Unit 1 - Lesson 3: Intervals (Bounded & Unbounded) and Set Operations (Union, Intersection, Difference, Complement)
Intervals serve as the universal mathematical notation for expressing connected subsets of the real number line, seamlessly capturing infinite continuum segments that cannot be exhaustively listed due to the density of R. They are classified into bounded intervals with finite lower and upper limits—such as closed, open, and half-open intervals—and unbounded intervals that extend indefinitely towards positive or negative infinity to define comprehensive domain sets. Students develop exceptional analytical rigor by performing fundamental set operations including union, intersection, set difference, and complementation, visually verified on dual-colored number lines with strictly open or closed endpoints in full compliance with official ministerial assessment guidelines.
1. Bounded Intervals
A bounded interval is a subset of R contained between two finite real numbers a, b (where a < b):
| Interval Type | Notation | Set-Builder Notation | Graphical Representation |
|---|---|---|---|
| Closed | [a, b] | {x : a ≤ x ≤ b} | Both endpoints shaded (solid circles) |
| Open | (a, b) | {x : a < x < b} | Both endpoints open (hollow circles) |
| Half-closed | [a, b) | {x : a ≤ x < b} | a shaded (solid), b open (hollow) |
| Half-open | (a, b] | {x : a < x ≤ b} | a open (hollow), b shaded (solid) |
Fundamental Rule: The lesser real number is always written first (a < b strictly).
2. Unbounded Intervals
Intervals extending indefinitely in one or both directions, utilizing infinity symbols ∞ and -∞:
[a, ∞): {x : x ≥ a} (starts at a closed, extends infinitely to the right).(a, ∞): {x : x > a} (starts at a open, extends infinitely to the right).(-∞, b]: {x : x ≤ b} (extends from negative infinity up to b closed).(-∞, b): {x : x < b} (extends from negative infinity up to b open).(-∞, ∞): Represents the entire real number line = R.
3. Set Operations on Intervals
Determined using graphical representation on the real number line:
- Union (A ∪ B): All elements belonging to A or B or both (from extreme left to extreme right).
- Intersection (A ∩ B): Only the overlapping region shared simultaneously by both intervals.
- Difference (A - B): Elements in A that do not belong to B (endpoints belonging to B are reversed from closed to open).
- Complement (A'): The remaining portion of R outside interval A (A' = R - A).
4. Intervals vs Sets vs Ordered Pairs
| Concept | Symbol | Meaning |
|---|---|---|
| Interval | [2, 5] | All real numbers from 2 to 5 inclusive (infinitely many numbers) |
| Discrete Set | {2, 5} | Only the two numbers 2 and 5 (exactly two elements) |
| Ordered Pair | (2, 5) | Coordinate point on the Cartesian plane (x = 2, y = 5) |
Example: [2, 5] - {2, 5} = (2, 5) (removing endpoints opens the closed interval).
Exercise 1: Interval Union and Intersection
Problem: Given A = [-2, 3] and B = [1, 5), find A ∪ B and A ∩ B.
• Plot A from -2 to 3, and B from 1 to 5 on the number line.
• Union (extreme left to extreme right): A ∪ B = [-2, 5).
• Intersection (common overlapping region): A ∩ B = [1, 3].
Exercise 2: Interval Difference
Problem: For A = [-2, 3] and B = [1, 5), find A - B and B - A.
• A - B (in A but not in B): [-2, 1) (1 is opened because 1 ∈ B).
• B - A (in B but not in A): (3, 5) (3 is opened because 3 ∈ A).
Exercise 3: Operations with Unbounded Intervals
Problem: If X = (-∞, 2] and Y = [-1, ∞), find X ∩ Y and X ∪ Y.
• Common overlap between the two rays extends from -1 to 2.
• X ∩ Y = [-1, 2].
• The union covers the entire real number line: X ∪ Y = (-∞, ∞) = R.
Exercise 4: Interval Complement
Problem: Find the complement of A = [-3, 4) in R.
• A' = R - [-3, 4).
• Left ray: (-∞, -3) (open at -3 because -3 ∈ A).
• Right ray: [4, ∞) (closed at 4 because 4 ∉ A).
• Thus, A' = (-∞, -3) ∪ [4, ∞).
Exercise 5: Subtracting Discrete Sets from Intervals
Problem: Simplify: [-1, 4] - {-1, 4}.
• [-1, 4] includes all real numbers from -1 to 4.
• Subtracting {-1, 4} removes only the two endpoints.
• Result: the open interval (-1, 4).
Exercise 6: Union of an Interval with a Discrete Element
Problem: Simplify: (2, 7) ∪ {2}.
• (2, 7) does not include 2.
• Adding {2} closes the left boundary.
• Result: [2, 7) (half-closed interval).
Exercise 7: Expressing R⁺ and R⁻ as Intervals
Problem: Express R⁺, R⁻, and non-negative real numbers as intervals.
• Positive real numbers R⁺ = (0, ∞) (zero is excluded).
• Negative real numbers R⁻ = (-∞, 0).
• Non-negative real numbers = [0, ∞) (zero is included).
Exercise 8: Interval Intersection with Natural Numbers
Problem: Find: [-2, 3) ∩ N.
• The set of natural numbers N = {0, 1, 2, 3, ...}.
• Integers inside [-2, 3) are: -2, -1, 0, 1, 2 (3 is open).
• The natural numbers among them are: {0, 1, 2}.
Tap any card to reveal its meaning, or tap the audio icon to listen:
Click on an option to check your answer and view the detailed explanation: