Function Basic: Background You’ll Need 1

  • Use set-builder, inequality, and interval notations to describe sets of numbers on a number line

Set Notation: Inequality, Set-Builder, and Interval Notations

In mathematics, we often need to describe sets of numbers that satisfy certain conditions. There are several ways to represent these sets, each with its own advantages and uses in different contexts. This page introduces three common notations for describing sets of numbers: inequality notation, set-builder notation, and interval notation.

Inequality notation uses the symbols [latex]<[/latex], [latex]>[/latex], [latex]\le[/latex], and [latex]\ge[/latex] to describe ranges of numbers.

  • [latex]x > 3[/latex] means all numbers greater than [latex]3[/latex]
  • [latex]2 \le x < 5[/latex] means all numbers greater than or equal to [latex]2[/latex] and less than [latex]5[/latex]

Set-builder notation uses curly braces [latex]{}[/latex] and a vertical bar [latex]|[/latex] to describe sets based on their properties.

  • [latex]{x \mid x > 3}[/latex] means the set of all [latex]x[/latex] such that [latex]x[/latex] is greater than [latex]3[/latex]
  • [latex]{x \mid 2 \le x < 5}[/latex] means the set of all [latex]x[/latex] such that [latex]x[/latex] is greater than or equal to [latex]2[/latex] and less than [latex]5[/latex]

Interval notation uses parentheses [latex]([/latex] [latex])[/latex] and square brackets [latex][[/latex] [latex]][/latex] to represent continuous ranges of numbers.

  • [latex](3, \infty)[/latex] means all numbers greater than [latex]3[/latex]
  • [latex][2, 5)[/latex] means all numbers greater than or equal to [latex]2[/latex] and less than [latex]5[/latex]

The table below compares inequality notation, set-builder notation, and interval notation.

Inequality Notation Set-builder Notation Interval Notation
1 [latex]5 \lt h \le 10[/latex] [latex]\{h | 5 < h \le 10\}[/latex] [latex](5,10][/latex]
2 [latex]5 \le  h<10[/latex] [latex]\{h | 5 \le h < 10\}[/latex] [latex][5,10)[/latex]
3 [latex]5 \lt h\lt 10[/latex] [latex]\{h | 5 < h < 10\}[/latex] [latex](5,10)[/latex]
4 [latex]h<10[/latex] [latex]\{h | h < 10\}[/latex] [latex](-\infty,10)[/latex]
5 [latex]h>10[/latex] [latex]\{h | h > 10\}[/latex] [latex](10,\infty)[/latex]
6 All real numbers [latex]\mathbf{R}[/latex] [latex](−\infty,\infty)[/latex]

Special Cases and Symbols

  • [latex]\infty[/latex] (infinity) is used in interval notation to represent unbounded intervals
  • The empty set is represented as [latex]\emptyset[/latex] or [latex]\{\}[/latex] in set-builder notation, and as [latex][  ][/latex] in interval notation
  • The union of sets is represented by the symbol [latex]\cup[/latex]
Describe the intervals of values shown below using inequality notation, set-builder notation, and interval notation.Line graph of 1<=x<=3 and 5<x.To describe the values, [latex]x[/latex], included in the intervals shown, we would say, ” [latex]x[/latex] is a real number greater than or equal to 1 and less than or equal to 3, or a real number greater than 5.”

Inequality [latex]1\le x\le 3\text{ or }x>5[/latex]
Set-builder notation [latex]\left\{x|1\le x\le 3 \text{ or }x>5\right\}[/latex]
Interval notation [latex]\left[1,3\right]\cup \left(5,\infty \right)[/latex]

Remember that, when writing or reading interval notation, using a square bracket means the boundary is included in the set. Using a parenthesis means the boundary is not included in the set.