交集jiāojí

Core Concept

The intersection of two sets A and B, written as A ∩ B, is the set containing all elements that are common to both A and B.

Mathematical Definition

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

An element belongs to the intersection if and only if it belongs to both sets.

Visual Representation

In a Venn diagram, the intersection is the overlapping region of two circles.

    A         B
  (   [###]   )

The shaded region [###] represents A ∩ B.

Key Properties

1. Commutative Property

$A \cap B = B \cap A$

2. Associative Property

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

3. Identity Property

$A \cap U = A$ (where U is the universal set)

4. Idempotent Property

$A \cap A = A$

5. Intersection with Empty Set

$A \cap \emptyset = \emptyset$

6. Distributive Property

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

Worked Examples

Example 1: Finite Sets

Given: A = {1, 2, 3, 4, 5}, B = {3, 4, 5, 6, 7}

Find: A ∩ B

Solution: Elements in both sets: 3, 4, 5

Answer: A ∩ B = {3, 4, 5}

Example 2: Interval Intersection

Given: A = [-2, 5], B = [1, 8]

Find: A ∩ B

Solution: The overlap of both intervals is [1, 5]

Answer: A ∩ B = [1, 5]

Example 3: Set-Builder Notation

Given: A = {x | x > 2}, B = {x | x < 7}

Find: A ∩ B

Solution: Elements greater than 2 AND less than 7

Answer: A ∩ B = {x | 2 < x < 7} = (2, 7)

CSCA Practice Problems

💡 Note: The following practice problems are designed based on the CSCA exam syllabus.

Example 1: Basic (Difficulty ★☆☆☆☆)

If A = {a, b, c, d} and B = {c, d, e, f}, find A ∩ B.

Options:

• A. {a, b, c, d, e, f}
• B. {c, d}
• C. {a, b}
• D. {e, f}

Solution: Common elements: c, d

Answer: B

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

Given A = {x | x² - 4x + 3 ≤ 0} and B = {x | x > 2}, find A ∩ B.

Solution:

First, solve the inequality for A: $x² - 4x + 3 ≤ 0$ $(x-1)(x-3) ≤ 0$ $A = [1, 3]$

Then find intersection with B = (2, +∞): $A \cap B = [1, 3] \cap (2, +\infty) = (2, 3]$

Answer: (2, 3]

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

If A ∩ B = A, what is the relationship between sets A and B?

Solution:

If A ∩ B = A, then every element of A must also be in B.

This means A ⊆ B (A is a subset of B).

Answer: A ⊆ B

Common Mistakes

❌ Mistake 1: Confusing Intersection with Union

Wrong: A ∩ B includes all elements from both sets ✗

Correct: A ∩ B includes only common elements ✓

❌ Mistake 2: Forgetting Empty Intersection

Wrong: Two sets always have a non-empty intersection ✗

Correct: If sets have no common elements, A ∩ B = ∅ ✓

❌ Mistake 3: Interval Notation Errors

Wrong: [1, 5] ∩ [3, 8] = [1, 8] ✗

Correct: [1, 5] ∩ [3, 8] = [3, 5] ✓

Study Tips

1. ✅ "AND" thinking: Intersection means AND - element must satisfy BOTH conditions
2. ✅ Draw Venn diagrams: Visual representation helps avoid mistakes
3. ✅ Check boundary points: For intervals, verify endpoints carefully
4. ✅ Practice with inequalities: Many CSCA problems combine sets with inequalities

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💡 Exam Tip: Intersection problems often appear with quadratic inequalities. Always solve the inequality first, then find the intersection!