Shading Sets | 集合的阴影表示

📚 Shading Sets | 集合的阴影表示

In IGCSE Edexcel Mathematics, shading sets on Venn diagrams is a fundamental skill that helps visualise relationships between collections of objects. This article explains how to shade regions corresponding to union, intersection, complement and difference, with clear step-by-step guidance.

在爱德思IGCSE数学中，在维恩图上给集合区域涂阴影是一项基本技能，它能帮助我们直观地看到多个集合之间的关系。本文将讲解如何为并集、交集、补集和差集对应的区域涂阴影，并给出清晰的分步指导。

1. Sets and Venn Diagrams | 集合与维恩图

A set is a well-defined collection of objects, called elements. The universal set, denoted by ξ, contains all elements under consideration. A Venn diagram uses circles to represent sets inside a rectangle that represents ξ.

集合是一个明确定义的对象总体，其中的对象称为元素。全集用符号 ξ 表示，包含所讨论的所有元素。维恩图用矩形表示全集，用圆表示集合。

For two sets A and B, the rectangle represents ξ, and the two circles overlap to show elements that belong to both sets. Regions outside the circles represent elements not in A or B.

对于两个集合 A 和 B，矩形表示全集 ξ，两个圆相交的部分表示同时属于两个集合的元素。圆外部的区域表示既不属于 A 也不属于 B 的元素。

ξ = {all possible elements}, A ⊆ ξ, B ⊆ ξ

When shading, always start by drawing the universal set as a rectangle, then draw circles for each set. Label each circle clearly.

涂阴影时，先画一个矩形表示全集，然后为每个集合画一个圆，并清晰标注。

2. Set Notation You Must Know | 必须掌握的集合符号

The following symbols are essential for shading problems:

以下符号对于阴影问题至关重要：

A ∪ B – union of A and B (elements in A or B or both) | A 和 B 的并集（属于 A 或 B 的所有元素）
A ∩ B – intersection of A and B (elements in both A and B) | A 和 B 的交集（同时属于 A 和 B 的元素）
A′ or Aᶜ – complement of A (elements not in A) | A 的补集（不属于 A 的元素）
A − B or A \ B – elements in A but not in B | A 与 B 的差集（在 A 中但不在 B 中的元素）
ξ – universal set | 全集
∅ – empty set (no elements) | 空集（没有元素）
∈ – is an element of | 属于
∉ – is not an element of | 不属于

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

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

Shading a set means colouring every region that represents elements in that set, and leaving all other regions white.

给集合涂阴影，就是把集合中所有元素所在的区域涂上颜色，其余区域留白。

3. Shading the Union A ∪ B | 并集 A ∪ B 的阴影

The union of two sets A and B is the set of all elements that belong to A, or to B, or to both. To shade A ∪ B, colour the entire region inside either circle.

两个集合 A 和 B 的并集是所有属于 A 或属于 B 或属于两者的元素的集合。要涂 A ∪ B 的阴影，就把任意一个圆内的全部区域涂色。

This includes the left circle, the right circle, and the overlapping middle section. The only unshaded region is outside both circles.

这包括左边的圆、右边的圆以及中间重叠的部分。唯一不涂色的区域是两个圆外部的部分。

A ∪ B = {x : x ∈ A 或 x ∈ B}

When in doubt, test with a specific element: if it is in A, shade it; if it is in B, shade it; if it is in both, shade it once.

如果拿不准，可以用一个具体元素测试：如果它在 A 中就涂色，在 B 中也涂色，两个都在只涂一次。

4. Shading the Intersection A ∩ B | 交集 A ∩ B 的阴影

The intersection of A and B is the set of elements that belong to both A and B. Only the overlapping lens-shaped region is shaded.

A 和 B 的交集是同时属于 A 和 B 的元素的集合。只有中间重叠的透镜状区域才被涂色。

The rest of circle A and circle B remain white, because those parts contain elements not simultaneously in both sets.

圆 A 和圆 B 的其余部分保持白色，因为那些部分包含的元素不同时属于两个集合。

A ∩ B = {x : x ∈ A 且 x ∈ B}

For three sets, A ∩ B ∩ C is the small region in the middle where all three circles overlap.

对于三个集合，A ∩ B ∩ C 是三个圆共同重叠的中间小区域。

5. Shading the Complement A′ | 补集 A′ 的阴影

The complement of set A, written A′ or Aᶜ, contains all elements of the universal set that are not in A. Shade everything outside circle A but inside the rectangle ξ.

集合 A 的补集记作 A′ 或 Aᶜ，包含全集中所有不在 A 中的元素。涂色时，将圆 A 外部但矩形 ξ 内部的所有区域涂色。

Note: the rectangle itself is the boundary of ξ. Do not shade outside the rectangle.

注意：矩形本身是全集 ξ 的边界，不要在矩形外部涂色。

A′ = {x : x ∈ ξ 且 x ∉ A}

If A and B are two sets, then (A ∪ B)′ represents the complement of the union, which is the region outside both circles.

如果 A 和 B 是两个集合，那么 (A ∪ B)′ 表示并集的补集，即两个圆外部的区域。

6. Shading the Difference A − B | 差集 A − B 的阴影

The difference A − B (also written A \ B) is the set of elements that are in A but not in B. Shade the part of circle A that does not overlap circle B.

差集 A − B（也可写作 A \ B）是那些在 A 中但不在 B 中的元素的集合。涂色时，涂圆 A 中不与圆 B 重叠的部分。

Equivalently, A − B = A ∩ B′. You can find it by shading A and then erasing the intersection A ∩ B.

等价地，A − B = A ∩ B′。你可以先涂 A，然后擦除交集 A ∩ B。

A − B = {x : x ∈ A 且 x ∉ B}

Note that A − B and B − A are different: B − A shades the part of B that does not overlap A.

注意 A − B 和 B − A 是不同的：B − A 涂的是 B 中不与 A 重叠的部分。

7. Shading Regions with Three Sets | 三个集合的阴影

With three sets A, B, C, the Venn diagram has eight distinct regions. You can shade complex expressions step by step.

当有三个集合 A、B、C 时，维恩图有八个不同区域。你可以分步来涂复杂表达式。

For example, to shade (A ∩ B) ∪ C, first shade the intersection A ∩ B, then add the whole circle C.

例如，要涂 (A ∩ B) ∪ C，先涂交集 A ∩ B，然后再加上整个圆 C。

To shade A ∩ (B ∪ C), first shade the union B ∪ C, then keep only the part that also lies inside circle A.

要涂 A ∩ (B ∪ C)，先涂并集 B ∪ C，然后只保留圆 A 内的那部分。

总区域数 = 2ⁿ，其中 n 为集合个数。三个集合时 n=3，所以有 8 个区域。

Remember that brackets change the operation order, just like in arithmetic.

记住，括号会改变运算顺序，就像算术中一样。

8. De Morgan’s Laws and Shading | 德摩根定律与阴影

De Morgan’s Laws connect complements of unions and intersections:

德摩根定律把并集和交集的补集联系起来：

(A ∪ B)′ = A′ ∩ B′

(A ∩ B)′ = A′ ∪ B′

The first law says the region outside both A and B is exactly the same as the region that is both outside A and outside B. Shading either expression gives the same white area inside the rectangle but outside both circles.

第一条定律说，A 和 B 两者外部的区域，恰好等同于既在 A 外部又在 B 外部的区域。给任一表达式涂色，都会得到矩形内但两个圆外部的相同白色区域。

The second law says the region not in the overlap of A and B is everything outside the overlap, which includes the individual circles minus the middle lens. Shade both A and B but leave the middle white.

第二条定律说，不在 A 和 B 共同重叠中的区域是重叠外部的所有部分，包括两个圆各自的区域减去中间透镜部分。给 A 和 B 都涂色，但中间保持白色。

These laws are useful for simplifying expressions before shading.

这些定律有助于在涂色前简化表达式。

9. Step-by-Step Shading Strategy | 分步涂色策略

Follow this reliable method for any shading question:

对于任何涂阴影问题，可遵循以下可靠步骤：

Identify the universal set and draw the rectangle.

确定全集并画出矩形。
Draw and label all circles representing sets.

画出所有代表集合的圆并标注。
If the expression has brackets, resolve the innermost operation first.

如果表达式有括号，先处理最内层运算。
Shade each simplified component lightly, then combine by union (add all shaded parts), intersection (keep only common parts), or complement (swap shaded and white).

先轻涂每个简化后的部分，然后通过并集（将涂色部分合并）、交集（只保留共同部分）或补集（交换涂色与白色区域）来组合。
Finally, darken the final shaded region clearly and erase any stray marks.

最后，用深色清晰标出最终阴影区域，擦去多余痕迹。

Practice with past Edexcel IGCSE questions to build speed.

用爱德思IGCSE的历年真题练习，以提升速度。

10. Common Mistakes to Avoid | 常见错误与避免方法

Many students make the same errors when shading sets. Avoid them by checking each region carefully:

许多学生在涂集合阴影时会犯同样的错误。仔细检查每个区域可以避免这些错误：

Shading outside the universal set rectangle – never do this. ξ defines the boundary.

在全集矩形外部涂色——绝对不要这样做。ξ 定义了边界。
Forgetting that A′ includes the areas outside A but still inside ξ.

忘记 A′ 包含 A 外部但仍在 ξ 内部的区域。
Confusing A − B with B − A. Always check which set comes first.

混淆 A − B 与 B − A。始终检查哪个集合在前。
For (A ∪ B)′ shading the union instead of its complement. The final shade is outside both circles, not inside.

对于 (A ∪ B)′，错涂成了并集而不是补集。最终阴影应在两个圆外部，而不是内部。
Not using brackets: (A ∩ B) ∪ C is not the same as A ∩ (B ∪ C).

没有正确使用括号：(A ∩ B) ∪ C 与 A ∩ (B ∪ C) 不同。

At the end of each problem, mentally test one point from each region to confirm whether it should be shaded.

每道题做完后，从每个区域取一个代表点在心里测试，确认它是否应该被涂色。

11. Worked Examples | 例题讲解

Example 1: Shade (A ∩ B)′ on a two-set Venn diagram.

例1: 在双集合维恩图上涂 (A ∩ B)′。

First, identify A ∩ B (the middle overlap). Then take the complement: shade everything except that overlap. So the shaded region is both circles except the lens, plus the area outside both circles.

首先确定 A ∩ B（中间重叠部分），然后取补集：除了这个重叠区域外全部涂色。所以阴影区域是两个圆除透镜外的所有部分，以及两圆外部的区域。

Example 2: Shade (A ∪ B) − C for three sets.

例2: 对三个集合涂 (A ∪ B) − C。

First shade A ∪ B (the whole of circles A and B). Then remove the part that lies in C. The result is the A and B regions that do not belong to C.

先涂 A ∪ B（圆 A 和 B 的全部），然后去掉 C 中的部分。结果就是 A 和 B 中不属于 C 的区域。

Example 3: Shade A ∩ (B ∪ C)′.

例3: 涂 A ∩ (B ∪ C)′。

First find (B ∪ C)′: the region outside both B and C. Then intersect with A: keep only the part of that outside region that is also inside A. This often results in a crescent-shaped piece of A.

先求 (B ∪ C)′：B 和 C 外部的区域。然后与 A 求交：只保留该外部区域中位于 A 内的部分。这通常得到 A 的一个月牙形区域。

12. Practice Questions | 巩固练习

Try these problems on your own, then check by reasoning through each region:

请独立完成以下练习，然后通过逐个区域推理来检查：

Shade A ∩ B′ on a two-set diagram.

在双集合图中涂 A ∩ B′。
Shade (A ∩ B) ∩ C′ on a three-set diagram.

在三集合图中涂 (A ∩ B) ∩ C′。
Shade (A′ ∪ B)′ and simplify it using De Morgan’s Laws.

涂 (A′ ∪ B)′ 并用德摩根定律化简。
Shade (A − B) ∪ (B − A), the symmetric difference of A and B.

涂 (A − B) ∪ (B − A)，即 A 和 B 的对称差。
Given ξ = {1,2,3,4,5,6}, A = {1,3,5}, B = {2,4,6}, shade A′ ∩ B′ and list the elements.

已知 ξ = {1,2,3,4,5,6}，A = {1,3,5}，B = {2,4,6}，涂 A′ ∩ B′ 并列出元素。

Answer to question 5: A′ ∩ B′ = { } (empty set), because A′ = {2,4,6} and B′ = {1,3,5}, and they have no common elements.

第5题答案：A′ ∩ B′ = { }（空集），因为 A′ = {2,4,6}，B′ = {1,3,5}，两者没有共同元素。

For question 3, (A′ ∪ B)′ = A ∩ B′ by De Morgan’s Laws, so shade the A-only region.

第3题中，根据德摩根定律，(A′ ∪ B)′ = A ∩ B′，所以只涂 A 独有的区域。

Mastering shading sets becomes easier with constant practice. Always state what you are shading, follow the operation order, and verify each region systematically.

通过持续练习，掌握集合阴影会变得容易。始终说明你要涂的内容，遵循运算顺序，并系统地验证每个区域。

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Shading Sets | 集合的阴影表示