区间qūjiān

Core Concept

An interval is a set of real numbers between two endpoints. Interval notation provides a compact way to represent continuous subsets of real numbers.

Four Types of Bounded Intervals

Type	Notation	Set-Builder	Description
Closed	$[a, b]$	$\{x \mid a \leq x \leq b\}$	Includes both endpoints
Open	$(a, b)$	$\{x \mid a < x < b\}$	Excludes both endpoints
Half-open (left)	$(a, b]$	$\{x \mid a < x \leq b\}$	Excludes left, includes right
Half-open (right)	$[a, b)$	$\{x \mid a \leq x < b\}$	Includes left, excludes right

Unbounded Intervals

Notation	Set-Builder	Description
$[a, +\infty)$	$\{x \mid x \geq a\}$	All numbers ≥ a
$(a, +\infty)$	$\{x \mid x > a\}$	All numbers > a
$(-\infty, b]$	$\{x \mid x \leq b\}$	All numbers ≤ b
$(-\infty, b)$	$\{x \mid x < b\}$	All numbers < b
$(-\infty, +\infty)$	$\mathbb{R}$	All real numbers

Important: Infinity symbols always use parentheses (never brackets) because ±∞ are not real numbers.

Bracket Rules

• Square bracket [ ]: The endpoint IS included (≤ or ≥)
• Round bracket ( ): The endpoint is NOT included (< or >)
• Infinity: Always uses round bracket

Interval Operations

Intersection of Intervals

$[1, 5] \cap [3, 8] = [3, 5]$

Take the overlap region:

• Left endpoint: max(1, 3) = 3
• Right endpoint: min(5, 8) = 5

Union of Intervals

$[1, 4] \cup [3, 7] = [1, 7]$

If intervals overlap, combine them.

$[1, 2] \cup [4, 5]$ cannot be simplified (disjoint intervals)

Complement of Intervals

If $A = [2, 5]$ and $U = \mathbb{R}$: $\complement_{\mathbb{R}} A = (-\infty, 2) \cup (5, +\infty)$

Note: Closed endpoint → open, open endpoint → closed.

Worked Examples

Example 1: Converting Inequality to Interval

Given: $-3 < x \leq 7$

Answer: $(-3, 7]$

Example 2: Converting Interval to Inequality

Given: $[-2, 4)$

Answer: $-2 \leq x < 4$

Example 3: Interval Intersection

Given: $A = [-1, 4]$, $B = (2, 6)$

Find: $A \cap B$

Solution:

• Left endpoint: max(-1, 2) = 2, bracket: ( (open, from B)
• Right endpoint: min(4, 6) = 4, bracket: ] (closed, from A)

Answer: $(2, 4]$

Example 4: Interval Union

Given: $A = (-\infty, 3)$, $B = [1, +\infty)$

Find: $A \cup B$

Solution: Since 1 < 3, the intervals overlap covering all reals.

Answer: $(-\infty, +\infty) = \mathbb{R}$

CSCA Practice Problems

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

Example 1: Basic (Difficulty ★☆☆☆☆)

Express the inequality $2 \leq x < 6$ in interval notation.

Options:

• A. $(2, 6]$
• B. $[2, 6)$
• C. $(2, 6)$
• D. $[2, 6]$

Solution:

• $x \geq 2$: use [
• $x < 6$: use )

Answer: B

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

Find $[-3, 2] \cap (0, 5]$.

Solution:

• Left endpoint: max(-3, 0) = 0, bracket: ( (from B, open)
• Right endpoint: min(2, 5) = 2, bracket: ] (from A, closed)

Answer: $(0, 2]$

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

If $A = \{x \mid x^2 - 5x + 4 \leq 0\}$, express A in interval notation.

Solution:

Factor the quadratic: $x^2 - 5x + 4 = (x-1)(x-4) \leq 0$

The parabola opens upward, so the inequality holds between roots: $1 \leq x \leq 4$

Answer: $[1, 4]$

Length of an Interval

For a bounded interval $[a, b]$: $\text{Length} = b - a$

Example: Length of $[-2, 5]$ is $5 - (-2) = 7$

Common Mistakes

❌ Mistake 1: Using Brackets with Infinity

Wrong: $[2, +\infty]$ ✗

Correct: $[2, +\infty)$ ✓

❌ Mistake 2: Wrong Endpoint After Intersection

Wrong: $[1, 5] \cap [3, 8] = [1, 8]$ ✗

Correct: $[1, 5] \cap [3, 8] = [3, 5]$ ✓

❌ Mistake 3: Forgetting to Change Bracket Type for Complement

Wrong: Complement of $[a, b]$ is $(-\infty, a] \cup [b, +\infty)$ ✗

Correct: Complement of $[a, b]$ is $(-\infty, a) \cup (b, +\infty)$ ✓

❌ Mistake 4: Assuming Empty Intersection

Two intervals only have empty intersection if they don't overlap at all.

$[1, 3] \cap [3, 5] = \{3\}$ (single point, not empty!)

Study Tips

1. ✅ Remember bracket meanings: [ ] includes, ( ) excludes
2. ✅ ∞ always gets ( ): Never use brackets with infinity
3. ✅ Intersection = overlap: Take the common region
4. ✅ Complement flips brackets: [ ↔ ) and ] ↔ (

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💡 Exam Tip: When solving quadratic inequalities, always sketch the parabola to correctly identify the solution interval!