{"id":128,"date":"2023-06-21T13:22:36","date_gmt":"2023-06-21T13:22:36","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/chapter\/standard-notation-for-domain-and-range\/"},"modified":"2023-08-21T21:54:05","modified_gmt":"2023-08-21T21:54:05","slug":"standard-notation-for-domain-and-range","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/chapter\/standard-notation-for-domain-and-range\/","title":{"raw":"\u25aa   Standard Notation for Defining Sets","rendered":"\u25aa   Standard Notation for Defining Sets"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Write sets\u00a0using set-builder, inequality, and interval notation<\/li>\r\n \t<li>Describe sets on the real number line\u00a0using set builder, interval, and inequality notation<\/li>\r\n<\/ul>\r\n<\/div>\r\nThere are several ways to define sets of numbers or mathematical objects. The reason we are introducing this here is because we often need to define the sets of numbers that make up the inputs and outputs of a function.\r\n\r\nHow we write sets that make up the domain and range of functions often depends on how the relation or function are defined or presented to us. \u00a0For example, we can use lists to describe the domain of functions that are given as sets of ordered pairs. If we are given an equation or graph, we might use inequalities or intervals to describe domain and range.\r\n\r\nIn this section, we will introduce the standard notation used to define sets, and give you a chance to practice writing sets in three ways, inequality notation, set-builder notation, and interval notation.\r\n\r\nConsider the set [latex]\\left\\{x|10\\le x&lt;30\\right\\}[\/latex], which describes the behavior of [latex]x[\/latex] in set-builder notation. The braces [latex]\\{\\}[\/latex] are read as \"the set of,\" and the vertical bar [latex]|[\/latex] is read as \"such that,\" so we would read [latex]\\left\\{x|10\\le x&lt;30\\right\\}[\/latex] as \"the set of <em>x<\/em>-values such that 10 is less than or equal to [latex]x[\/latex], and [latex]x[\/latex] is less than 30.\"\r\n\r\nThe table below compares inequality notation, set-builder notation, and interval notation.\r\n<table style=\"height: 312px;\">\r\n<thead>\r\n<tr style=\"height: 30px;\">\r\n<th style=\"height: 30px; width: 194px;\"><\/th>\r\n<th style=\"height: 30px; width: 148px;\">Inequality Notation<\/th>\r\n<th style=\"height: 30px; width: 118px;\">Set-builder Notation<\/th>\r\n<th style=\"height: 30px; width: 80px;\">Interval Notation<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr style=\"height: 45px;\">\r\n<td style=\"height: 45px; width: 194px;\"><a href=\"https:\/\/courses.candelalearning.com\/precalcone1xmommaster\/wp-content\/uploads\/sites\/1226\/2015\/08\/1.png\"><img class=\"size-full wp-image-12492 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18193534\/1.png\" alt=\"1\" width=\"265\" height=\"60\" \/><\/a><\/td>\r\n<td style=\"height: 45px; width: 148px;\">[latex]5&lt;h\\le10[\/latex]<\/td>\r\n<td style=\"height: 45px; width: 118px;\">[latex]\\{h | 5 &lt; h \\le 10\\}[\/latex]<\/td>\r\n<td style=\"height: 45px; width: 80px;\">[latex](5,10][\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 48px;\">\r\n<td style=\"height: 48px; width: 194px;\"><a href=\"https:\/\/courses.candelalearning.com\/precalcone1xmommaster\/wp-content\/uploads\/sites\/1226\/2015\/08\/2.png\"><img class=\"size-full wp-image-12493 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18193535\/2.png\" alt=\"2\" width=\"281\" height=\"75\" \/><\/a><\/td>\r\n<td style=\"height: 48px; width: 148px;\">[latex]5\\le h&lt;10[\/latex]<\/td>\r\n<td style=\"height: 48px; width: 118px;\">[latex]\\{h | 5 \\le h &lt; 10\\}[\/latex]<\/td>\r\n<td style=\"height: 48px; width: 80px;\">[latex][5,10)[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 49px;\">\r\n<td style=\"height: 49px; width: 194px;\"><a href=\"https:\/\/courses.candelalearning.com\/precalcone1xmommaster\/wp-content\/uploads\/sites\/1226\/2015\/08\/3.png\"><img class=\"size-full wp-image-12494 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18193537\/3.png\" alt=\"3\" width=\"283\" height=\"76\" \/><\/a><\/td>\r\n<td style=\"height: 49px; width: 148px;\">[latex]5&lt;h&lt;10[\/latex]<\/td>\r\n<td style=\"height: 49px; width: 118px;\">[latex]\\{h | 5 &lt; h &lt; 10\\}[\/latex]<\/td>\r\n<td style=\"height: 49px; width: 80px;\">[latex](5,10)[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 51px;\">\r\n<td style=\"height: 51px; width: 194px;\"><a href=\"https:\/\/courses.candelalearning.com\/precalcone1xmommaster\/wp-content\/uploads\/sites\/1226\/2015\/08\/4.png\"><img class=\"size-full wp-image-12495 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18193538\/4.png\" alt=\"4\" width=\"271\" height=\"76\" \/><\/a><\/td>\r\n<td style=\"height: 51px; width: 148px;\">[latex]h&lt;10[\/latex]<\/td>\r\n<td style=\"height: 51px; width: 118px;\">[latex]\\{h | h &lt; 10\\}[\/latex]<\/td>\r\n<td style=\"height: 51px; width: 80px;\">[latex](-\\infty,10)[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 44px;\">\r\n<td style=\"height: 44px; width: 194px;\"><a href=\"https:\/\/courses.candelalearning.com\/precalcone1xmommaster\/wp-content\/uploads\/sites\/1226\/2015\/08\/5.png\"><img class=\"size-full wp-image-12496 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18193540\/5.png\" alt=\"5\" width=\"310\" height=\"66\" \/><\/a><\/td>\r\n<td style=\"height: 44px; width: 148px;\">[latex]h&gt;10[\/latex]<\/td>\r\n<td style=\"height: 44px; width: 118px;\">[latex]\\{h | h &gt; 10\\}[\/latex]<\/td>\r\n<td style=\"height: 44px; width: 80px;\">[latex](10,\\infty)[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 45px;\">\r\n<td style=\"height: 45px; width: 194px;\"><a href=\"https:\/\/courses.candelalearning.com\/precalcone1xmommaster\/wp-content\/uploads\/sites\/1226\/2015\/08\/6.png\"><img class=\"size-full wp-image-12497 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18193542\/6.png\" alt=\"6\" width=\"359\" height=\"67\" \/><\/a><\/td>\r\n<td style=\"height: 45px; width: 148px;\">All real numbers<\/td>\r\n<td style=\"height: 45px; width: 118px;\">[latex]\\mathbf{R}[\/latex]<\/td>\r\n<td style=\"height: 45px; width: 80px;\">[latex](\u2212\\infty,\\infty)[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nTo combine two intervals using inequality notation or set-builder notation, we use the word \"or.\" As we saw in earlier examples, we use the union symbol, [latex]\\cup [\/latex], to combine two unconnected intervals. For example, the union of the sets [latex]\\left\\{2,3,5\\right\\}[\/latex]\u00a0and [latex]\\left\\{4,6\\right\\}[\/latex]\u00a0is the set [latex]\\left\\{2,3,4,5,6\\right\\}[\/latex]. It is the set of all elements that belong to one <em>or<\/em> 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\r\n<p style=\"text-align: center;\">[latex]\\left\\{x|\\text{ }|x|\\ge 3\\right\\}=\\left(-\\infty ,-3\\right]\\cup \\left[3,\\infty \\right)[\/latex]<\/p>\r\nThis video describes how to use interval notation to describe a set.\r\n\r\nhttps:\/\/www.youtube.com\/watch?v=hqg85P0ZMZ4\r\n\r\nThis video describes how to use Set-Builder notation to describe a set.\r\n\r\nhttps:\/\/www.youtube.com\/watch?v=rPcGeaDRnyc&amp;feature=youtu.be\r\n<div class=\"textbox\">\r\n<h3>A General Note: Set-Builder Notation and Interval Notation<\/h3>\r\nSet-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,\r\n<p style=\"text-align: center;\">[latex]\\left\\{x|4&lt;x\\le 12\\right\\}[\/latex]<\/p>\r\n<strong>Interval notation<\/strong> 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,\r\n<p style=\"text-align: center;\">[latex]\\left(4,12\\right][\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox\">\r\n<h3>How To: Given a line graph, describe the set of values using interval notation.<\/h3>\r\n<ol>\r\n \t<li>Identify the intervals to be included in the set by determining where the heavy line overlays the real line.<\/li>\r\n \t<li>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).<\/li>\r\n \t<li>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).<\/li>\r\n \t<li>Use the union symbol [latex]\\cup [\/latex] to combine all intervals into one set.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Describing Sets on the Real-Number Line<\/h3>\r\nDescribe the intervals of values shown below using inequality notation, set-builder notation, and interval notation.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18193544\/CNX_Precalc_Figure_01_02_0042.jpg\" alt=\"Line graph of 1&lt;=x&lt;=3 and 5&lt;x.\" width=\"487\" height=\"50\" \/>\r\n[reveal-answer q=\"169160\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"169160\"]\r\n\r\nTo 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.\"\r\n<table summary=\"..\">\r\n<tbody>\r\n<tr>\r\n<td><strong>Inequality<\/strong><\/td>\r\n<td>[latex]1\\le x\\le 3\\hspace{2mm}\\text{or}\\hspace{2mm}x&gt;5[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong>Set-builder notation<\/strong><\/td>\r\n<td>[latex]\\left\\{x|1\\le x\\le 3\\hspace{2mm}\\text{or}\\hspace{2mm}x&gt;5\\right\\}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong>Interval notation<\/strong><\/td>\r\n<td>[latex]\\left[1,3\\right]\\cup \\left(5,\\infty \\right)[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nRemember 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.\r\n\r\n[\/hidden-answer]\r\n<iframe id=\"mom18class=\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=108347&amp;theme=oea&amp;iframe_resize_id=mom18\" width=\"100%\" height=\"500\"><\/iframe>\r\n<iframe id=\"mom20\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=3190&amp;theme=oea&amp;iframe_resize_id=mom20\" width=\"100%\" height=\"440\"><\/iframe>\r\n<iframe id=\"mom30\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=3191&amp;theme=oea&amp;iframe_resize_id=mom30\" width=\"100%\" height=\"350\"><\/iframe>\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nGiven the graph below, specify the graphed set in\r\n<ol>\r\n \t<li>words<\/li>\r\n \t<li>set-builder notation<\/li>\r\n \t<li>interval notation<\/li>\r\n<\/ol>\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18193546\/CNX_Precalc_Figure_01_02_0052.jpg\" alt=\"Line graph of -2&lt;=x, -1&lt;=x&lt;3.\" width=\"487\" height=\"50\" \/>\r\n\r\n[reveal-answer q=\"102737\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"102737\"]\r\n\r\nWords: values that are less than or equal to \u20132, or values that are greater than or equal to \u20131 and less than 3.\r\n\r\nSet-builder notation: [latex]\\left\\{x|x\\le -2\\hspace{2mm}\\text{or}\\hspace{2mm}-1\\le x&lt;3\\right\\}[\/latex];\r\n\r\nInterval notation: [latex]\\left(-\\infty ,-2\\right]\\cup \\left[-1,3\\right)[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nThe table below gives\u00a0a summary of interval notation.\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18193529\/CNX_Precalc_Figure_01_02_029n2.jpg\" alt=\"Summary of interval notation. Row 1, Inequality: x is greater than a. Interval notation: open parenthesis, a, infinity, close parenthesis. Row 2, Inequality: x is less than a. Interval notation: open parenthesis, negative infinity, a, close parenthesis. Row 3, Inequality x is greater than or equal to a. Interval notation: open bracket, a, infinity, close parenthesis. Row 4, Inequality: x less than or equal to a. Interval notation: open parenthesis, negative infinity, a, close bracket. Row 5, Inequality: a is less than x is less than b. Interval notation: open parenthesis, a, b, close parenthesis. Row 6, Inequality: a is less than or equal to x is less than b. Interval notation: Open bracket, a, b, close parenthesis. Row 7, Inequality: a is less than x is less than or equal to b. Interval notation: Open parenthesis, a, b, close bracket. Row 8, Inequality: a, less than or equal to x is less than or equal to b. Interval notation: open bracket, a, b, close bracket.\" width=\"975\" height=\"905\" \/>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Write sets\u00a0using set-builder, inequality, and interval notation<\/li>\n<li>Describe sets on the real number line\u00a0using set builder, interval, and inequality notation<\/li>\n<\/ul>\n<\/div>\n<p>There are several ways to define sets of numbers or mathematical objects. The reason we are introducing this here is because we often need to define the sets of numbers that make up the inputs and outputs of a function.<\/p>\n<p>How we write sets that make up the domain and range of functions often depends on how the relation or function are defined or presented to us. \u00a0For example, we can use lists to describe the domain of functions that are given as sets of ordered pairs. If we are given an equation or graph, we might use inequalities or intervals to describe domain and range.<\/p>\n<p>In this section, we will introduce the standard notation used to define sets, and give you a chance to practice writing sets in three ways, inequality notation, set-builder notation, and interval notation.<\/p>\n<p>Consider the set [latex]\\left\\{x|10\\le x<30\\right\\}[\/latex], which describes the behavior of [latex]x[\/latex] in set-builder notation. The braces [latex]\\{\\}[\/latex] are read as &#8220;the set of,&#8221; and the vertical bar [latex]|[\/latex] is read as &#8220;such that,&#8221; so we would read [latex]\\left\\{x|10\\le x<30\\right\\}[\/latex] as &#8220;the set of <em>x<\/em>-values such that 10 is less than or equal to [latex]x[\/latex], and [latex]x[\/latex] is less than 30.&#8221;<\/p>\n<p>The table below compares inequality notation, set-builder notation, and interval notation.<\/p>\n<table style=\"height: 312px;\">\n<thead>\n<tr style=\"height: 30px;\">\n<th style=\"height: 30px; width: 194px;\"><\/th>\n<th style=\"height: 30px; width: 148px;\">Inequality Notation<\/th>\n<th style=\"height: 30px; width: 118px;\">Set-builder Notation<\/th>\n<th style=\"height: 30px; width: 80px;\">Interval Notation<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr style=\"height: 45px;\">\n<td style=\"height: 45px; width: 194px;\"><a href=\"https:\/\/courses.candelalearning.com\/precalcone1xmommaster\/wp-content\/uploads\/sites\/1226\/2015\/08\/1.png\"><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-12492 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18193534\/1.png\" alt=\"1\" width=\"265\" height=\"60\" \/><\/a><\/td>\n<td style=\"height: 45px; width: 148px;\">[latex]5<h\\le10[\/latex]<\/td>\n<td style=\"height: 45px; width: 118px;\">[latex]\\{h | 5 < h \\le 10\\}[\/latex]<\/td>\n<td style=\"height: 45px; width: 80px;\">[latex](5,10][\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 48px;\">\n<td style=\"height: 48px; width: 194px;\"><a href=\"https:\/\/courses.candelalearning.com\/precalcone1xmommaster\/wp-content\/uploads\/sites\/1226\/2015\/08\/2.png\"><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-12493 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18193535\/2.png\" alt=\"2\" width=\"281\" height=\"75\" \/><\/a><\/td>\n<td style=\"height: 48px; width: 148px;\">[latex]5\\le h<10[\/latex]<\/td>\n<td style=\"height: 48px; width: 118px;\">[latex]\\{h | 5 \\le h < 10\\}[\/latex]<\/td>\n<td style=\"height: 48px; width: 80px;\">[latex][5,10)[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 49px;\">\n<td style=\"height: 49px; width: 194px;\"><a href=\"https:\/\/courses.candelalearning.com\/precalcone1xmommaster\/wp-content\/uploads\/sites\/1226\/2015\/08\/3.png\"><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-12494 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18193537\/3.png\" alt=\"3\" width=\"283\" height=\"76\" \/><\/a><\/td>\n<td style=\"height: 49px; width: 148px;\">[latex]5<h<10[\/latex]<\/td>\n<td style=\"height: 49px; width: 118px;\">[latex]\\{h | 5 < h < 10\\}[\/latex]<\/td>\n<td style=\"height: 49px; width: 80px;\">[latex](5,10)[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 51px;\">\n<td style=\"height: 51px; width: 194px;\"><a href=\"https:\/\/courses.candelalearning.com\/precalcone1xmommaster\/wp-content\/uploads\/sites\/1226\/2015\/08\/4.png\"><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-12495 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18193538\/4.png\" alt=\"4\" width=\"271\" height=\"76\" \/><\/a><\/td>\n<td style=\"height: 51px; width: 148px;\">[latex]h<10[\/latex]<\/td>\n<td style=\"height: 51px; width: 118px;\">[latex]\\{h | h < 10\\}[\/latex]<\/td>\n<td style=\"height: 51px; width: 80px;\">[latex](-\\infty,10)[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 44px;\">\n<td style=\"height: 44px; width: 194px;\"><a href=\"https:\/\/courses.candelalearning.com\/precalcone1xmommaster\/wp-content\/uploads\/sites\/1226\/2015\/08\/5.png\"><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-12496 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18193540\/5.png\" alt=\"5\" width=\"310\" height=\"66\" \/><\/a><\/td>\n<td style=\"height: 44px; width: 148px;\">[latex]h>10[\/latex]<\/td>\n<td style=\"height: 44px; width: 118px;\">[latex]\\{h | h > 10\\}[\/latex]<\/td>\n<td style=\"height: 44px; width: 80px;\">[latex](10,\\infty)[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 45px;\">\n<td style=\"height: 45px; width: 194px;\"><a href=\"https:\/\/courses.candelalearning.com\/precalcone1xmommaster\/wp-content\/uploads\/sites\/1226\/2015\/08\/6.png\"><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-12497 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18193542\/6.png\" alt=\"6\" width=\"359\" height=\"67\" \/><\/a><\/td>\n<td style=\"height: 45px; width: 148px;\">All real numbers<\/td>\n<td style=\"height: 45px; width: 118px;\">[latex]\\mathbf{R}[\/latex]<\/td>\n<td style=\"height: 45px; width: 80px;\">[latex](\u2212\\infty,\\infty)[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>To combine two intervals using inequality notation or set-builder notation, we use the word &#8220;or.&#8221; As we saw in earlier examples, we use the union symbol, [latex]\\cup[\/latex], to combine two unconnected intervals. For example, the union of the sets [latex]\\left\\{2,3,5\\right\\}[\/latex]\u00a0and [latex]\\left\\{4,6\\right\\}[\/latex]\u00a0is the set [latex]\\left\\{2,3,4,5,6\\right\\}[\/latex]. It is the set of all elements that belong to one <em>or<\/em> 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<\/p>\n<p style=\"text-align: center;\">[latex]\\left\\{x|\\text{ }|x|\\ge 3\\right\\}=\\left(-\\infty ,-3\\right]\\cup \\left[3,\\infty \\right)[\/latex]<\/p>\n<p>This video describes how to use interval notation to describe a set.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Interval Notation\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/hqg85P0ZMZ4?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>This video describes how to use Set-Builder notation to describe a set.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-2\" title=\"Set-Builder Notation\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/rPcGeaDRnyc?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<div class=\"textbox\">\n<h3>A General Note: Set-Builder Notation and Interval Notation<\/h3>\n<p>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, &#8220;the set of all [latex]x[\/latex] such that the statement about [latex]x[\/latex] is true.&#8221; For example,<\/p>\n<p style=\"text-align: center;\">[latex]\\left\\{x|4<x\\le 12\\right\\}[\/latex]<\/p>\n<p><strong>Interval notation<\/strong> 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,<\/p>\n<p style=\"text-align: center;\">[latex]\\left(4,12\\right][\/latex]<\/p>\n<\/div>\n<div class=\"textbox\">\n<h3>How To: Given a line graph, describe the set of values using interval notation.<\/h3>\n<ol>\n<li>Identify the intervals to be included in the set by determining where the heavy line overlays the real line.<\/li>\n<li>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).<\/li>\n<li>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).<\/li>\n<li>Use the union symbol [latex]\\cup[\/latex] to combine all intervals into one set.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Describing Sets on the Real-Number Line<\/h3>\n<p>Describe the intervals of values shown below using inequality notation, set-builder notation, and interval notation.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18193544\/CNX_Precalc_Figure_01_02_0042.jpg\" alt=\"Line graph of 1&lt;=x&lt;=3 and 5&lt;x.\" width=\"487\" height=\"50\" \/><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q169160\">Show Solution<\/span><\/p>\n<div id=\"q169160\" class=\"hidden-answer\" style=\"display: none\">\n<p>To describe the values, [latex]x[\/latex], included in the intervals shown, we would say, &#8221; [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.&#8221;<\/p>\n<table summary=\"..\">\n<tbody>\n<tr>\n<td><strong>Inequality<\/strong><\/td>\n<td>[latex]1\\le x\\le 3\\hspace{2mm}\\text{or}\\hspace{2mm}x>5[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><strong>Set-builder notation<\/strong><\/td>\n<td>[latex]\\left\\{x|1\\le x\\le 3\\hspace{2mm}\\text{or}\\hspace{2mm}x>5\\right\\}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><strong>Interval notation<\/strong><\/td>\n<td>[latex]\\left[1,3\\right]\\cup \\left(5,\\infty \\right)[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>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.<\/p>\n<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"mom18class=\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=108347&amp;theme=oea&amp;iframe_resize_id=mom18\" width=\"100%\" height=\"500\"><\/iframe><br \/>\n<iframe loading=\"lazy\" id=\"mom20\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=3190&amp;theme=oea&amp;iframe_resize_id=mom20\" width=\"100%\" height=\"440\"><\/iframe><br \/>\n<iframe loading=\"lazy\" id=\"mom30\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=3191&amp;theme=oea&amp;iframe_resize_id=mom30\" width=\"100%\" height=\"350\"><\/iframe><\/p>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Given the graph below, specify the graphed set in<\/p>\n<ol>\n<li>words<\/li>\n<li>set-builder notation<\/li>\n<li>interval notation<\/li>\n<\/ol>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18193546\/CNX_Precalc_Figure_01_02_0052.jpg\" alt=\"Line graph of -2&lt;=x, -1&lt;=x&lt;3.\" width=\"487\" height=\"50\" \/><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q102737\">Show Solution<\/span><\/p>\n<div id=\"q102737\" class=\"hidden-answer\" style=\"display: none\">\n<p>Words: values that are less than or equal to \u20132, or values that are greater than or equal to \u20131 and less than 3.<\/p>\n<p>Set-builder notation: [latex]\\left\\{x|x\\le -2\\hspace{2mm}\\text{or}\\hspace{2mm}-1\\le x<3\\right\\}[\/latex];\n\nInterval notation: [latex]\\left(-\\infty ,-2\\right]\\cup \\left[-1,3\\right)[\/latex]\n\n<\/div>\n<\/div>\n<\/div>\n<p>The table below gives\u00a0a summary of interval notation.<br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18193529\/CNX_Precalc_Figure_01_02_029n2.jpg\" alt=\"Summary of interval notation. Row 1, Inequality: x is greater than a. Interval notation: open parenthesis, a, infinity, close parenthesis. Row 2, Inequality: x is less than a. Interval notation: open parenthesis, negative infinity, a, close parenthesis. Row 3, Inequality x is greater than or equal to a. Interval notation: open bracket, a, infinity, close parenthesis. Row 4, Inequality: x less than or equal to a. Interval notation: open parenthesis, negative infinity, a, close bracket. Row 5, Inequality: a is less than x is less than b. Interval notation: open parenthesis, a, b, close parenthesis. Row 6, Inequality: a is less than or equal to x is less than b. Interval notation: Open bracket, a, b, close parenthesis. Row 7, Inequality: a is less than x is less than or equal to b. Interval notation: Open parenthesis, a, b, close bracket. Row 8, Inequality: a, less than or equal to x is less than or equal to b. Interval notation: open bracket, a, b, close bracket.\" width=\"975\" height=\"905\" \/><\/p>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-128\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Original<\/div><ul class=\"citation-list\"><li>Revision and Adaptation. <strong>Provided by<\/strong>: Lumen Learning. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>College Algebra. <strong>Authored by<\/strong>: Abramson, Jay et al.. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2\">http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: Download for free at http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2<\/li><li>Question ID 108347. <strong>Authored by<\/strong>: Coulston,Charles R. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: IMathAS Community License CC-BY + GPL<\/li><li>Question ID 3190, 3191. <strong>Authored by<\/strong>: Anderson,Tophe. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: IMathAS Community License CC-BY + GPL<\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":395986,"menu_order":13,"template":"","meta":{"_candela_citation":"[{\"type\":\"original\",\"description\":\"Revision and Adaptation\",\"author\":\"\",\"organization\":\"Lumen Learning\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"College Algebra\",\"author\":\"Abramson, Jay et al.\",\"organization\":\"OpenStax\",\"url\":\"http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"Download for free at http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2\"},{\"type\":\"cc\",\"description\":\"Question ID 108347\",\"author\":\"Coulston,Charles R\",\"organization\":\"\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"IMathAS Community License CC-BY + 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