Interval Notation
Intervals express solution sets: (a,b) open, [a,b] closed, [a,b) half-open. [ ] = include endpoint, ( ) = exclude. Union ∪ combines, intersection ∩ overlaps. |x−a|<b → (a−b,a+b).
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[ ] = include endpoint, ( ) = exclude. (−∞,∞)=ℝ. Union of disjoint intervals: (−∞,2]∪[5,∞). |x−a|<b means distance from a is less than b.
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Why: Interval notation compactly describes solution sets of inequalities. Used in calculus (domain, convergence), optimization (feasible regions), and real analysis.
How: (a,b): a<x<b. [a,b]: a≤x≤b. [a,b): a≤x<b. (−∞,a]: x≤a. Union: A∪B = all in A or B. Intersection: A∩B = all in both. |x−a|<b → (a−b,a+b).
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Interval Notation — Convert & Operate
Interval, inequality, set-builder. Union, intersection, complement, |x−a|<b.
Endpoint Values
Included vs Excluded Portion
Calculation Steps
For educational and informational purposes only. Verify with a qualified professional.
🧮 Fascinating Math Facts
|x−3|<5 → x∈(−2,8): all points within 5 of 3.
— Absolute value
ℝ = (−∞,∞). Empty set: ∅.
— Special
Key Takeaways
- • Parentheses ( ) mean the endpoint is not included (open). Brackets [ ] mean the endpoint is included (closed).
- • Infinity always gets parentheses: (−∞, 3) and [5, ∞) — we never "include" infinity.
- • Empty set ∅: when two intervals have no overlap, their intersection is ∅.
- • Universal set: (−∞, ∞) or ℝ denotes all real numbers.
- • De Morgan's laws for intervals: complement of (A ∪ B) = (complement of A) ∩ (complement of B).
- • |x − a| < b is equivalent to the open interval (a−b, a+b).
Did You Know?
How It Works
Interval notation uses brackets [ ] for closed endpoints (included) and parentheses ( ) for open endpoints (excluded). [2,5] means 2 ≤ x ≤ 5; (2,5) means 2 < x < 5; [2,5) means 2 ≤ x < 5. For union, we write A ∪ B; for intersection, A ∩ B. The complement of an interval in ℝ is everything outside it. |x−a|<b means a−b < x < a+b.
Notation Conventions
(a,b) open [a,b] closed (a,b] half-open [a,b) half-open (−∞,a) x<a [a,∞) x≥a
Expert Tips
Parentheses vs Brackets
Remember: ( = open = excluded; [ = closed = included. On a number line, closed = filled dot, open = hollow dot.
Infinity Always Open
Always use ( with ±∞: (−∞,3), [5,∞). We never write ]∞ or [−∞ because infinity is not a number.
Empty Set
If two intervals don't overlap, their intersection is ∅. Example: [0,1] ∩ [2,3] = ∅.
De Morgan for Intervals
Complement of (A ∪ B) = (complement of A) ∩ (complement of B). Useful for simplifying compound intervals.
Reference Table — Interval Types
| Notation | Inequality | Endpoints |
|---|---|---|
| (a,b) | a < x < b | Both excluded |
| [a,b] | a ≤ x ≤ b | Both included |
| (a,b] | a < x ≤ b | Left open, right closed |
| [a,b) | a ≤ x < b | Left closed, right open |
| (-∞,a) | x < a | Unbounded left |
| [a,∞) | x ≥ a | Unbounded right |
FAQ
What is the difference between ( and [ ?
( means open—endpoint not included. [ means closed—endpoint included. So (2,5) excludes 2 and 5; [2,5] includes both.
Why do we use parentheses with infinity?
Infinity is not a real number, so we cannot "include" it. We always write (−∞,a) or (a,∞) with parentheses.
What is the empty set?
∅ denotes a set with no elements. When two intervals don't overlap, their intersection is ∅.
How do I convert |x−2| < 3 to interval notation?
|x−2| < 3 means −3 < x−2 < 3, so −1 < x < 5. In interval notation: (−1, 5).
What is set-builder notation?
Set-builder notation describes a set by a condition: {x ∈ ℝ : a ≤ x ≤ b} means "all real x such that a ≤ x ≤ b".
What are De Morgan's laws for intervals?
Complement of (A∪B) = (complement of A)∩(complement of B), and similarly for intersection. Useful for simplifying complements.
Quick Reference
Disclaimer: This calculator is for educational use. Interval notation conventions may vary slightly in different mathematical contexts.
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