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
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] |
[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.beA 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]
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- 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.