definitions given in your formula booklet. When in doubt, use ⊆ for “subset or equal” because it is unambiguous in most contexts.
有些教材用 ⊂ 表示“子集”而不是“真子集”。在 IB 中符号通常是一致的，但你应该查看公式手册中的定义。拿不准时，使用 ⊆ 表示“子集或相等”，因为它在大多数语境中含义明确。
6. Union and Intersection: ∪ and ∩ | 并集与交集
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 intersection, written A ∩ B, contains exactly those elements that belong to both A and B.
两个集合 A 和 B 的并集记作 A ∪ B，是所有属于 A 或 B（或同时属于两者）的元素组成的集合。交集记作 A ∩ B，恰好包含那些同时属于 A 和 B 的元素。
For example, if A = {1, 2, 3} and B = {2, 3, 4}, then A ∪ B = {1, 2, 3, 4} and A ∩ B = {2, 3}. Note that union includes duplicates only once.
例如，若 A = {1, 2, 3}，B = {2, 3, 4}，则 A ∪ B = {1, 2, 3, 4}，A ∩ B = {2, 3}。注意并集中重复元素只出现一次。
x ∈ A ∪ B ⇔ x ∈ A or x ∈ B | x ∈ A ∩ B ⇔ x ∈ A and x ∈ B
x ∈ A ∪ B ⇔ x ∈ A 或 x ∈ B | x ∈ A ∩ B ⇔ x ∈ A 且 x ∈ B
7. Difference and Complement: \ and ‘ | 差集与补集
The difference or relative complement of B in A, written A \ B or A − B, is the set of elements that belong to A but not to B. In IB, the complement of a set A with respect to the universal set U is written A’ or Aᶜ.
B 在 A 中的差集或相对补集记作 A \ B 或 A − B，表示属于 A 但不属于 B 的元素集合。在 IB 中，集合 A 关于全集 U 的补集记作 A’ 或 Aᶜ。
For U = {1, 2, 3, 4, 5} and A = {2, 4}, the complement A’ = {1, 3, 5}. Also, A \ B and A ∩ B’ are identical.
设 U = {1, 2, 3, 4, 5}，A = {2, 4}，则补集 A’ = {1, 3, 5}。此外，A \ B 与 A ∩ B’ 是相同的。
8. Empty Set and Universal Set: ∅ and U | 空集与全集
The empty set, written ∅ or {}, is the set with no elements. The universal set, written U, is the set containing all objects under consideration. The choice of U depends on the context.
空集记作 ∅ 或 {}，是不含任何元素的集合。全集记作 U，是所讨论范围内包含所有对象的集合。全集的选择取决于具体情境。
For every set A, we have ∅ ⊆ A and A ⊆ U. Also, A ∪ ∅ = A and A ∩ ∅ = ∅. These identities are easy to remember and frequently tested.
对任意集合 A，有 ∅ ⊆ A 且 A ⊆ U。同时，A ∪ ∅ = A，A ∩ ∅ = ∅。这些恒等式容易记忆，且经常被考查。
Be careful with the set {∅}: it is not empty because it contains one element, namely ∅. This distinction is a classic trap in IB questions.
注意 {∅} 这个集合：它并不是空集，因为它含有一个元素，即 ∅。这个区别是 IB 考试中的经典陷阱。
9. Cardinality: n(A) and |A| | 集合的基数
The cardinality of a finite set A is the number of distinct elements it contains. We write n(A) or |A|. For example, if A = {3, 5, 7}, then n(A) = 3.
有限集合 A 的基数是它所含互异元素的个数，记作 n(A) 或 |A|。例如，若 A = {3, 5, 7}，则 n(A) = 3。
The cardinality of the empty set is n(∅) = 0. For two finite sets, the formula n(A ∪ B) = n(A) + n(B) − n(A ∩ B) is very useful in probability and counting problems.
空集的基数为 n(∅) = 0。对两个有限集合，公式 n(A ∪ B) = n(A) + n(B) − n(A ∩ B) 在概率和计数问题中非常有用。
For the power set P(A), which is the set of all subsets of A, the cardinality is given by n(P(A)) = 2ⁿ, where n = n(A). For instance, if n(A) = 2, then A has 4 subsets.
对于幂集 P(A)，也就是 A 的所有子集构成的集合，其基数为 n(P(A)) = 2ⁿ，其中 n = n(A)。例如，若 n(A) = 2，则 A 有 4 个子集。
10. Venn Diagrams and Set Relations | 维恩图与集合关系
Venn diagrams represent sets as regions in a rectangle. The rectangle is the universal set U, and circles or ellipses represent sets. The visual relationships help determine union, intersection, difference, and complement.
维恩图用矩形表示全集 U，用圆形或椭圆形表示集合。通过区域的图像关系，可以直观判断并、交、差和补集。
When A ⊆ B, the circle for A lies entirely inside the circle for B. When two sets overlap, the overlapping region represents A ∩ B. The entire shaded area of both circles represents A ∪ B.
当 A ⊆ B 时，代表 A 的圆完全落在代表 B 的圆内。当两个集合有交集时，重叠区域表示 A ∩ B。两个圆覆盖的总阴影区域表示 A ∪ B。
To find A’, shade the region outside the circle of A but inside the rectangle U. Practice drawing these diagrams for all combinations of two or three sets; they are invaluable for checking identities.
要求 A’，就阴影化 A 的圆之外、矩形 U 之内的区域。练习画两集合或三集合的所有情形，对验证恒等式非常有帮助。
11. Common Set Identities | 常用集合恒等式
De Morgan’s Laws connect complement, union, and intersection. They are frequently used to simplify set expressions in exams and are valid for any sets A and B within a universal set U.
德摩根定律将补集、并集和交集联系起来。它们在考试中常用于化简集合表达式，且对全集 U 内的任意集合 A 和 B 都成立。
(A ∪ B)’ = A’ ∩ B’
(A ∩ B)’ = A’ ∪ B’
Other useful identities include A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C) and A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C). These are the distributive laws for sets.
其他有用恒等式包括 A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C) 以及 A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)。这些是集合的分配律。
You should be ready to prove simple identities by showing each side is a subset of the other, or by using Venn diagrams. Both methods are accepted in IB.
你应该准备好通过证明两边互为子集，或借助维恩图来证明简单恒等式。这两种方法在 IB 中均可接受。
12. Exam Tips: Avoid Common Mistakes | 考试注意事项：避免常见错误
First, distinguish ∈ from ⊆. The symbol ∈ relates an element to a set, while ⊆ relates two sets. For example, 2 ∈ {1, 2} is correct, but 2 ⊆ {1, 2} is false.
首先，区分 ∈ 和 ⊆。符号 ∈ 表示元素与集合的关系，而 ⊆ 表示集合与集合的关系。例如，2 ∈ {1, 2} 正确，但 2 ⊆ {1, 2} 错误。
Second, always state the universal set when using complement notation. The complement of A depends on U: if U = {1, 2, 3} then {1, 3}’ = {2}, but if U = ℤ then {1, 3}’ is infinite.
其次，使用补集符号时必须说明全集。A 的补集依赖于 U：若 U = {1, 2, 3}，则 {1, 3}’ = {2}；但若 U = ℤ，则 {1, 3}’ 是无限集。
Third, be careful with the empty set in subset problems. The empty set is a subset of every set, and ∅ ∈ {∅} is true, but ∅ ∉ { } because { } contains no elements at all.
第三，在子集问题中要小心空集。空集是任何集合的子集，∅ ∈ {∅} 成立，但 ∅ ∉ {}，因为 {} 根本不含任何元素。
Finally, read set-builder notation carefully. The condition after the colon or vertical bar is the defining rule. Check whether the variable is restricted to integers, real numbers, or another set before writing your final answer.
最后，仔细阅读描述法表示。冒号或竖线后的条件就是定义规则。在写出最终答案前，检查变量是限于整数、实数还是其他集合。
Published by TutorHao | Mathematics Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
屏轩国际教育cambridge primary/secondary checkpoint, cat4, ukiset,ukcat,igcse,alevel,PAT,STEP,MAT, ibdp,ap,ssat,sat,sat2课程辅导，国外大学本科硕士研究生博士课程论文辅导Cancel reply