集合jíhé

Core Concept

A set is one of the most fundamental concepts in mathematics, referring to a collection of distinct objects with certain properties. Each object in a set is called an element.

Basic Concepts

• Element: Each object in a set
• Membership: Element $a$ is in set $A$, written $a \in A$
• Non-membership: Element $a$ is not in set $A$, written $a \notin A$

Set Notation

1. Roster Method: $A = \{1, 2, 3, 4, 5\}$

2. Set-Builder Notation: $B = \{x \mid x \text{ is a positive integer less than 10}\}$

3. Venn Diagrams: Visual representation using circles

Special Sets

• Empty set: Contains no elements, denoted $\emptyset$ or $\{\}$
• Natural numbers: $\mathbb{N} = \{0, 1, 2, 3, ...\}$
• Integers: $\mathbb{Z} = \{..., -2, -1, 0, 1, 2, ...\}$
• Rational numbers: $\mathbb{Q}$
• Real numbers: $\mathbb{R}$

Set Relationships

Subset

If every element of $A$ is also in $B$, then $A$ is a subset of $B$, written $A \subseteq B$.

Proper subset: $A \subseteq B$ and $A \neq B$, written $A \subset B$.

Equality

$A = B$ if and only if $A \subseteq B$ and $B \subseteq A$.

Set Operations

1. Union

$A \cup B = \{x \mid x \in A \text{ or } x \in B\}$

2. Intersection

$A \cap B = \{x \mid x \in A \text{ and } x \in B\}$

3. Complement

In universal set $U$: $\complement_U A = \{x \mid x \in U \text{ and } x \notin A\}$

4. Difference

$A - B = \{x \mid x \in A \text{ and } x \notin B\}$

CSCA Practice Problems

💡 Note: The following practice problems are designed based on the CSCA exam syllabus and Chinese standardized test formats to help students familiarize themselves with question types and problem-solving approaches.

Example 1: Basic (Difficulty ★☆☆☆☆)

Given $A = \{1, 2, 3\}$ and $B = \{2, 3, 4\}$, find $A \cup B$.

Solution:

Union contains all elements in $A$ or $B$: $A \cup B = \{1, 2, 3, 4\}$

Answer: $\{1, 2, 3, 4\}$

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Example 2: Intermediate (Difficulty ★★★☆☆)

Given universal set $U = \{1, 2, 3, 4, 5\}$ and $A = \{1, 3, 5\}$, find $\complement_U A$.

Solution:

Complement contains elements in $U$ but not in $A$: $\complement_U A = \{2, 4\}$

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Example 3: Advanced (Difficulty ★★★★☆)

Given $A = \{x \mid x^2 - 3x + 2 = 0\}$ and $B = \{x \mid x < 3\}$, find $A \cap B$.

Solution:

Solve $x^2 - 3x + 2 = 0$: $x = 1 \text{ or } x = 2$

So $A = \{1, 2\}$

$A \cap B = \{1, 2\} \cap (-\infty, 3) = \{1, 2\}$

Common Mistakes

❌ Mistake 1: $\{0\} = \emptyset$

Correction: $\{0\}$ contains one element (zero), not empty! Empty set is $\emptyset$ or $\{\}$.

❌ Mistake 2: Sets have order

Correction: Sets are unordered, $\{1, 2, 3\} = \{3, 2, 1\}$.

❌ Mistake 3: Sets can have duplicates

Correction: Set elements are distinct, $\{1, 1, 2\} = \{1, 2\}$.

Study Tips

1. ✅ Understand concept: Sets are unordered collections of distinct objects
2. ✅ Master notation: Roster and set-builder notation
3. ✅ Practice operations: Union, intersection, complement are fundamental
4. ✅ Use diagrams: Venn diagrams help visualize relationships

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💡 Exam Tip: Sets are the foundation of high school math. Set problems in CSCA are relatively simple, but concepts must be solid!