Inequality Set Notation Interval Notation Calculator

Introduction

In mathematics, there are several ways to represent sets of numbers. The three most common notations are inequality notation, set notation, and interval notation. Each has its own advantages and is used in different contexts.

Understanding these notations is essential for working with real numbers, solving equations, and graphing functions. This guide will explain each notation, show how to convert between them, and provide practical examples.

Understanding the Notations

Inequality Notation

Inequality notation uses symbols to describe ranges of numbers. The most common symbols are:

• < - less than
• > - greater than
• ≤ - less than or equal to
• ≥ - greater than or equal to

For example, x > 3 means all real numbers greater than 3. Inequality notation is often used in algebra and calculus to describe domains and ranges of functions.

Set Notation

Set notation uses curly braces to list elements or describe properties. For example:

• {1, 2, 3} - a set containing the numbers 1, 2, and 3
• {x | x > 3} - a set of all real numbers greater than 3

Set notation is versatile and can be used to describe both finite and infinite sets. It's commonly used in set theory and discrete mathematics.

Interval Notation

Interval notation uses parentheses and brackets to indicate ranges on the number line. The most common symbols are:

• ( ) - parentheses indicate that the endpoint is not included
• [ ] - brackets indicate that the endpoint is included

For example, (3, ∞) means all real numbers greater than 3. Interval notation is commonly used in calculus and real analysis to describe domains and ranges of functions.

Conversion Between Notations

Converting between these notations is straightforward once you understand the relationships between them. Here's a quick guide:

Inequality to Set Notation

To convert an inequality to set notation, simply replace the inequality symbol with a vertical bar and enclose the result in curly braces. For example:

• x > 3 becomes {x | x > 3}
• x ≤ 5 becomes {x | x ≤ 5}

Inequality to Interval Notation

To convert an inequality to interval notation, use the following rules:

• < becomes (
• > becomes )
• ≤ becomes [
• ≥ becomes ]

For example:

• x > 3 becomes (3, ∞)
• x ≤ 5 becomes (-∞, 5]

Set to Interval Notation

To convert set notation to interval notation, identify the range described by the set and use the appropriate parentheses or brackets. For example:

• {x | 3 < x < 5} becomes (3, 5)
• {x | x ≥ 2} becomes [2, ∞)

Worked Examples

Example 1: Converting Inequality to Set and Interval Notation

Given the inequality x > 4, convert it to set notation and interval notation.

Set notation: {x | x > 4}

Interval notation: (4, ∞)

Example 2: Converting Set to Interval Notation

Given the set {x | -2 ≤ x < 3}, convert it to interval notation.

Interval notation: [-2, 3)

Example 3: Converting Interval to Set Notation

Given the interval (-∞, 0], convert it to set notation.

Set notation: {x | x ≤ 0}