{"id":6782,"date":"2021-12-30T21:30:01","date_gmt":"2021-12-30T21:30:01","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/frontrange-mathforliberalartscorequisite1\/?post_type=chapter&#038;p=6782"},"modified":"2022-01-06T22:15:51","modified_gmt":"2022-01-06T22:15:51","slug":"1-3-set-operations","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/frontrange-mathforliberalartscorequisite1\/chapter\/1-3-set-operations\/","title":{"raw":"1.3 Set Operations","rendered":"1.3 Set Operations"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3 style=\"padding-left: 30px;\">Learning Objectives<\/h3>\r\n<ul>\r\n \t<li>Universal Set<\/li>\r\n \t<li>Using sets and proper notation, perform the operations of\r\n<ul>\r\n \t<li>complement<\/li>\r\n \t<li>union<\/li>\r\n \t<li>intersection<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li>Order of Set Operations<\/li>\r\n \t<li>Cardinality of Set Operations<\/li>\r\n \t<li>Equal and Equivalent Sets<\/li>\r\n<\/ul>\r\n<\/div>\r\nCommonly, sets interact. For example, you and a new roommate decide to have a house party, and you both invite your circle of friends. At this party, two sets are being combined, though it might turn out that there are some friends that were in both sets.\r\n<div class=\"textbox examples\">\r\n<h3>Math vocabulary and notation<\/h3>\r\nIt takes repetition and practice to obtain new vocabulary in a language that is new to you. Math is no different in many respects than learning a new language, with all its vocabulary, syntax, and spelling conventions. The symbols in this section may be completely unfamiliar to you. If so, you'll need to spend time with them, employing flashcards and writing them out by hand in context.\r\n\r\nGive yourself time to learn and appreciate the language of mathematics!\r\n\r\n<\/div>\r\n<div><\/div>\r\n<div><\/div>\r\n<div class=\"textbox\">\r\n<h3>Universal Set<\/h3>\r\n<span style=\"background-color: initial; font-size: 0.9em;\">A <\/span><strong style=\"background-color: initial; font-size: 0.9em;\">universal set<\/strong><span style=\"background-color: initial; font-size: 0.9em;\"> is a set that contains all the elements we are interested in. This would have to be defined by the context.<\/span>\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\n<ol>\r\n \t<li>If we were discussing searching for books, the universal set might be all the books in the library.<\/li>\r\n \t<li>If we were grouping your Facebook friends, the universal set would be all your Facebook friends.<\/li>\r\n \t<li>If you were working with sets of numbers, the universal set might be all whole numbers, all integers, or all real numbers<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox\">\r\n<h3>Complement<\/h3>\r\nThe <strong>complement<\/strong> of a set <em>A<\/em> contains everything that is <em>not<\/em> in the set <em>A<\/em>.\u00a0The complement is notated <em>A\u2019.<\/em>\r\n\r\nA complement is relative to the universal set, so\u00a0<em>A'<\/em><em>\u00a0<\/em>contains all the elements in the universal set that are not in <em>A<\/em>.\r\n\r\n<\/div>\r\n<div class=\"textbox examples\">\r\n<h3>a new use for a superscript<\/h3>\r\nNotice in the descriptions of the notation introduced above that the\u00a0<strong>complement<\/strong> of a set is denoted A\u2032. This superscript is\u00a0not an exponent, it is a superscript (superscripts can be for variables). It is a symbol that denotes\u00a0<strong>the complement of a set<\/strong>.\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nSuppose the universal set is <em>U<\/em> = all whole numbers from 1 to 9. If <em>A<\/em> = {1, 2, 4}, then\u00a0<em>A' <\/em>= {3, 5, 6, 7, 8, 9}.\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n[ohm_question]238270[\/ohm_question]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox\">\r\n<h3>Union and Intersection<\/h3>\r\nThe <strong>union<\/strong> of two sets contains all the elements contained in either set (or both sets).\u00a0The union is notated <em>A <\/em>\u22c3<em> B.\u00a0<\/em>More formally, <em>x <\/em>\u220a <em>A <\/em>\u22c3 <em>B<\/em> if <em>x <\/em>\u2208 <em>A<\/em> or <em>x <\/em>\u2208 <em>B<\/em> (or both)\r\n\r\nThe <strong>intersection <\/strong>of two sets contains only the elements that are in both sets.\u00a0The intersection is notated <em>A <\/em>\u22c2<em> B.\u00a0<\/em>More formally, <em>x <\/em>\u2208 <em>A <\/em>\u22c2 <em>B<\/em> if <em>x <\/em>\u2208 <em>A<\/em> and <em>x <\/em>\u2208 <em>B.<\/em>\r\n\r\n&nbsp;\r\n\r\n<\/div>\r\n<div class=\"textbox examples\">\r\n<h3>intersection and union symbols<\/h3>\r\nThe intersection [latex]\\cap[\/latex]\u00a0and union [latex]\\cup[\/latex] symbols look a little like letters in the alphabet. In fact, that's a trick for remembering them.\r\n\r\nThe union symbol looks like\u00a0 a capital U, for\u00a0<em>union<\/em>.\r\n\r\nThe intersection symbol looks a little like a big lower-case n, for <em>in-tersect<\/em>\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nConsider the sets:\r\n\r\n<em>A<\/em> = {red, green, blue}\r\n<em>B<\/em> = {red, yellow, orange}\r\n<em>C<\/em> = {red, orange, yellow, green, blue, purple}\r\n\r\nFind the following:\r\n<ol>\r\n \t<li>Find <em>A <\/em>\u22c3<em> B<\/em><\/li>\r\n \t<li>Find <em>A <\/em>\u22c2<em> B<\/em><\/li>\r\n \t<li>Find <em>A'<\/em>\u22c2<em> C<\/em><\/li>\r\n<\/ol>\r\n[reveal-answer q=\"691926\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"691926\"]\r\n<ol>\r\n \t<li>The union contains all the elements in either set: <em>A <\/em>\u22c3<em> B<\/em> = {red, green, blue, yellow, orange}\u00a0Notice we only list red once.<\/li>\r\n \t<li>The intersection contains all the elements in both sets: <em>A <\/em>\u22c2<em> B<\/em> = {red}<\/li>\r\n \t<li>Here we\u2019re looking for all the elements that are <em>not<\/em> in set <em>A<\/em> and are also in <em>C<\/em>.\u00a0<em>A' <\/em>\u22c2<em> C<\/em> = {orange, yellow, purple}<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\nNotice that in the example above, it would be hard to just ask for <em>A'<\/em>, since everything from the color fuchsia to puppies and peanut butter are included in the complement of the set. For this reason, complements are usually only used with intersections, or when we have a universal set in place.\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n[ohm_question]125865[\/ohm_question]\r\n\r\n<\/div>\r\n&nbsp;\r\n<h2>Order of Set Operations<\/h2>\r\nAs we saw earlier with the expression\u00a0<em>A'<\/em><em>\u00a0<\/em>\u22c2<em> C<\/em>, set operations can be grouped together. Grouping symbols can be used like they are with arithmetic \u2013 to force an order of operations.\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nSuppose <em>H<\/em> = {cat, dog, rabbit, mouse}, <em>F<\/em> = {dog, cow, duck, pig, rabbit}, and\u00a0<em>W<\/em> = {duck, rabbit, deer, frog, mouse}\r\n<ol>\r\n \t<li>Find (<em>H <\/em>\u22c2<em> F<\/em>) \u22c3<em> W<\/em><\/li>\r\n \t<li>Find <em>H <\/em>\u22c2 (<em>F<\/em> \u22c3<em> W<\/em>)<\/li>\r\n \t<li>Find (<em>H <\/em>\u22c2<em> F<\/em>)' \u22c2<em> W<\/em><\/li>\r\n<\/ol>\r\n[reveal-answer q=\"444204\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"444204\"]\r\n<ol>\r\n \t<li>We start with the intersection: <em>H <\/em>\u22c2<em> F<\/em> = {dog, rabbit}.\u00a0Now we union that result with <em>W<\/em>: (<em>H <\/em>\u22c2<em> F<\/em>) \u22c3<em> W<\/em> = {dog, duck, rabbit, deer, frog, mouse}<\/li>\r\n \t<li>We start with the union: <em>F<\/em> \u22c3<em> W<\/em> = {dog, cow, rabbit, duck, pig, deer, frog, mouse}.\u00a0Now we intersect that result with <em>H<\/em>: <em>H <\/em>\u22c2 (<em>F<\/em> \u22c3<em> W<\/em>) = {dog, rabbit, mouse}<\/li>\r\n \t<li>We start with the intersection: <em>H <\/em>\u22c2<em> F<\/em> = {dog, rabbit}.\u00a0Now we want to find the elements of <em>W<\/em> that are <em>not<\/em> in <em>H <\/em>\u22c2<em> F.\u00a0<\/em>(<em>H <\/em>\u22c2<em> F)'<\/em>\u00a0\u22c2<em> W<\/em> = {duck, deer, frog, mouse}<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<h2><\/h2>\r\n<h2>How does Cardinality apply to Set Operations<\/h2>\r\n<div class=\"textbox exercises\">\r\n<h3>Exercises<\/h3>\r\nLet <em>A<\/em> = {1, 2, 3, 4, 5, 6} and <em>B<\/em> = {2, 4, 6, 8}.\r\n\r\nWhat is the cardinality of <em>B<\/em>? <em>A<\/em> \u22c3<em> B<\/em>, <em>A <\/em>\u22c2<em> B<\/em>?\r\n[reveal-answer q=\"100844\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"100844\"]\r\n\r\nThe cardinality of <em>B<\/em> is 4, since there are 4 elements in the set.\r\n\r\nThe cardinality of <em>A<\/em> \u22c3<em> B<\/em> is 7, since <em>A<\/em> \u22c3<em> B<\/em> = {1, 2, 3, 4, 5, 6, 8}, which contains 7 elements.\r\n\r\nThe cardinality of <em>A <\/em>\u22c2<em> B<\/em> is 3, since <em>A <\/em>\u22c2<em> B<\/em> = {2, 4, 6}, which contains 3 elements.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n[ohm_question]109846[\/ohm_question]\r\n\r\n<\/div>\r\nhttps:\/\/youtu.be\/wErcETeKvrU\r\n<h2>Equal and Equivalent Sets<\/h2>\r\nTwo sets are <strong>equal<\/strong> when they have the same elements. They might not be in the same order.\r\n\r\nFor example: {pink, purple, blue, green} is equal to {green, blue, purple, pink}\r\n\r\nTwo sets are <strong>equivalent<\/strong> when they have the same number of elements, not necessarily the same elements.\r\n\r\nFor example: {June, July, August} is equivalent to {4,8,9} because they each have 3 elements.\r\n<p style=\"text-align: center;\"><strong><span style=\"color: #ff0000;\">This is the end of the section. Close this tab and proceed to the corresponding assignment.<\/span><\/strong><\/p>","rendered":"<div class=\"textbox learning-objectives\">\n<h3 style=\"padding-left: 30px;\">Learning Objectives<\/h3>\n<ul>\n<li>Universal Set<\/li>\n<li>Using sets and proper notation, perform the operations of\n<ul>\n<li>complement<\/li>\n<li>union<\/li>\n<li>intersection<\/li>\n<\/ul>\n<\/li>\n<li>Order of Set Operations<\/li>\n<li>Cardinality of Set Operations<\/li>\n<li>Equal and Equivalent Sets<\/li>\n<\/ul>\n<\/div>\n<p>Commonly, sets interact. For example, you and a new roommate decide to have a house party, and you both invite your circle of friends. At this party, two sets are being combined, though it might turn out that there are some friends that were in both sets.<\/p>\n<div class=\"textbox examples\">\n<h3>Math vocabulary and notation<\/h3>\n<p>It takes repetition and practice to obtain new vocabulary in a language that is new to you. Math is no different in many respects than learning a new language, with all its vocabulary, syntax, and spelling conventions. The symbols in this section may be completely unfamiliar to you. If so, you&#8217;ll need to spend time with them, employing flashcards and writing them out by hand in context.<\/p>\n<p>Give yourself time to learn and appreciate the language of mathematics!<\/p>\n<\/div>\n<div><\/div>\n<div><\/div>\n<div class=\"textbox\">\n<h3>Universal Set<\/h3>\n<p><span style=\"background-color: initial; font-size: 0.9em;\">A <\/span><strong style=\"background-color: initial; font-size: 0.9em;\">universal set<\/strong><span style=\"background-color: initial; font-size: 0.9em;\"> is a set that contains all the elements we are interested in. This would have to be defined by the context.<\/span><\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<ol>\n<li>If we were discussing searching for books, the universal set might be all the books in the library.<\/li>\n<li>If we were grouping your Facebook friends, the universal set would be all your Facebook friends.<\/li>\n<li>If you were working with sets of numbers, the universal set might be all whole numbers, all integers, or all real numbers<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox\">\n<h3>Complement<\/h3>\n<p>The <strong>complement<\/strong> of a set <em>A<\/em> contains everything that is <em>not<\/em> in the set <em>A<\/em>.\u00a0The complement is notated <em>A\u2019.<\/em><\/p>\n<p>A complement is relative to the universal set, so\u00a0<em>A&#8217;<\/em><em>\u00a0<\/em>contains all the elements in the universal set that are not in <em>A<\/em>.<\/p>\n<\/div>\n<div class=\"textbox examples\">\n<h3>a new use for a superscript<\/h3>\n<p>Notice in the descriptions of the notation introduced above that the\u00a0<strong>complement<\/strong> of a set is denoted A\u2032. This superscript is\u00a0not an exponent, it is a superscript (superscripts can be for variables). It is a symbol that denotes\u00a0<strong>the complement of a set<\/strong>.<\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Suppose the universal set is <em>U<\/em> = all whole numbers from 1 to 9. If <em>A<\/em> = {1, 2, 4}, then\u00a0<em>A&#8217; <\/em>= {3, 5, 6, 7, 8, 9}.<\/p>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm238270\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=238270&theme=oea&iframe_resize_id=ohm238270&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox\">\n<h3>Union and Intersection<\/h3>\n<p>The <strong>union<\/strong> of two sets contains all the elements contained in either set (or both sets).\u00a0The union is notated <em>A <\/em>\u22c3<em> B.\u00a0<\/em>More formally, <em>x <\/em>\u220a <em>A <\/em>\u22c3 <em>B<\/em> if <em>x <\/em>\u2208 <em>A<\/em> or <em>x <\/em>\u2208 <em>B<\/em> (or both)<\/p>\n<p>The <strong>intersection <\/strong>of two sets contains only the elements that are in both sets.\u00a0The intersection is notated <em>A <\/em>\u22c2<em> B.\u00a0<\/em>More formally, <em>x <\/em>\u2208 <em>A <\/em>\u22c2 <em>B<\/em> if <em>x <\/em>\u2208 <em>A<\/em> and <em>x <\/em>\u2208 <em>B.<\/em><\/p>\n<p>&nbsp;<\/p>\n<\/div>\n<div class=\"textbox examples\">\n<h3>intersection and union symbols<\/h3>\n<p>The intersection [latex]\\cap[\/latex]\u00a0and union [latex]\\cup[\/latex] symbols look a little like letters in the alphabet. In fact, that&#8217;s a trick for remembering them.<\/p>\n<p>The union symbol looks like\u00a0 a capital U, for\u00a0<em>union<\/em>.<\/p>\n<p>The intersection symbol looks a little like a big lower-case n, for <em>in-tersect<\/em><\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Consider the sets:<\/p>\n<p><em>A<\/em> = {red, green, blue}<br \/>\n<em>B<\/em> = {red, yellow, orange}<br \/>\n<em>C<\/em> = {red, orange, yellow, green, blue, purple}<\/p>\n<p>Find the following:<\/p>\n<ol>\n<li>Find <em>A <\/em>\u22c3<em> B<\/em><\/li>\n<li>Find <em>A <\/em>\u22c2<em> B<\/em><\/li>\n<li>Find <em>A&#8217;<\/em>\u22c2<em> C<\/em><\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q691926\">Show Solution<\/span><\/p>\n<div id=\"q691926\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>The union contains all the elements in either set: <em>A <\/em>\u22c3<em> B<\/em> = {red, green, blue, yellow, orange}\u00a0Notice we only list red once.<\/li>\n<li>The intersection contains all the elements in both sets: <em>A <\/em>\u22c2<em> B<\/em> = {red}<\/li>\n<li>Here we\u2019re looking for all the elements that are <em>not<\/em> in set <em>A<\/em> and are also in <em>C<\/em>.\u00a0<em>A&#8217; <\/em>\u22c2<em> C<\/em> = {orange, yellow, purple}<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<p>Notice that in the example above, it would be hard to just ask for <em>A&#8217;<\/em>, since everything from the color fuchsia to puppies and peanut butter are included in the complement of the set. For this reason, complements are usually only used with intersections, or when we have a universal set in place.<\/p>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm125865\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=125865&theme=oea&iframe_resize_id=ohm125865&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<h2>Order of Set Operations<\/h2>\n<p>As we saw earlier with the expression\u00a0<em>A&#8217;<\/em><em>\u00a0<\/em>\u22c2<em> C<\/em>, set operations can be grouped together. Grouping symbols can be used like they are with arithmetic \u2013 to force an order of operations.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Suppose <em>H<\/em> = {cat, dog, rabbit, mouse}, <em>F<\/em> = {dog, cow, duck, pig, rabbit}, and\u00a0<em>W<\/em> = {duck, rabbit, deer, frog, mouse}<\/p>\n<ol>\n<li>Find (<em>H <\/em>\u22c2<em> F<\/em>) \u22c3<em> W<\/em><\/li>\n<li>Find <em>H <\/em>\u22c2 (<em>F<\/em> \u22c3<em> W<\/em>)<\/li>\n<li>Find (<em>H <\/em>\u22c2<em> F<\/em>)&#8217; \u22c2<em> W<\/em><\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q444204\">Show Solution<\/span><\/p>\n<div id=\"q444204\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>We start with the intersection: <em>H <\/em>\u22c2<em> F<\/em> = {dog, rabbit}.\u00a0Now we union that result with <em>W<\/em>: (<em>H <\/em>\u22c2<em> F<\/em>) \u22c3<em> W<\/em> = {dog, duck, rabbit, deer, frog, mouse}<\/li>\n<li>We start with the union: <em>F<\/em> \u22c3<em> W<\/em> = {dog, cow, rabbit, duck, pig, deer, frog, mouse}.\u00a0Now we intersect that result with <em>H<\/em>: <em>H <\/em>\u22c2 (<em>F<\/em> \u22c3<em> W<\/em>) = {dog, rabbit, mouse}<\/li>\n<li>We start with the intersection: <em>H <\/em>\u22c2<em> F<\/em> = {dog, rabbit}.\u00a0Now we want to find the elements of <em>W<\/em> that are <em>not<\/em> in <em>H <\/em>\u22c2<em> F.\u00a0<\/em>(<em>H <\/em>\u22c2<em> F)&#8217;<\/em>\u00a0\u22c2<em> W<\/em> = {duck, deer, frog, mouse}<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<h2><\/h2>\n<h2>How does Cardinality apply to Set Operations<\/h2>\n<div class=\"textbox exercises\">\n<h3>Exercises<\/h3>\n<p>Let <em>A<\/em> = {1, 2, 3, 4, 5, 6} and <em>B<\/em> = {2, 4, 6, 8}.<\/p>\n<p>What is the cardinality of <em>B<\/em>? <em>A<\/em> \u22c3<em> B<\/em>, <em>A <\/em>\u22c2<em> B<\/em>?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q100844\">Show Solution<\/span><\/p>\n<div id=\"q100844\" class=\"hidden-answer\" style=\"display: none\">\n<p>The cardinality of <em>B<\/em> is 4, since there are 4 elements in the set.<\/p>\n<p>The cardinality of <em>A<\/em> \u22c3<em> B<\/em> is 7, since <em>A<\/em> \u22c3<em> B<\/em> = {1, 2, 3, 4, 5, 6, 8}, which contains 7 elements.<\/p>\n<p>The cardinality of <em>A <\/em>\u22c2<em> B<\/em> is 3, since <em>A <\/em>\u22c2<em> B<\/em> = {2, 4, 6}, which contains 3 elements.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm109846\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=109846&theme=oea&iframe_resize_id=ohm109846&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Sets: cardinality\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/wErcETeKvrU?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h2>Equal and Equivalent Sets<\/h2>\n<p>Two sets are <strong>equal<\/strong> when they have the same elements. They might not be in the same order.<\/p>\n<p>For example: {pink, purple, blue, green} is equal to {green, blue, purple, pink}<\/p>\n<p>Two sets are <strong>equivalent<\/strong> when they have the same number of elements, not necessarily the same elements.<\/p>\n<p>For example: {June, July, August} is equivalent to {4,8,9} because they each have 3 elements.<\/p>\n<p style=\"text-align: center;\"><strong><span style=\"color: #ff0000;\">This is the end of the section. 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