集合的运算

CSCA 集合的运算 study guide aligned to the official syllabus. Practice 数学 questions on aicsca.com.

Syllabus Alignment

This study guide follows the official CSCA exam syllabus for international undergraduate applicants.

Who It Is For

International students preparing for CSCA Math, Physics, Chemistry, or Chinese exams.

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Tutorial Content

Operations on Sets

Set operations are a mandatory topic in the CSCA exam. Mastering definitions of **Union**, **Intersection**, and **Complement**, along with their representations in Venn Diagrams and on the number line, is key to solving these problems.

1. Three Basic Operations

Let $A$ and $B$ be two sets, and $U$ be the universal set.

#### (1) Union

* **Definition**: The set of elements that belong to $A$ **OR** $B$.

* **Symbol**: $A \cup B$

* **Mnemonic**: "Combine A and B, counting duplicates only once."

* **Venn Diagram**:


#### (2) Intersection

* **Definition**: The set of elements that belong to **BOTH** $A$ **AND** $B$.

* **Symbol**: $A \cap B$

* **Mnemonic**: "The common part shared by A and B." If $A \cap B = \varnothing$, the sets are **Disjoint**.

* **Venn Diagram**:


#### (3) Complement

* **Definition**: The set of elements in the universal set $U$ that are **NOT** in $A$.

* **Symbol**: $\complement_U A$ (Common in Chinese textbooks) or $A^c$.

* **Mnemonic**: "Remove A from U; what remains is the complement."

* **Venn Diagram**:


2. Important Laws of Operations

The CSCA exam often uses these laws to simplify set problems:

* **Commutative Laws**: $A \cup B = B \cup A$, $A \cap B = B \cap A$

* **Distributive Laws**:

* $A \cap (B \cup C) = (A \cap B) \cup (A \cap C)$

* $A \cup (B \cap C) = (A \cup B) \cap (A \cup C)$

* **De Morgan's Laws ★ Key Point**:

* $\complement_U (A \cup B) = (\complement_U A) \cap (\complement_U B)$ (Complement of Union = Intersection of Complements)

* $\complement_U (A \cap B) = (\complement_U A) \cup (\complement_U B)$ (Complement of Intersection = Union of Complements)

3. Worked Examples

**Example 1: Interval Operations**

Let set $A = \{x | -2 \le x < 3\}$ and $B = \{x | 1 < x \le 5\}$. Find $A \cup B$ and $A \cap B$.

**Tip**: Draw a number line!


* **Union ($A \cup B$)**: The total range covered. From left endpoint $-2$ to right endpoint $5$. $\Rightarrow [-2, 5]$

* **Intersection ($A \cap B$)**: The overlapping section where both lines exist. From $1$ (exclusive) to $3$ (exclusive). $\Rightarrow (1, 3)$

**Example 2: Discrete Set Operations**

Given $U = \{1,2,3,4,5,6\}$, $A = \{1, 4\}$, $B = \{1, 3, 5\}$. Find $\complement_U (A \cup B)$.

**Solution**:

1. First, find the union: $A \cup B = \{1, 3, 4, 5\}$

2. Next, find the complement: Remove $\{1, 3, 4, 5\}$ from $U$, leaving $\{2, 6\}$.

Thus, $\complement_U (A \cup B) = \{2, 6\}$.