{"id":27,"date":"2023-01-25T16:33:53","date_gmt":"2023-01-25T16:33:53","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/chapter\/set-theory-basics-learn-it-page-1\/"},"modified":"2024-10-18T20:50:08","modified_gmt":"2024-10-18T20:50:08","slug":"set-theory-basics-learn-it-1","status":"web-only","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/chapter\/set-theory-basics-learn-it-1\/","title":{"raw":"Set Theory Basics: Learn It 1","rendered":"Set Theory Basics: Learn It 1"},"content":{"raw":"<section class=\"textbox learningGoals\">\r\n<ul>\r\n\t<li>Understand the concepts of sets, including empty sets, subsets, and proper subsets, and use correct set notation<\/li>\r\n\t<li>Describe and perform set operations (union, intersection, complement, and difference) using proper set notation<\/li>\r\n\t<li>Create and interpret Venn diagrams to represent and analyze set relationships and operations<\/li>\r\n\t<li>Apply the concepts of sets, subsets, and cardinality properties to solve real-life problems<\/li>\r\n<\/ul>\r\n<\/section>\r\n<p>A movie lover might own a collection of movie posters, while a music lover might keep a collection of vinyl records. Any collection of items can form a <strong>set<\/strong>.<\/p>\r\n<section class=\"textbox keyTakeaway\">\r\n<div>\r\n<h3>set<\/h3>\r\n<p>A <strong>set<\/strong> is a collection of distinct objects, called <strong>elements<\/strong> of the set.<\/p>\r\n<p>&nbsp;<\/p>\r\n<p>A set can be defined by describing the contents, or by listing the elements of the set, enclosed in curly brackets separated by commas.<\/p>\r\n<p>&nbsp;<\/p>\r\n<p>Repeated elements in a set are only listed once. A set simply specifies the contents; order is not important.<\/p>\r\n<p>&nbsp;<\/p>\r\n<center>The set represented by [latex]\\{1, 2, 3\\}[\/latex] is equivalent to the set [latex]\\{3, 1, 2\\}[\/latex].<\/center><\/div>\r\n<\/section>\r\n<section class=\"textbox example\">Some examples of sets defined by describing the contents:\r\n\r\n<ol>\r\n\t<li>The set of all even numbers<\/li>\r\n\t<li>The set of all books written about travel to Chile<\/li>\r\n<\/ol>\r\n<p>Some examples of sets defined by listing the elements of the set:<\/p>\r\n<ol>\r\n\t<li>[latex]\\{1, 3, 9, 12\\}[\/latex]<\/li>\r\n\t<li>[latex]\\{\\text{red, orange, yellow, green, blue, indigo, purple}\\}[\/latex]<\/li>\r\n<\/ol>\r\n<\/section>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]2199[\/ohm2_question]<\/section>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]7414[\/ohm2_question]<\/section>\r\n<section class=\"textbox keyTakeaway\">\r\n<div>\r\n<h3>set notation<\/h3>\r\n<p>Commonly, we will use a variable to represent a set, to make it easier to refer to that set later.<\/p>\r\n<p>&nbsp;<\/p>\r\n<p>The symbol [latex]\\in[\/latex]\u00a0means \u201cis an element of\u201d.<\/p>\r\n<p>&nbsp;<\/p>\r\n<p>A set that contains no elements, [latex]\\{ \\}[\/latex], is called the <strong>empty set<\/strong> or null set and is notated [latex]\\emptyset[\/latex].<\/p>\r\n<\/div>\r\n<\/section>\r\n<section class=\"textbox example\">Let [latex]A=\\{1,2,3,4\\}[\/latex]. To notate that [latex]2[\/latex] is element of the set, we'd write [latex]2 \\in A[\/latex].<\/section>\r\n<p>Sometimes a collection might not contain all the elements of a set. For example, Marta owns ninety-five Pok\u00e9mon cards. While Marta\u2019s collection is a set, we can also say it is a <strong>subset<\/strong> of the larger set of all Pok\u00e9mon cards.<\/p>\r\n<section class=\"textbox keyTakeaway\">\r\n<div>\r\n<h3>subset<\/h3>\r\n<p>A <strong>subset<\/strong> of a set [latex]A[\/latex] is a set that consists solely of elements drawn from set [latex]A[\/latex]. It may include all, some, or none of set [latex]A[\/latex]'s elements. In other words, every element in the subset is also an element of set [latex]A[\/latex].<\/p>\r\n<p>&nbsp;<\/p>\r\n<center>If [latex]B[\/latex] is a subset of [latex]A[\/latex], we write [latex]B \\subseteq A[\/latex]<\/center>\r\n<p>&nbsp;<\/p>\r\n<p>A <strong>proper subset<\/strong> of a set [latex]A[\/latex] is a subset that contains some, but not all, of the elements of set [latex]A[\/latex]. It is a subset that is not identical to the original set, meaning it must contain fewer elements than set [latex]A[\/latex].<\/p>\r\n<p>&nbsp;<\/p>\r\n<center>If [latex]B[\/latex] is a proper subset of [latex]A[\/latex], we write [latex]B \\subset A[\/latex]<\/center><\/div>\r\n<\/section>\r\n<section class=\"textbox seeExample\">Given the\u00a0set: [latex]A = \\{a, b, c, d\\}[\/latex]. List all of the subsets of [latex]A[\/latex].<br \/>\r\n[reveal-answer q=\"417709\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"417709\"]<center>[latex]\\{\\} (\\text{or } \\emptyset), \\{a\\}, \\{b\\}, \\{c\\}, \\{d\\}, \\{a,b\\}, \\{a,c\\}, \\{a,d\\}, \\{b,c\\}, \\{b,d\\}, \\{c,d\\}, \\{a,b,c\\},[\/latex][latex] \\{a,b,d\\}, \\{a,c,d\\}, \\{b,c,d\\}, \\{a,b,c,d\\}[\/latex]<\/center><p><\/p><p>You can see that there are [latex]16[\/latex] subsets, [latex]15[\/latex] of which are proper subsets.<\/p>[\/hidden-answer]<\/section>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]2200[\/ohm2_question]<\/section>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]2201[\/ohm2_question]<\/section>\r\n<p>Listing the sets is fine if you have only a few elements. However, if we were to list all of the subsets of a set containing many elements, it would be quite tedious. Instead, in the next example we will consider each element of the set separately.<\/p>\r\n<section class=\"textbox seeExample\">Given the\u00a0set: [latex]A = \\{a, b, c, d\\}[\/latex], there are four elements.\r\n\r\n<p>For the first element, [latex]a[\/latex], either it\u2019s in the set or it\u2019s not. Thus there are [latex]2[\/latex] choices for that first element.<\/p>\r\n<p>Similarly, there are two choices for [latex]b[\/latex] \u2014 either it\u2019s in the set or it\u2019s not.<\/p>\r\n<p>Using just those two elements, list all the possible subsets of the set [latex]\\{a,b\\}[\/latex].<\/p>\r\n\r\n[reveal-answer q=\"417739\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"417739\"]<center>[latex]\\{\\} (\\text{or } \\emptyset)[\/latex] \u2014 both elements are not in the set<br \/>\r\n[latex]\\{a\\}[\/latex] \u2014 [latex]a[\/latex] is in; [latex]b[\/latex] is not in the set<br \/>\r\n[latex]\\{b\\}[\/latex] \u2014 [latex]a[\/latex] is not in the set; [latex]b[\/latex] is in<br \/>\r\n[latex]\\{a,b\\}[\/latex] \u2014 [latex]a[\/latex] is in; [latex]b[\/latex] is in<\/center><p><\/p><p>Two choices for [latex]a[\/latex] times the two for [latex]b[\/latex] gives us [latex]2^2=4[\/latex] subsets.<\/p><br \/>\r\n[\/hidden-answer]<\/section>\r\n<section class=\"textbox recall\">\r\n<p><strong>Exponential Notation<\/strong><\/p>\r\n<p>Recall that the expression [latex]a^{n}[\/latex] states that some real number [latex]a[\/latex] is to be used as a factor [latex]n[\/latex] times.<\/p>\r\n<center>Ex. [latex]2^{5} = 2 \\cdot 2 \\cdot 2 \\cdot 2 \\cdot 2 = 32[\/latex]<\/center><\/section>\r\n<p>You may be asked to find the number of subsets and proper subsets of a given set. To do so, you need to look at the number of elements in the given set. In general, if you have [latex]n[\/latex] elements in your set, then there are [latex]2^{n}[\/latex] subsets and [latex]2^{n}\u22121[\/latex] proper subsets.<\/p>\r\n<section class=\"textbox keyTakeaway\">\r\n<div>\r\n<h3>number of subsets and proper subsets set<\/h3>\r\n<p>If you have [latex]n[\/latex] elements in your set:<\/p>\r\n<ul>\r\n\t<li>Number of subsets: [latex]2^{n}[\/latex]<\/li>\r\n\t<li>Number of proper subsets: [latex]2^{n}\u22121[\/latex]<\/li>\r\n<\/ul>\r\n<\/div>\r\n<\/section>\r\n<section class=\"textbox seeExample\">Determine the total number of subsets and proper subsets of the set [latex]\\{1,3,5,7\\}[\/latex].<br \/>\r\n[reveal-answer q=\"857946\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"857946\"]For this set, there are [latex]4[\/latex] elements which means [latex]n = 4[\/latex].To determine the number of subsets we must use [latex]2^n[\/latex]:\r\n\r\n<div style=\"text-align: center;\">[latex]\\begin{array}{ccc}\\hfill 2^n = 2^4 = 16\\\\ \\end{array}[\/latex]<\/div>\r\n<p><\/p>\r\n<p>To determine the number of proper subsets we must use [latex]2^n - 1[\/latex]:<\/p>\r\n<div style=\"text-align: center;\">[latex]\\begin{array}{ccc}\\hfill 2^n - 1 = 2^4 - 1 = 16 - 1 = 15\\\\ \\end{array}[\/latex]<\/div>\r\n<p><\/p>\r\n<p>There are [latex]16[\/latex] subsets and [latex]15[\/latex] proper subsets.<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]2202[\/ohm2_question]<\/section>\r\n<section class=\"textbox connectIt\">\r\n<p><strong>Subsets of real numbers<\/strong><\/p>\r\n<p>The idea of subsets can also be applied to the sets of real numbers. For example, the set of all whole numbers is a subset of the set of all integers. The set of integers in turn is contained within the set of rational numbers.<\/p>\r\n<p>We say <em>the integers are a subset of the rational numbers<\/em>. In fact that the integers are a <em>proper subset<\/em> of the rational numbers.<\/p>\r\n<\/section>","rendered":"<section class=\"textbox learningGoals\">\n<ul>\n<li>Understand the concepts of sets, including empty sets, subsets, and proper subsets, and use correct set notation<\/li>\n<li>Describe and perform set operations (union, intersection, complement, and difference) using proper set notation<\/li>\n<li>Create and interpret Venn diagrams to represent and analyze set relationships and operations<\/li>\n<li>Apply the concepts of sets, subsets, and cardinality properties to solve real-life problems<\/li>\n<\/ul>\n<\/section>\n<p>A movie lover might own a collection of movie posters, while a music lover might keep a collection of vinyl records. Any collection of items can form a <strong>set<\/strong>.<\/p>\n<section class=\"textbox keyTakeaway\">\n<div>\n<h3>set<\/h3>\n<p>A <strong>set<\/strong> is a collection of distinct objects, called <strong>elements<\/strong> of the set.<\/p>\n<p>&nbsp;<\/p>\n<p>A set can be defined by describing the contents, or by listing the elements of the set, enclosed in curly brackets separated by commas.<\/p>\n<p>&nbsp;<\/p>\n<p>Repeated elements in a set are only listed once. A set simply specifies the contents; order is not important.<\/p>\n<p>&nbsp;<\/p>\n<div style=\"text-align: center;\">The set represented by [latex]\\{1, 2, 3\\}[\/latex] is equivalent to the set [latex]\\{3, 1, 2\\}[\/latex].<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\">Some examples of sets defined by describing the contents:<\/p>\n<ol>\n<li>The set of all even numbers<\/li>\n<li>The set of all books written about travel to Chile<\/li>\n<\/ol>\n<p>Some examples of sets defined by listing the elements of the set:<\/p>\n<ol>\n<li>[latex]\\{1, 3, 9, 12\\}[\/latex]<\/li>\n<li>[latex]\\{\\text{red, orange, yellow, green, blue, indigo, purple}\\}[\/latex]<\/li>\n<\/ol>\n<\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm2199\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=2199&theme=lumen&iframe_resize_id=ohm2199&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm7414\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=7414&theme=lumen&iframe_resize_id=ohm7414&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section class=\"textbox keyTakeaway\">\n<div>\n<h3>set notation<\/h3>\n<p>Commonly, we will use a variable to represent a set, to make it easier to refer to that set later.<\/p>\n<p>&nbsp;<\/p>\n<p>The symbol [latex]\\in[\/latex]\u00a0means \u201cis an element of\u201d.<\/p>\n<p>&nbsp;<\/p>\n<p>A set that contains no elements, [latex]\\{ \\}[\/latex], is called the <strong>empty set<\/strong> or null set and is notated [latex]\\emptyset[\/latex].<\/p>\n<\/div>\n<\/section>\n<section class=\"textbox example\">Let [latex]A=\\{1,2,3,4\\}[\/latex]. To notate that [latex]2[\/latex] is element of the set, we&#8217;d write [latex]2 \\in A[\/latex].<\/section>\n<p>Sometimes a collection might not contain all the elements of a set. For example, Marta owns ninety-five Pok\u00e9mon cards. While Marta\u2019s collection is a set, we can also say it is a <strong>subset<\/strong> of the larger set of all Pok\u00e9mon cards.<\/p>\n<section class=\"textbox keyTakeaway\">\n<div>\n<h3>subset<\/h3>\n<p>A <strong>subset<\/strong> of a set [latex]A[\/latex] is a set that consists solely of elements drawn from set [latex]A[\/latex]. It may include all, some, or none of set [latex]A[\/latex]&#8216;s elements. In other words, every element in the subset is also an element of set [latex]A[\/latex].<\/p>\n<p>&nbsp;<\/p>\n<div style=\"text-align: center;\">If [latex]B[\/latex] is a subset of [latex]A[\/latex], we write [latex]B \\subseteq A[\/latex]<\/div>\n<p>&nbsp;<\/p>\n<p>A <strong>proper subset<\/strong> of a set [latex]A[\/latex] is a subset that contains some, but not all, of the elements of set [latex]A[\/latex]. It is a subset that is not identical to the original set, meaning it must contain fewer elements than set [latex]A[\/latex].<\/p>\n<p>&nbsp;<\/p>\n<div style=\"text-align: center;\">If [latex]B[\/latex] is a proper subset of [latex]A[\/latex], we write [latex]B \\subset A[\/latex]<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox seeExample\">Given the\u00a0set: [latex]A = \\{a, b, c, d\\}[\/latex]. List all of the subsets of [latex]A[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q417709\">Show Solution<\/button><\/p>\n<div id=\"q417709\" class=\"hidden-answer\" style=\"display: none\">\n<div style=\"text-align: center;\">[latex]\\{\\} (\\text{or } \\emptyset), \\{a\\}, \\{b\\}, \\{c\\}, \\{d\\}, \\{a,b\\}, \\{a,c\\}, \\{a,d\\}, \\{b,c\\}, \\{b,d\\}, \\{c,d\\}, \\{a,b,c\\},[\/latex][latex]\\{a,b,d\\}, \\{a,c,d\\}, \\{b,c,d\\}, \\{a,b,c,d\\}[\/latex]<\/div>\n<\/p>\n<p>You can see that there are [latex]16[\/latex] subsets, [latex]15[\/latex] of which are proper subsets.<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm2200\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=2200&theme=lumen&iframe_resize_id=ohm2200&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm2201\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=2201&theme=lumen&iframe_resize_id=ohm2201&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<p>Listing the sets is fine if you have only a few elements. However, if we were to list all of the subsets of a set containing many elements, it would be quite tedious. Instead, in the next example we will consider each element of the set separately.<\/p>\n<section class=\"textbox seeExample\">Given the\u00a0set: [latex]A = \\{a, b, c, d\\}[\/latex], there are four elements.<\/p>\n<p>For the first element, [latex]a[\/latex], either it\u2019s in the set or it\u2019s not. Thus there are [latex]2[\/latex] choices for that first element.<\/p>\n<p>Similarly, there are two choices for [latex]b[\/latex] \u2014 either it\u2019s in the set or it\u2019s not.<\/p>\n<p>Using just those two elements, list all the possible subsets of the set [latex]\\{a,b\\}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q417739\">Show Solution<\/button><\/p>\n<div id=\"q417739\" class=\"hidden-answer\" style=\"display: none\">\n<div style=\"text-align: center;\">[latex]\\{\\} (\\text{or } \\emptyset)[\/latex] \u2014 both elements are not in the set<br \/>\n[latex]\\{a\\}[\/latex] \u2014 [latex]a[\/latex] is in; [latex]b[\/latex] is not in the set<br \/>\n[latex]\\{b\\}[\/latex] \u2014 [latex]a[\/latex] is not in the set; [latex]b[\/latex] is in<br \/>\n[latex]\\{a,b\\}[\/latex] \u2014 [latex]a[\/latex] is in; [latex]b[\/latex] is in<\/div>\n<\/p>\n<p>Two choices for [latex]a[\/latex] times the two for [latex]b[\/latex] gives us [latex]2^2=4[\/latex] subsets.<\/p>\n<p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox recall\">\n<p><strong>Exponential Notation<\/strong><\/p>\n<p>Recall that the expression [latex]a^{n}[\/latex] states that some real number [latex]a[\/latex] is to be used as a factor [latex]n[\/latex] times.<\/p>\n<div style=\"text-align: center;\">Ex. [latex]2^{5} = 2 \\cdot 2 \\cdot 2 \\cdot 2 \\cdot 2 = 32[\/latex]<\/div>\n<\/section>\n<p>You may be asked to find the number of subsets and proper subsets of a given set. To do so, you need to look at the number of elements in the given set. In general, if you have [latex]n[\/latex] elements in your set, then there are [latex]2^{n}[\/latex] subsets and [latex]2^{n}\u22121[\/latex] proper subsets.<\/p>\n<section class=\"textbox keyTakeaway\">\n<div>\n<h3>number of subsets and proper subsets set<\/h3>\n<p>If you have [latex]n[\/latex] elements in your set:<\/p>\n<ul>\n<li>Number of subsets: [latex]2^{n}[\/latex]<\/li>\n<li>Number of proper subsets: [latex]2^{n}\u22121[\/latex]<\/li>\n<\/ul>\n<\/div>\n<\/section>\n<section class=\"textbox seeExample\">Determine the total number of subsets and proper subsets of the set [latex]\\{1,3,5,7\\}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q857946\">Show Solution<\/button><\/p>\n<div id=\"q857946\" class=\"hidden-answer\" style=\"display: none\">For this set, there are [latex]4[\/latex] elements which means [latex]n = 4[\/latex].To determine the number of subsets we must use [latex]2^n[\/latex]:<\/p>\n<div style=\"text-align: center;\">[latex]\\begin{array}{ccc}\\hfill 2^n = 2^4 = 16\\\\ \\end{array}[\/latex]<\/div>\n<\/p>\n<p>To determine the number of proper subsets we must use [latex]2^n - 1[\/latex]:<\/p>\n<div style=\"text-align: center;\">[latex]\\begin{array}{ccc}\\hfill 2^n - 1 = 2^4 - 1 = 16 - 1 = 15\\\\ \\end{array}[\/latex]<\/div>\n<\/p>\n<p>There are [latex]16[\/latex] subsets and [latex]15[\/latex] proper subsets.<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm2202\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=2202&theme=lumen&iframe_resize_id=ohm2202&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section class=\"textbox connectIt\">\n<p><strong>Subsets of real numbers<\/strong><\/p>\n<p>The idea of subsets can also be applied to the sets of real numbers. For example, the set of all whole numbers is a subset of the set of all integers. The set of integers in turn is contained within the set of rational numbers.<\/p>\n<p>We say <em>the integers are a subset of the rational numbers<\/em>. In fact that the integers are a <em>proper subset<\/em> of the rational numbers.<\/p>\n<\/section>\n","protected":false},"author":15,"menu_order":4,"template":"","meta":{"_candela_citation":"[{\"type\":\"original\",\"description\":\"Revision and Adaptation\",\"author\":\"\",\"organization\":\"Lumen Learning\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"Sets\",\"author\":\"David Lippman\",\"organization\":\"\",\"url\":\"http:\/\/www.opentextbookstore.com\/mathinsociety\/\",\"project\":\"Math in Society\",\"license\":\"cc-by-sa\",\"license_terms\":\"\"}]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":24,"module-header":"learn_it","content_attributions":[{"type":"original","description":"Revision and Adaptation","author":"","organization":"Lumen Learning","url":"","project":"","license":"cc-by","license_terms":""},{"type":"cc","description":"Sets","author":"David Lippman","organization":"","url":"http:\/\/www.opentextbookstore.com\/mathinsociety\/","project":"Math in Society","license":"cc-by-sa","license_terms":""}],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters\/27"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/users\/15"}],"version-history":[{"count":60,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters\/27\/revisions"}],"predecessor-version":[{"id":15008,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters\/27\/revisions\/15008"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/parts\/24"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters\/27\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/media?parent=27"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapter-type?post=27"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/contributor?post=27"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/license?post=27"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}