IGCSE Maths: Three-Set Venn Diagram Problem Solving | IGCSE数学：三集合文氏图问题解法

📚 IGCSE Maths: Three-Set Venn Diagram Problem Solving | IGCSE数学：三集合文氏图问题解法

Three-set Venn diagrams are a cornerstone of the Edexcel IGCSE Mathematics syllabus. They test your ability to organise complex information, interpret overlapping conditions, and apply logical reasoning systematically. Mastering this topic is essential for success in the Statistics and Probability components of the exam.

三集合文氏图是Edexcel IGCSE数学大纲中的核心内容。它考察你组织复杂信息、理解重叠条件以及系统运用逻辑推理的能力。掌握这一主题对于在考试的统计与概率部分取得好成绩至关重要。

---

1. Understanding the Three-Set Venn Diagram | 理解三集合文氏图

A three-set Venn diagram consists of three overlapping circles, typically labelled A, B, and C, enclosed within a universal set rectangle denoted by ξ. The circles divide the universal set into 8 distinct regions, each representing a different combination of membership in the three sets.

三集合文氏图由三个重叠的圆组成，通常标记为A、B、C，并被包含在一个记为ξ的全集矩形内。这些圆将全集划分为8个不同的区域，每个区域代表三个集合中元素归属的不同组合。

• The universal set ξ contains all elements under consideration. 全集ξ包含所考虑的所有元素。

• Each region corresponds to a unique intersection or exclusion of sets. 每个区域对应于集合交集或排除的唯一组合。

• Elements may belong to one, two, all three, or none of the sets. 元素可能属于一个、两个、三个或不属于任何集合。

---

2. Labelling the 8 Regions | 标注8个区域

Proper labelling is the foundation of successful problem-solving. Each of the 8 regions has a precise meaning that must be recognised instantly during the exam.

正确标注是成功解题的基础。8个区域中的每一个都有精确的含义，在考试中必须能够立即识别。

Region | 区域	Meaning | 含义
A ∩ B ∩ C	In all three sets | 属于所有三个集合
A ∩ B only	In A and B, but not C | 属于A和B，但不属于C
A ∩ C only	In A and C, but not B | 属于A和C，但不属于B
B ∩ C only	In B and C, but not A | 属于B和C，但不属于A
A only	In A only, not B or C | 仅属于A，不属于B或C
B only	In B only, not A or C | 仅属于B，不属于A或C
C only	In C only, not A or B | 仅属于C，不属于A或B
Outside all circles	In ξ but in none of A, B, C | 属于ξ，但不在A、B、C中

---

3. The Standard Formula | 标准公式

A common error is memorising formulas without understanding their derivation. However, one inclusion-exclusion formula for three sets appears repeatedly in IGCSE questions.

常见的错误是死记公式而不理解其推导过程。然而，一个关于三集合的容斥公式在IGCSE题目中反复出现。

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

This formula accounts for double-counting and ensures each element in the union is counted exactly once. It is particularly useful when working with total counts rather than diagram regions.

这个公式处理了重复计数的问题，确保并集中的每个元素都恰好被计数一次。当处理总数而非图形区域时，它特别有用。

---

4. Problem Type 1: Direct Application | 问题类型1：直接应用

In direct application problems, the number of elements in each region is provided, and you must calculate specific counts or percentages. These questions test your ability to interpret the diagram correctly.

在直接应用类问题中，每个区域的元素数量已经给出，你需要计算特定的数量或百分比。这类问题考察你正确解读图表的能力。

Example | 示例：In a class of 40 students, 20 play football (F), 18 play cricket (C), 15 play tennis (T), 8 play both F and C, 6 play both F and T, 5 play both C and T, and 3 play all three. How many play none?

示例：一个班有40名学生，20人踢足球（F），18人打板球（C），15人打网球（T），8人同时踢足球和打板球，6人同时踢足球和打网球，5人同时打板球和网球，3人三项都参加。有多少人一项都不参加？

Solution: n(F ∪ C ∪ T) = 20 + 18 + 15 − 8 − 6 − 5 + 3 = 37. Therefore, 40 − 37 = 3 students play none.

解法：n(F ∪ C ∪ T) = 20 + 18 + 15 − 8 − 6 − 5 + 3 = 37。因此，40 − 37 = 3名学生一项都不参加。

---

5. Problem Type 2: Working Inwards | 问题类型2：从内向外逐步填充

The most common IGCSE question type gives a partially completed diagram and asks you to fill in the missing numbers. The key strategy is to start with the innermost region (the triple intersection) and work outwards.

最常见的IGCSE题型是给出一个部分完成的图形，要求你填入缺失的数字。关键策略是从最内层的区域（三交集）开始，然后向外逐步推进。

• Step 1: Enter the value for A ∩ B ∩ C. 第1步：填入A ∩ B ∩ C的值。

• Step 2: Subtract from each paired intersection. 第2步：从每个两两交集中减去它。

• Step 3: Subtract from each total to find “only” values. 第3步：从每个总数中减去，找出”仅属于”的值。

• Step 4: Sum all regions and subtract from ξ. 第4步：将所有区域相加，然后从ξ中减去。

Example | 示例：n(A) = 30, n(B) = 25, n(C) = 20, n(A ∩ B) = 10, n(A ∩ C) = 8, n(B ∩ C) = 6, n(A ∩ B ∩ C) = 4, n(ξ) = 70.

示例：n(A) = 30，n(B) = 25，n(C) = 20，n(A ∩ B) = 10，n(A ∩ C) = 8，n(B ∩ C) = 6，n(A ∩ B ∩ C) = 4，n(ξ) = 70。

The “only” values are: A only = 30 − 10 − 8 + 4 = 16; B only = 25 − 10 − 6 + 4 = 13; C only = 20 − 8 − 6 + 4 = 10. The double-only regions: A∩B only = 10 − 4 = 6; A∩C only = 8 − 4 = 4; B∩C only = 6 − 4 = 2. Sum = 16 + 13 + 10 + 6 + 4 + 2 + 4 = 55. Outside = 70 − 55 = 15.

“仅属于”的值分别为：A仅 = 30 − 10 − 8 + 4 = 16；B仅 = 25 − 10 − 6 + 4 = 13；C仅 = 20 − 8 − 6 + 4 = 10。仅属于两个交集的区域：A∩B仅 = 10 − 4 = 6；A∩C仅 = 8 − 4 = 4；B∩C仅 = 6 − 4 = 2。总和 = 16 + 13 + 10 + 6 + 4 + 2 + 4 = 55。外部 = 70 − 55 = 15。

---

6. Problem Type 3: Algebraic Approach | 问题类型3：代数方法

When a problem involves ratios, unknown numbers, or conditions such as “n(A) = 2 × n(B)”, assigning algebraic variables to regions and solving equations becomes necessary.

当问题涉及比例、未知数量或”n(A) = 2 × n(B)”等条件时，就需要为各个区域分配代数变量并解方程。

Technique | 技巧：Let x represent the triple intersection. Express all other regions in terms of x using the given relationships, then use the total of ξ to form an equation.

技巧：设x代表三交集。利用给定的关系将其他所有区域用x表示，然后利用ξ的总数构造方程。

Example: In a survey, n(A ∩ B ∩ C) = x, and every other region is twice the size of the corresponding region in a standard proportional layout. If n(ξ) = 100 and each circle contains exactly 50 elements, set up equations: 3 × 50 − [sum of intersections] + x = 100, then solve for x.

示例：在一项调查中，n(A ∩ B ∩ C) = x，其他每个区域都是标准比例布局中对应区域的两倍。如果n(ξ) = 100且每个圆恰好包含50个元素，建立方程：3 × 50 − [交集之和] + x = 100，然后解出x。

Let the three double intersections be a, b, c. Then: 150 − (a + b + c − x) = 100, so a + b + c − x = 50.

设三个两两交集为a、b、c。则：150 − (a + b + c − x) = 100，所以a + b + c − x = 50。

---

7. Shading and Describing Regions | 阴影区域与描述

Exam questions often require you to shade a given set expression or to describe a shaded region using set notation. This tests your spatial reasoning and recall of set operations.

考试题目通常要求你为给定的集合表达式画阴影，或者用集合符号描述一个阴影区域。这考察你的空间推理能力以及对集合运算的掌握。

• A ∪ B ∪ C: shade all three circles entirely. A ∪ B ∪ C：将三个圆全部涂满。

• A ∩ B ∩ C: shade only the central triple overlap. A ∩ B ∩ C：只涂中心的三角形重叠部分。

• (A ∪ B) ∩ C: shade where C meets either A or B. (A ∪ B) ∩ C：涂C与A或B相交的部分。

• A′ ∩ B ∩ C: shade B ∩ C but exclude the part inside A. A′ ∩ B ∩ C：涂B ∩ C但排除A内的部分。

When describing a shaded region, first identify which sets and combinations are present, then express using ∩, ∪, and ′ notation precisely.

在描述阴影区域时，首先确定存在哪些集合及其组合，然后用∩、∪和′符号精确地表达。

---

8. Critical Exam Traps | 常见考试陷阱

Several common errors repeatedly cost students marks in three-set Venn diagram questions. Being aware of them allows you to avoid these pitfalls.

有几个常见错误反复让学生在三分集合文氏图问题中失分。了解这些错误能够帮助你避免陷入这些陷阱。

Mistake | 常见错误	Correct Approach | 正确做法
Writing n(A∩B) in the triple centre | 把n(A∩B)写进三交集中心	Always subtract n(A∩B∩C) first | 务必先减去n(A∩B∩C)
Forgetting elements outside all sets | 忘记所有集合之外的元素	Check that all 8 regions sum to n(ξ) | 检查8个区域之和等于n(ξ)
Using the formula for disjoint sets incorrectly | 错误地将公式用于不相交集合	Verify intersections are not zero before applying | 在应用前确认交集不为零
Confusing ∪ and ∩ in word problems | 在应用题中混淆∪和∩	Translate “or” as ∪, “and” as ∩ | 将”或”翻译为∪，”且”翻译为∩

---

9. Worked Example: Full Exam Question | 例题精解：完整考试题

Let us work through a complete exam-style question from start to finish, applying all the strategies discussed.

让我们从开始到结束完整地解答一道考试风格的题目，应用前面讨论的所有策略。

Question | 题目：A group of 80 people were asked about pets. 35 own dogs (D), 30 own cats (C), 25 own birds (B), 12 own both D and C, 10 own both D and B, 8 own both C and B, 5 own all three.

题目：调查了80人的宠物饲养情况。35人养狗（D），30人养猫（C），25人养鸟（B），12人同时养狗和猫，10人同时养狗和鸟，8人同时养猫和鸟，5人三种都养。

(a) How many own exactly two types of pets? (b) How many own only dogs? (c) How many own none?

（a）有多少人恰好养两种宠物？（b）有多少人只养狗？（c）有多少人什么也没养？

Solution | 解答：The double-only regions are: D∩C only = 12 − 5 = 7; D∩B only = 10 − 5 = 5; C∩B only = 8 − 5 = 3. Exactly two = 7 + 5 + 3 = 15.

解答：仅双交集的区域为：D∩C仅 = 12 − 5 = 7；D∩B仅 = 10 − 5 = 5；C∩B仅 = 8 − 5 = 3。恰好两种 = 7 + 5 + 3 = 15。

D only = 35 − 7 − 5 − 5 = 18. Sum = 18 + 7 + 5 + 3 + 5 + C-only + B-only. Since n(C) = 30: C only = 30 − 7 − 3 − 5 = 15. Since n(B) = 25: B only = 25 − 5 − 3 − 5 = 12. Sum = 18 + 7 + 5 + 3 + 5 + 15 + 12 = 65. None = 80 − 65 = 15.

D仅 = 35 − 7 − 5 − 5 = 18。总和 = 18 + 7 + 5 + 3 + 5 + C仅 + B仅。因为n(C) = 30：C仅 = 30 − 7 − 3 − 5 = 15。因为n(B) = 25：B仅 = 25 − 5 − 3 − 5 = 12。总和 = 18 + 7 + 5 + 3 + 5 + 15 + 12 = 65。什么都没有 = 80 − 65 = 15。

---

10. Probability from Venn Diagrams | 文氏图中的概率计算

Venn diagrams provide an intuitive way to calculate probabilities. The probability of an event is simply the count in the relevant region divided by the total in the universal set.

文氏图提供了一种直观的概率计算方法。事件的概率就是相关区域中的数量除以全集中的总数。

P(A) = n(A) / n(ξ) ; P(A ∩ B) = n(A ∩ B) / n(ξ) ; P(A ∪ B) = n(A ∪ B) / n(ξ)

Conditional probability is also accessible: P(A|B) = P(A ∩ B) / P(B), which corresponds to the proportion within B that also lies in A.

条件概率也可以由此求出：P(A|B) = P(A ∩ B) / P(B)，即B中同时属于A的元素所占的比例。

Example: Using the previous pet question data, P(owns exactly two pets) = 15/80 = 3/16. The probability that someone owns a dog given they own a cat is 12/30 = 2/5.

示例：使用前面宠物问题的数据，P(恰好养两种宠物) = 15/80 = 3/16。已知某人养猫，他养狗的概率是12/30 = 2/5。

---

11. Strategic Tips for the Edexcel Exam | Edexcel考试策略技巧

These strategies will help you maximise your marks on Venn diagram questions, whether they appear in Paper 1 or Paper 2.

以下策略将帮助你在文氏图题目中最大化得分，无论它们出现在试卷1还是试卷2中。

• Always redraw the diagram if the printed one is too small. 如果印刷的图形太小，总是重画一个。

• Write each value in the correct region; never leave numbers in the margin. 将每个值写在正确的区域中；不要把数字写在页边空白处。

• Show your working for partial marks. 展示你的计算过程以获得步骤分。

• Check that every region is filled; an empty region may indicate an error. 检查每个区域都已填充；空白区域可能表明有错误。

• Verify that the sum of all 8 regions equals n(ξ) before finalising. 在最终确定之前，验证8个区域的总和等于n(ξ)。

• In probability questions, simplify fractions where possible. 在概率问题中，尽可能化简分数。

---

12. Summary and Final Checklist | 总结与最终检查清单

Three-set Venn diagram problems are systematic and well-suited to methodical problem-solving. By mastering the structure of the diagram, understanding each of the 8 regions, and practising the three main problem types, you can approach even the most complex questions with confidence.

三集合文氏图问题是系统性的，非常适合有条理的解题方法。通过掌握图的结构、理解8个区域中的每一个，以及练习三种主要问题类型，你可以自信地应对最复杂的问题。

Final checklist: label regions carefully; start from the triple intersection; subtract before dividing; use algebra for unknowns; check totals against ξ; connect to probability when required.

最终检查清单：仔细标注区域；从三交集开始；先减后分；对未知数使用代数方法；对照ξ检查总数；在需要时与概率联系。

With regular practice, these problems become routine, and you can secure full marks in this topic area.

通过定期练习，这些问题会变得常规化，你可以在这一主题领域获得满分。

Published by TutorHao | IGCSE Mathematics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）