Set Union & Combining Sets | 集合并集与集合合并

📚 Set Union & Combining Sets | 集合并集与集合合并

In IGCSE Mathematics, one of the fundamental operations you will perform on sets is the union of two or more sets. The union combines every element from the involved sets into a single larger set, removing any duplicates.

在IGCSE数学中，你对集合进行的基本运算之一就是两个或多个集合的并集。并集将所涉及集合中的每一个元素合并到一个更大的集合中，并去除任何重复项。

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1. What is a Set? | 什么是集合？

A set is a well-defined collection of distinct objects, called elements or members. Sets are usually denoted by capital letters such as A, B, or C.

集合是一个定义明确的、由不同对象（称为元素或成员）组成的整体。集合通常用大写字母A、B或C表示。

For example: A = {1, 2, 3, 4} and B = {3, 4, 5, 6}.

例如：A = {1, 2, 3, 4}，B = {3, 4, 5, 6}。

• Each object in a set is called an element. | 集合中的每个对象称为一个元素。

• Sets are written using curly brackets { }. | 集合使用大括号 { } 表示。

• The order of elements does not matter. | 元素的顺序不影响集合本身。

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2. Definition of Union | 并集的定义

The union of sets A and B, written as A ∪ B, is the set that contains all elements that are in A, or in B, or in both. In other words, it combines the elements of both sets.

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A和B的元素的集合。换句话说，它将两个集合的元素合并在一起。

Using the example above: A ∪ B = {1, 2, 3, 4, 5, 6}.

使用上面的例子：A ∪ B = {1, 2, 3, 4, 5, 6}。

A ∪ B = { x | x ∈ A or x ∈ B }

Notice that the common elements 3 and 4 are listed only once. The union does not count duplicates twice.

注意，共同的元素3和4只列出一次。并集不会将重复元素计算两次。

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3. The Union Symbol ∪ | 并集符号 ∪

The symbol ∪ represents the union operation. It resembles the letter U, which can help you remember that it means “union” or “together”.

符号∪表示并集运算。它的形状类似字母U，可以帮助你记住它表示”联合”或”合并”。

• A ∪ B is read as “A union B”. | A ∪ B读作”A并B”。

• The union symbol points upward, like a cup collecting elements from both sets. | 并集符号开口朝上，如同一个杯子收集两个集合中的元素。

Do not confuse ∪ with ∩ (intersection), which points downward and represents only the common elements.

不要将∪与∩（交集）混淆，交集的符号开口朝下，仅表示共同元素。

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4. Finding the Union by Listing | 通过列举法求并集

When sets are given in list form, you can find their union by writing down every element from both sets, then removing any repeated elements.

当集合以列举形式给出时，你可以通过写下两个集合中的所有元素、然后删除任何重复元素来找到它们的并集。

Example: A = {a, b, c}, B = {c, d, e}. Find A ∪ B.

示例：A = {a, b, c}，B = {c, d, e}。求A ∪ B。

Step 1: List all elements from A: a, b, c.
Step 2: List elements from B that are not already listed: d, e.
Step 3: Combine them: A ∪ B = {a, b, c, d, e}.

步骤1：列出A中的所有元素：a, b, c。
步骤2：列出B中尚未列出的元素：d, e。
步骤3：合并它们：A ∪ B = {a, b, c, d, e}。

Always double-check that no element is repeated in your final answer.

务必反复检查最终答案中没有重复的元素。

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5. Union of Sets Using Venn Diagrams | 用韦恩图表示并集

A Venn diagram is a visual way to represent sets. In a Venn diagram, the union of two sets is represented by shading both circles entirely, including the overlapping region.

韦恩图是一种直观表示集合的方法。在韦恩图中，两个集合的并集通过将两个圆全部涂上阴影来表示，包括重叠区域。

• Draw two overlapping circles, one for A and one for B. | 画出两个相交的圆，一个代表A，一个代表B。

• The overlapping middle region represents A ∩ B. | 中间相交区域代表A ∩ B。

• For A ∪ B, shade the entire area inside both circles. | 对于A ∪ B，将两个圆内部的全部区域涂上阴影。

The shaded area of A ∪ B includes the left circle, the right circle, and the overlap. In exam questions, you may be asked to identify or shade this region.

A ∪ B的阴影区域包括左圆、右圆以及重叠部分。在考试题目中，你可能会被要求识别或画出这个区域。

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6. Union with Disjoint Sets | 不相交集合的并集

When two sets have no elements in common, they are called disjoint sets. Their Venn diagram shows two separate circles that do not overlap.

当两个集合没有共同元素时，它们被称为不相交集合。它们的韦恩图显示为两个不重叠的独立圆。

Example: A = {1, 2}, B = {3, 4}. Since they share no common elements, A ∩ B = ∅ (empty set).

示例：A = {1, 2}，B = {3, 4}。由于它们没有共同元素，A ∩ B = ∅（空集）。

Their union is still defined: A ∪ B = {1, 2, 3, 4}. The union of disjoint sets is simply all elements placed together.

它们的并集仍然有定义：A ∪ B = {1, 2, 3, 4}。不相交集合的并集就是将所有元素放在一起。

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7. Union with the Empty Set | 与空集的并集

The empty set, denoted by ∅ or { }, contains no elements. When you take the union of any set with the empty set, the result is simply the original set.

空集，用∅或{ }表示，不含任何元素。当任何集合与空集取并集时，结果就是原集合本身。

A ∪ ∅ = A

This is because adding no elements to a set leaves the set unchanged. This property is called the identity property of the empty set for union.

这是因为向集合中不添加任何元素，集合保持不变。这个性质称为空集关于并集的恒等性质。

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8. The Universal Set and the Union | 全集与并集

The universal set, denoted by ξ or U, contains all possible elements under consideration. If A is a subset of the universal set, then A ∪ ξ = ξ.

全集，用ξ或U表示，包含所讨论的所有可能元素。如果A是全集的一个子集，那么A ∪ ξ = ξ。

This is because the universal set already contains every element that exists in the context, including all elements of A.

这是因为全集已经包含该情境中存在的所有元素，包括A的所有元素。

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

例如，如果ξ = {1, 2, 3, 4, 5}，A = {2, 4}，那么A ∪ ξ = {1, 2, 3, 4, 5} = ξ。

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9. Number of Elements in a Union | 并集的元素个数

The notation n(A) represents the number of elements in set A. To find the number of elements in A ∪ B, you can use a formula.

符号n(A)表示集合A中元素的数量。要求A ∪ B中元素的数量，你可以使用一个公式。

n(A ∪ B) = n(A) + n(B) − n(A ∩ B)

We subtract n(A ∩ B) because the common elements were counted twice, once in n(A) and once in n(B).

我们减去n(A ∩ B)，是因为共同元素被计算了两次，一次在n(A)中，一次在n(B)中。

Example: n(A) = 5, n(B) = 7, n(A ∩ B) = 2. Then n(A ∪ B) = 5 + 7 − 2 = 10.

示例：n(A) = 5，n(B) = 7，n(A ∩ B) = 2。那么n(A ∪ B) = 5 + 7 − 2 = 10。

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10. Union with Three Sets | 三个集合的并集

You can also find the union of three sets. The union A ∪ B ∪ C contains every element that is in A, or in B, or in C, without repetition.

你也可以求三个集合的并集。并集A ∪ B ∪ C包含所有属于A、属于B或属于C的元素，且不重复。

Example: A = {1, 2}, B = {2, 3}, C = {3, 4}. Then A ∪ B ∪ C = {1, 2, 3, 4}.

示例：A = {1, 2}，B = {2, 3}，C = {3, 4}。那么A ∪ B ∪ C = {1, 2, 3, 4}。

For three sets, the formula for the number of elements is more complex: n(A ∪ B ∪ C) = n(A) + n(B) + n(C) − n(A ∩ B) − n(A ∩ C) − n(B ∩ C) + n(A ∩ B ∩ C).

对于三个集合，元素数量的公式更为复杂：n(A ∪ B ∪ C) = n(A) + n(B) + n(C) − n(A ∩ B) − n(A ∩ C) − n(B ∩ C) + n(A ∩ B ∩ C)。

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11. Properties of Union | 并集的性质

The union operation follows several important properties that are useful for simplifying expressions.

并集运算遵循几个重要性质，这些性质有助于简化表达式。

Property | 性质	Expression | 表达式
Commutative | 交换律	A ∪ B = B ∪ A
Associative | 结合律	(A ∪ B) ∪ C = A ∪ (B ∪ C)
Idempotent | 幂等律	A ∪ A = A
Identity | 恒等律	A ∪ ∅ = A

These properties are useful when working with complex set expressions in exam questions.

在考试题目中处理复杂的集合表达式时，这些性质非常有用。

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12. Common Exam Tips | 常见考试技巧

Here are some essential tips to help you solve union problems accurately in your IGCSE exam.

以下是一些基本技巧，可帮助你在IGCSE考试中准确解答并集问题。

• Always check whether elements are repeated before writing the final set. | 在写出最终集合之前，务必检查元素是否有重复。

• Read the question carefully: “and” often suggests intersection, while “or” suggests union. | 仔细审题：”且”通常表示交集，而”或”表示并集。

• When using the formula n(A ∪ B), do not forget to subtract the overlap. | 使用公式n(A ∪ B)时，不要忘记减去重叠部分。

• Shade the entire region of both circles for a union in a Venn diagram. | 在韦恩图中表示并集时，要将两个圆的全部区域涂上阴影。

Practising these skills will build your confidence in handling set theory questions efficiently.

练习这些技能将增强你高效处理集合论题目的信心。

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