# Standard Notation for Defining Sets

### Learning Outcomes

- Write sets using set-builder, inequality, and interval notation
- Describe sets on the real number line using set builder, interval, and inequality notation

*x*-values such that 10 is less than or equal to [latex]x[/latex], and [latex]x[/latex] is less than 30." The table below compares inequality notation, set-builder notation, and interval notation.

Inequality Notation | Set-builder Notation | Interval Notation | |
---|---|---|---|

[latex]5<h\le10[/latex] | [latex]\{h | 5 < h \le 10\}[/latex] | [latex](5,10][/latex] | |

[latex]5\le h<10[/latex] | [latex]\{h | 5 \le h < 10\}[/latex] | [latex][5,10)[/latex] | |

[latex]5<h<10[/latex] | [latex]\{h | 5 < h < 10\}[/latex] | [latex](5,10)[/latex] | |

[latex]h<10[/latex] | [latex]\{h | h < 10\}[/latex] | [latex](-\infty,10)[/latex] | |

[latex]h>10[/latex] | [latex]\{h | h > 10\}[/latex] | [latex](10,\infty)[/latex] | |

All real numbers | [latex]\mathbf{R}[/latex] | [latex](−\infty,\infty)[/latex] |

*or*the other (or both) of the original two sets. For sets with a finite number of elements like these, the elements do not have to be listed in ascending order of numerical value. If the original two sets have some elements in common, those elements should be listed only once in the union set. For sets of real numbers on intervals, another example of a union is

[latex]\left\{x|\text{ }|x|\ge 3\right\}=\left(-\infty ,-3\right]\cup \left[3,\infty \right)[/latex]

This video describes how to use interval notation to describe a set. https://www.youtube.com/watch?v=hqg85P0ZMZ4 This video describes how to use Set-Builder notation to describe a set. https://www.youtube.com/watch?v=rPcGeaDRnyc&feature=youtu.be### A General Note: Set-Builder Notation and Interval Notation

Set-builder notation is a method of specifying a set of elements that satisfy a certain condition. It takes the form [latex]\left\{x|\text{statement about }x\right\}[/latex] which is read as, "the set of all [latex]x[/latex] such that the statement about [latex]x[/latex] is true." For example,[latex]\left\{x|4<x\le 12\right\}[/latex]

**Interval notation**is a way of describing sets that include all real numbers between a lower limit that may or may not be included and an upper limit that may or may not be included. The endpoint values are listed between brackets or parentheses. A square bracket indicates inclusion in the set, and a parenthesis indicates exclusion from the set. For example,

[latex]\left(4,12\right][/latex]

### How To: Given a line graph, describe the set of values using interval notation.

- Identify the intervals to be included in the set by determining where the heavy line overlays the real line.
- At the left end of each interval, use [ with each end value to be included in the set (solid dot) or ( for each excluded end value (open dot).
- At the right end of each interval, use ] with each end value to be included in the set (filled dot) or ) for each excluded end value (open dot).
- Use the union symbol [latex]\cup [/latex] to combine all intervals into one set.

### Example: Describing Sets on the Real-Number Line

Describe the intervals of values shown below using inequality notation, set-builder notation, and interval notation.Answer: 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\hspace{2mm}\text{or}\hspace{2mm}x>5[/latex] |

Set-builder notation |
[latex]\left\{x|1\le x\le 3\hspace{2mm}\text{or}\hspace{2mm}x>5\right\}[/latex] |

Interval notation |
[latex]\left[1,3\right]\cup \left(5,\infty \right)[/latex] |

### Try It

Given the graph below, specify the graphed set in- words
- set-builder notation
- interval notation

Answer: Words: values that are less than or equal to –2, or values that are greater than or equal to –1 and less than 3. Set-builder notation: [latex]\left\{x|x\le -2\hspace{2mm}\text{or}\hspace{2mm}-1\le x<3\right\}[/latex]; Interval notation: [latex]\left(-\infty ,-2\right]\cup \left[-1,3\right)[/latex]

## Licenses & Attributions

### CC licensed content, Original

- Revision and Adaptation.
**Provided by:**Lumen Learning**License:**CC BY: Attribution.

### CC licensed content, Shared previously

- College Algebra.
**Provided by:**OpenStax**Authored by:**Abramson, Jay et al..**Located at:**https://openstax.org/books/college-algebra/pages/1-introduction-to-prerequisites.**License:**CC BY: Attribution.**License terms:**Download for free at http://cnx.org/contents/[email protected]. - Question ID 108347.
**Authored by:**Coulston,Charles R.**License:**CC BY: Attribution.**License terms:**IMathAS Community License CC-BY + GPL. - Question ID 3190, 3191.
**Authored by:**Anderson,Tophe.**License:**CC BY: Attribution.**License terms:**IMathAS Community License CC-BY + GPL.