{"id":170,"date":"2016-01-25T22:05:39","date_gmt":"2016-01-25T22:05:39","guid":{"rendered":"https:\/\/courses.candelalearning.com\/math4libarts\/?post_type=chapter&#038;p=170"},"modified":"2021-02-05T23:51:17","modified_gmt":"2021-02-05T23:51:17","slug":"set-theory","status":"web-only","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/chapter\/set-theory\/","title":{"raw":"Set Theory","rendered":"Set Theory"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Describe memberships of sets and relationships between sets, including the empty set, subsets, and proper subset, while using correct set notation.<\/li>\r\n<\/ul>\r\n<\/div>\r\nAn art collector might own a collection of paintings, while a music lover might keep a collection of CDs. Any collection of items can form a <strong>set<\/strong>.\r\n<div class=\"textbox examples\">\r\n<h3>recall sets of real numbers<\/h3>\r\nRecall the sets of real numbers you studied previously. Each number contained in a set is an element of the set that contains it. For example, the number [latex]1[\/latex] is an element of the set of counting numbers. The number [latex]\\dfrac{2}{3}[\/latex] is an element of the set of rational numbers. And so on. The same idea applies to any set of distinct objects, as described below.\r\n\r\n<\/div>\r\n<div class=\"textbox\">\r\n<h3>Set<\/h3>\r\nA <strong>set<\/strong> is a collection of distinct objects, called <strong>elements<\/strong> of the set\r\n\r\nA set can be defined by describing the contents, or by listing the elements of the set, enclosed in curly brackets.\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nSome examples of sets defined by describing the contents:\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\nSome examples of sets defined by listing the elements of the set:\r\n<ol>\r\n \t<li>{1, 3, 9, 12}<\/li>\r\n \t<li>{red, orange, yellow, green, blue, indigo, purple}<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox\">\r\n<h3>Notation<\/h3>\r\nCommonly, we will use a variable to represent a set, to make it easier to refer to that set later.\r\n\r\nThe symbol \u2208\u00a0means \u201cis an element of\u201d.\r\n\r\nA set that contains no elements, { }, is called the <strong>empty set<\/strong> and is notated \u2205\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nLet <em>A <\/em>= {1, 2, 3, 4}\r\n\r\nTo notate that 2 is element of the set, we\u2019d write 2 \u2208 <em>A<\/em>\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Key Takeaways<\/h3>\r\n[ohm_question]125855[\/ohm_question]\r\n\r\n<\/div>\r\nA set simply specifies the contents; order is not important. The set represented by {1, 2, 3} is equivalent to the set {3, 1, 2}.\r\n<h2>Subsets<\/h2>\r\nSometimes a collection might not contain all the elements of a set. For example, Chris owns three Madonna albums. While Chris\u2019s collection is a set, we can also say it is a <strong>subset<\/strong> of the larger set of all Madonna albums.\r\n<div class=\"textbox examples\">\r\n<h3>Subsets of real numbers<\/h3>\r\nThe idea of subsets can also be applied to the sets of real numbers you studied previously. 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.\r\n\r\nWe say <em>the integers are a subset of the rational numbers<\/em>. You'll see below in fact that the integers are a <em>proper subset<\/em> of the rational numbers.\r\n\r\n<\/div>\r\n<div class=\"textbox\">\r\n<h3>Subset<\/h3>\r\nA <strong>subset<\/strong> of a set <em>A<\/em> is another set that contains only elements from the set <em>A<\/em>, but may not contain all the elements of <em>A<\/em>.\r\n\r\nIf <em>B<\/em> is a subset of <em>A<\/em>, we write <em>B<\/em> \u2286 <em>A<\/em>\r\n\r\nA <strong>proper subset<\/strong> is a subset that is not identical to the original set\u2014it contains fewer elements.\r\n\r\nIf <em>B<\/em> is a proper subset of <em>A<\/em>, we write <em>B<\/em> \u2282 <em>A<\/em>\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nConsider these three sets:\r\n\r\n<em>A<\/em> = the set of all even numbers\r\n<em>B<\/em> = {2, 4, 6}\r\n<em>C<\/em> = {2, 3, 4, 6}\r\n\r\nHere <em>B<\/em> \u2282 <em>A<\/em> since every element of <em>B<\/em> is also an even number, so is an element of <em>A<\/em>.\r\n\r\nMore formally, we could say <em>B<\/em> \u2282 <em>A<\/em> since if <em>x <\/em>\u2208\u00a0<em>B<\/em>, then <em>x <\/em>\u2208 <em>A<\/em>.\r\n\r\nIt is also true that <em>B<\/em> \u2282 <em>C<\/em>.\r\n\r\n<em>C<\/em> is not a subset of <em>A<\/em>, since C contains an element, 3, that is not contained in <em>A<\/em>\r\n\r\n<\/div>\r\nhttps:\/\/youtu.be\/5xthPHH4i_A?list=PL7138FAEC01D6F3F3\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nSuppose a set contains the plays \u201cMuch Ado About Nothing,\u201d \u201cMacBeth,\u201d and \u201cA Midsummer\u2019s Night Dream.\u201d What is a larger set this might be a subset of?\r\n[reveal-answer q=\"42047\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"42047\"]There are many possible answers here. One would be the set of plays by Shakespeare. This is also a subset of the set of all plays ever written. It is also a subset of all British literature.[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nConsider the\u00a0set\u00a0[latex]A = \\{1, 3, 5\\} [\/latex]. Which of the following sets is [latex]A [\/latex] a subset of?\r\n[latex]X = \\{1, 3, 7, 5\\} [\/latex]\r\n[latex]Y = \\{1, 3 \\} [\/latex]\r\n[latex]Z = \\{1, m, n, 3, 5\\}[\/latex]\r\n[reveal-answer q=\"3546\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"3546\"] [latex] X [\/latex] and [latex] Y [\/latex] [\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Exercises<\/h3>\r\nGiven the\u00a0set: <em>A<\/em> = {<em>a<\/em>, <em>b<\/em>, <em>c<\/em>, <em>d<\/em>}. List all of the subsets of <em>A\r\n<\/em>[reveal-answer q=\"706217\"]Show Solution[\/reveal-answer]<em>\r\n<\/em>[hidden-answer a=\"706217\"]{} (or \u00d8), {a}, {b}, {c}, {d}, {a,b}, {a,c}, {a,d}, {b,c}, {b,d}, {c,d}, {a,b,c}, {a,b,d}, {a,c,d},\u00a0{b,c,d}, {a,b,c,d}\r\n\r\nYou can see that there are 16 subsets, 15 of which are proper subsets.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nListing 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.\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nIn the previous example, there are four elements. For the first element, <em>a<\/em>, either it\u2019s in the set or it\u2019s not. Thus there are 2 choices for that first element. Similarly, there are two choices for <em>b<\/em>\u2014either it\u2019s in the set or it\u2019s not. Using just those two elements,\u00a0list all the possible subsets of the set {a,b}\r\n[reveal-answer q=\"857946\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"857946\"]\r\n\r\n{}\u2014both elements are not in the set\r\n{<em>a<\/em>}\u2014<em>a<\/em> is in;\u00a0<em>b<\/em> is not in the set\r\n{<em>b<\/em>}\u2014<em>a<\/em> is not in the set;\u00a0<em>b<\/em> is in\r\n{<em>a<\/em>,<em>b<\/em>}\u2014<em>a<\/em> is in; <em>b<\/em> is in\r\n\r\nTwo choices for <em>a<\/em>\u00a0times the two for <em>b<\/em>\u00a0gives us [latex]2^{2}=4[\/latex] subsets.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox examples\">\r\n<h3>Recall exponential notation<\/h3>\r\n<div>Recall that the expression [latex]a^{m}[\/latex] states that some real number [latex]a[\/latex] is to be used as a factor [latex]m[\/latex] times.<\/div>\r\n<div><\/div>\r\n<div>Ex. [latex]2^{5} = 2 \\cdot 2 \\cdot 2 \\cdot 2 \\cdot 2 = 32[\/latex]<\/div>\r\n<\/div>\r\nNow let\u2019s include <em>c,\u00a0<\/em>just for fun. List all the possible subsets of the new set {a,b,c}.\r\nAgain, either <em>c<\/em> is included or it isn\u2019t, which gives us two choices. The outcomes are {}, {<em>a<\/em>}, {<em>b<\/em>}, {<em>c<\/em>}, {<em>a<\/em>,<em>b<\/em>}, {<em>a<\/em>,<em>c<\/em>}, {<em>b<\/em>,<em>c<\/em>}, {<em>a<\/em>,<em>b<\/em>,<em>c<\/em>}. Note that there are [latex]2^{3}=8[\/latex] subsets.\r\n\r\nIf you include\u00a0four elements, there would be [latex]2^{4}=16[\/latex] subsets. 15 of those subsets are proper, 1 subset, namely {<em>a<\/em>,<em>b<\/em>,<em>c<\/em>,<em>d<\/em>}, is not.\r\n\r\nIn general, if you have <em>n<\/em> elements in your set, then there are [latex]2^{n}[\/latex] subsets and [latex]2^{n}\u22121[\/latex] proper subsets.\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n[ohm_question]132343[\/ohm_question]\r\n\r\n<\/div>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Describe memberships of sets and relationships between sets, including the empty set, subsets, and proper subset, while using correct set notation.<\/li>\n<\/ul>\n<\/div>\n<p>An art collector might own a collection of paintings, while a music lover might keep a collection of CDs. Any collection of items can form a <strong>set<\/strong>.<\/p>\n<div class=\"textbox examples\">\n<h3>recall sets of real numbers<\/h3>\n<p>Recall the sets of real numbers you studied previously. Each number contained in a set is an element of the set that contains it. For example, the number [latex]1[\/latex] is an element of the set of counting numbers. The number [latex]\\dfrac{2}{3}[\/latex] is an element of the set of rational numbers. And so on. The same idea applies to any set of distinct objects, as described below.<\/p>\n<\/div>\n<div class=\"textbox\">\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>A set can be defined by describing the contents, or by listing the elements of the set, enclosed in curly brackets.<\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>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>{1, 3, 9, 12}<\/li>\n<li>{red, orange, yellow, green, blue, indigo, purple}<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox\">\n<h3>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>The symbol \u2208\u00a0means \u201cis an element of\u201d.<\/p>\n<p>A set that contains no elements, { }, is called the <strong>empty set<\/strong> and is notated \u2205<\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Let <em>A <\/em>= {1, 2, 3, 4}<\/p>\n<p>To notate that 2 is element of the set, we\u2019d write 2 \u2208 <em>A<\/em><\/p>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Key Takeaways<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm125855\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=125855&theme=oea&iframe_resize_id=ohm125855&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>A set simply specifies the contents; order is not important. The set represented by {1, 2, 3} is equivalent to the set {3, 1, 2}.<\/p>\n<h2>Subsets<\/h2>\n<p>Sometimes a collection might not contain all the elements of a set. For example, Chris owns three Madonna albums. While Chris\u2019s collection is a set, we can also say it is a <strong>subset<\/strong> of the larger set of all Madonna albums.<\/p>\n<div class=\"textbox examples\">\n<h3>Subsets of real numbers<\/h3>\n<p>The idea of subsets can also be applied to the sets of real numbers you studied previously. 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>. You&#8217;ll see below in fact that the integers are a <em>proper subset<\/em> of the rational numbers.<\/p>\n<\/div>\n<div class=\"textbox\">\n<h3>Subset<\/h3>\n<p>A <strong>subset<\/strong> of a set <em>A<\/em> is another set that contains only elements from the set <em>A<\/em>, but may not contain all the elements of <em>A<\/em>.<\/p>\n<p>If <em>B<\/em> is a subset of <em>A<\/em>, we write <em>B<\/em> \u2286 <em>A<\/em><\/p>\n<p>A <strong>proper subset<\/strong> is a subset that is not identical to the original set\u2014it contains fewer elements.<\/p>\n<p>If <em>B<\/em> is a proper subset of <em>A<\/em>, we write <em>B<\/em> \u2282 <em>A<\/em><\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Consider these three sets:<\/p>\n<p><em>A<\/em> = the set of all even numbers<br \/>\n<em>B<\/em> = {2, 4, 6}<br \/>\n<em>C<\/em> = {2, 3, 4, 6}<\/p>\n<p>Here <em>B<\/em> \u2282 <em>A<\/em> since every element of <em>B<\/em> is also an even number, so is an element of <em>A<\/em>.<\/p>\n<p>More formally, we could say <em>B<\/em> \u2282 <em>A<\/em> since if <em>x <\/em>\u2208\u00a0<em>B<\/em>, then <em>x <\/em>\u2208 <em>A<\/em>.<\/p>\n<p>It is also true that <em>B<\/em> \u2282 <em>C<\/em>.<\/p>\n<p><em>C<\/em> is not a subset of <em>A<\/em>, since C contains an element, 3, that is not contained in <em>A<\/em><\/p>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Sets: basics\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/5xthPHH4i_A?list=PL7138FAEC01D6F3F3\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Suppose a set contains the plays \u201cMuch Ado About Nothing,\u201d \u201cMacBeth,\u201d and \u201cA Midsummer\u2019s Night Dream.\u201d What is a larger set this might be a subset of?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q42047\">Show Solution<\/span><\/p>\n<div id=\"q42047\" class=\"hidden-answer\" style=\"display: none\">There are many possible answers here. One would be the set of plays by Shakespeare. This is also a subset of the set of all plays ever written. It is also a subset of all British literature.<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Consider the\u00a0set\u00a0[latex]A = \\{1, 3, 5\\}[\/latex]. Which of the following sets is [latex]A[\/latex] a subset of?<br \/>\n[latex]X = \\{1, 3, 7, 5\\}[\/latex]<br \/>\n[latex]Y = \\{1, 3 \\}[\/latex]<br \/>\n[latex]Z = \\{1, m, n, 3, 5\\}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q3546\">Show Solution<\/span><\/p>\n<div id=\"q3546\" class=\"hidden-answer\" style=\"display: none\"> [latex]X[\/latex] and [latex]Y[\/latex] <\/div>\n<\/div>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Exercises<\/h3>\n<p>Given the\u00a0set: <em>A<\/em> = {<em>a<\/em>, <em>b<\/em>, <em>c<\/em>, <em>d<\/em>}. List all of the subsets of <em>A<br \/>\n<\/em><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q706217\">Show Solution<\/span><em><br \/>\n<\/em><\/p>\n<div id=\"q706217\" class=\"hidden-answer\" style=\"display: none\">{} (or \u00d8), {a}, {b}, {c}, {d}, {a,b}, {a,c}, {a,d}, {b,c}, {b,d}, {c,d}, {a,b,c}, {a,b,d}, {a,c,d},\u00a0{b,c,d}, {a,b,c,d}<\/p>\n<p>You can see that there are 16 subsets, 15 of which are proper subsets.<\/p>\n<\/div>\n<\/div>\n<\/div>\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<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>In the previous example, there are four elements. For the first element, <em>a<\/em>, either it\u2019s in the set or it\u2019s not. Thus there are 2 choices for that first element. Similarly, there are two choices for <em>b<\/em>\u2014either it\u2019s in the set or it\u2019s not. Using just those two elements,\u00a0list all the possible subsets of the set {a,b}<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q857946\">Show Solution<\/span><\/p>\n<div id=\"q857946\" class=\"hidden-answer\" style=\"display: none\">\n<p>{}\u2014both elements are not in the set<br \/>\n{<em>a<\/em>}\u2014<em>a<\/em> is in;\u00a0<em>b<\/em> is not in the set<br \/>\n{<em>b<\/em>}\u2014<em>a<\/em> is not in the set;\u00a0<em>b<\/em> is in<br \/>\n{<em>a<\/em>,<em>b<\/em>}\u2014<em>a<\/em> is in; <em>b<\/em> is in<\/p>\n<p>Two choices for <em>a<\/em>\u00a0times the two for <em>b<\/em>\u00a0gives us [latex]2^{2}=4[\/latex] subsets.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox examples\">\n<h3>Recall exponential notation<\/h3>\n<div>Recall that the expression [latex]a^{m}[\/latex] states that some real number [latex]a[\/latex] is to be used as a factor [latex]m[\/latex] times.<\/div>\n<div><\/div>\n<div>Ex. [latex]2^{5} = 2 \\cdot 2 \\cdot 2 \\cdot 2 \\cdot 2 = 32[\/latex]<\/div>\n<\/div>\n<p>Now let\u2019s include <em>c,\u00a0<\/em>just for fun. List all the possible subsets of the new set {a,b,c}.<br \/>\nAgain, either <em>c<\/em> is included or it isn\u2019t, which gives us two choices. The outcomes are {}, {<em>a<\/em>}, {<em>b<\/em>}, {<em>c<\/em>}, {<em>a<\/em>,<em>b<\/em>}, {<em>a<\/em>,<em>c<\/em>}, {<em>b<\/em>,<em>c<\/em>}, {<em>a<\/em>,<em>b<\/em>,<em>c<\/em>}. Note that there are [latex]2^{3}=8[\/latex] subsets.<\/p>\n<p>If you include\u00a0four elements, there would be [latex]2^{4}=16[\/latex] subsets. 15 of those subsets are proper, 1 subset, namely {<em>a<\/em>,<em>b<\/em>,<em>c<\/em>,<em>d<\/em>}, is not.<\/p>\n<p>In general, if you have <em>n<\/em> elements in your set, then there are [latex]2^{n}[\/latex] subsets and [latex]2^{n}\u22121[\/latex] proper subsets.<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm132343\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=132343&theme=oea&iframe_resize_id=ohm132343&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\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-170\">\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>Sets. <strong>Authored by<\/strong>: David Lippman. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/www.opentextbookstore.com\/mathinsociety\/\">http:\/\/www.opentextbookstore.com\/mathinsociety\/<\/a>. <strong>Project<\/strong>: Math in Society. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by-sa\/4.0\/\">CC BY-SA: Attribution-ShareAlike<\/a><\/em><\/li><li>Question ID 132343. <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><li>Question ID 125855. <strong>Authored by<\/strong>: Bohart, Jenifer. <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":276,"menu_order":14,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Sets\",\"author\":\"David Lippman\",\"organization\":\"\",\"url\":\"http:\/\/www.opentextbookstore.com\/mathinsociety\/\",\"project\":\"Math in Society\",\"license\":\"cc-by-sa\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"Question ID 132343\",\"author\":\"\",\"organization\":\"lumen 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