Interval Notation Calculator

Interval Notation Basics

Interval notation uses brackets [ ] for inclusive boundaries (closed endpoints) and parentheses ( ) for exclusive boundaries (open endpoints). The general form is [a, b] where a and b are the endpoints of the interval. The left endpoint always comes first, and the right endpoint comes second. For example, [2, 5] represents all real numbers from 2 to 5, including both 2 and 5.

[a, b] = \x \mid a ≤ x ≤ b\

Types of Intervals

There are four main types of finite intervals:

1. Closed interval [a, b]: Includes both endpoints a and b. All values x where a ≤ x ≤ b.

2. Open interval (a, b): Excludes both endpoints. All values x where a < x < b.

3. Half-open interval [a, b): Includes left endpoint a but excludes right endpoint b. All values x where a ≤ x < b.

4. Half-open interval (a, b]: Excludes left endpoint a but includes right endpoint b. All values x where a < x ≤ b.

The choice of interval type depends on whether the boundary values are part of the solution set.

\beginalign* \textClosed: & [a, b] = \x \mid a ≤ x ≤ b\ \\ \textOpen: & (a, b) = \x \mid a < x < b\ \\ \textHalf-open: & [a, b) = \x \mid a ≤ x < b\ \\ \textHalf-open: & (a, b] = \x \mid a < x ≤ b\ \endalign*

Infinite Intervals

Infinite intervals extend indefinitely in one or both directions using the infinity symbol (∞). Since infinity is not a real number, intervals involving infinity always use parentheses on the infinite end:

• (-∞, a] or (-∞, a): All numbers less than or equal to (or less than) a • [a, ∞) or (a, ∞): All numbers greater than or equal to (or greater than) a • (-∞, ∞): All real numbers

Note that ∞ and -∞ are never enclosed in square brackets because they are not actual values that can be included.

\beginalign* (-∞, a] &= \x \mid x ≤ a\ \\ (-∞, a) &= \x \mid x < a\ \\ [a, ∞) &= \x \mid x ≥ a\ \\ (a, ∞) &= \x \mid x > a\ \\ (-∞, ∞) &= \mathbbR \endalign*

Union and Intersection of Intervals

Multiple intervals can be combined using set operations:

Union (∪): Combines all values from both intervals. The result includes any value that belongs to at least one of the intervals. For example, [1, 3] ∪ [2, 5] = [1, 5].

Intersection (∩): Finds values common to both intervals. The result includes only values that belong to both intervals. For example, [1, 4] ∩ [2, 5] = [2, 4].

Disjoint intervals: When intervals do not overlap, their union is written as the combination of both intervals. For example, [1, 2] ∪ [4, 5] remains [1, 2] ∪ [4, 5].

\beginalign* A \cup B &= \x \mid x \in A \text or x \in B\ \\ A \cap B &= \x \mid x \in A \text and x \in B\ \\ [1,3] \cup [2,5] &= [1,5] \\ [1,4] \cap [2,5] &= [2,4] \endalign*

Converting Between Notations

Converting between interval notation and inequality notation:

• [a, b] ↔ a ≤ x ≤ b • (a, b) ↔ a < x < b • [a, b) ↔ a ≤ x < b • (a, b] ↔ a < x ≤ b • [a, ∞) ↔ x ≥ a • (a, ∞) ↔ x > a • (-∞, b] ↔ x ≤ b • (-∞, b) ↔ x < b

The key is to match square brackets with ≤ or ≥, and parentheses with < or >.

\beginalign* [a,b] &\Leftrightarrow a ≤ x ≤ b \\ (a,b) &\Leftrightarrow a < x < b \\ [a,b) &\Leftrightarrow a ≤ x < b \\ (a,b] &\Leftrightarrow a < x ≤ b \endalign*

Applications in Mathematics

Interval notation is widely used in various areas of mathematics:

1. Domain and Range: Expressing the set of valid input values (domain) and output values (range) of functions.

2. Solution Sets: Representing solutions to inequalities and systems of inequalities.

3. Calculus: Defining intervals for continuity, differentiability, and integration.

4. Real Analysis: Describing open and closed sets, neighborhoods, and convergence intervals.

5. Optimization: Specifying constraint regions and feasible solution spaces.

Interval notation provides a clear, concise way to communicate mathematical concepts across these applications.

\textExample: Domain of f(x) = √(x-2) \text is [2, ∞)

Worked Example

Inputs

• conversionType: Inequality to Interval
• leftBound: 2
• rightBound: 5
• leftInclusive: Inclusive [
• rightInclusive: Inclusive ]

Result: Interval: [2, 5]

Explanation

The interval [2, 5] represents all real numbers x such that 2 ≤ x ≤ 5

Second Scenario

Inputs

• conversionType: Inequality to Interval
• leftBound: 2.4
• rightBound: 5
• leftInclusive: Inclusive [
• rightInclusive: Inclusive ]

Result: Interval: [2, 5]

Explanation

This scenario uses different inputs (conversionType = Inequality to Interval, leftBound = 2.4, rightBound = 5, leftInclusive = Inclusive [, rightInclusive = Inclusive ]) to show how changing one variable affects the interval notation result. Run the calculator above with these values to get the exact updated output with step-by-step work.