Interval Notation Converter

What Is Interval Notation? Interval notation is a shorthand for describing sets of real numbers that form a continuous range. It uses brackets [ ] for closed endpoints (included) and parentheses ( ) for open endpoints (excluded). Example: [2, 5) means “all real numbers x where 2 ≤ x < 5” — including 2 but not 5. This notation was standardized in the early 20th century and is universally used in mathematics.

Types of Intervals Closed interval [a, b]: includes both endpoints. Inequality: a ≤ x ≤ b. Open interval (a, b): excludes both endpoints. Inequality: a < x < b. Half-open [a, b): left endpoint included, right excluded. Inequality: a ≤ x < b. Half-open (a, b]: left excluded, right included. Inequality: a < x ≤ b. Infinite intervals: (−∞, b], [a, ∞), (−∞, ∞) for all reals. Infinity is always written with a parenthesis — it is never reached, never “included.”

Set-Builder Notation Set-builder notation: {x ∈ ℝ | condition}. Example: {x ∈ ℝ | 2 ≤ x < 5} is the same as [2, 5). The | symbol means “such that.” ℝ = the real numbers. Both notations appear in calculus, analysis, and linear algebra.

Number Line Representation Closed endpoint: filled circle ● at the point. Open endpoint: open circle ○ at the point. The interval is shaded between the endpoints.

Set Operations on Intervals Union (A ∪ B): all numbers in either A or B. Example: (1,3) ∪ [5,7] — numbers between 1 and 3 (exclusive) OR between 5 and 7 (inclusive). Intersection (A ∩ B): only numbers in both A and B. Example: [1,4] ∩ (3,6) = (3,4] — numbers that satisfy both conditions. Complement: all numbers NOT in the interval. Example: complement of (−∞, 3] is (3, +∞).

Disjoint Intervals Two intervals are disjoint if their intersection is empty. Example: (1,2) and (3,4) are disjoint — no x is in both. Disjoint intervals are often written with ∪: (−∞,2) ∪ (5,∞).

Applications Domain of functions: sqrt(x−3) has domain [3, ∞) — x must be ≥ 3. Continuity: a function is continuous on [a,b] means it is continuous at every point in the closed interval. Convergence: a power series converges on (−R, R) where R is the radius of convergence. IVT and MVT: both require closed intervals [a,b] for their conclusions.