{"id":28,"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-2\/"},"modified":"2024-10-18T20:50:09","modified_gmt":"2024-10-18T20:50:09","slug":"set-theory-basics-learn-it-2","status":"web-only","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/chapter\/set-theory-basics-learn-it-2\/","title":{"raw":"Set Theory Basics: Learn It 2","rendered":"Set Theory Basics: Learn It 2"},"content":{"raw":"<h2>Set Operations<\/h2>\r\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>\r\n<section class=\"textbox keyTakeaway\">\r\n<div>\r\n<h3>set operations: union, intersection, complement, and difference<\/h3>\r\n<ul>\r\n\t<li>The <strong>union<\/strong> of two sets contains all the elements contained in either set (or both sets).\r\n\r\n\r\n<ul>\r\n\t<li>The union is notated [latex]A \\cup B[\/latex].\u00a0More formally, [latex]x \\in A \\cup B[\/latex] if [latex]x \\in A[\/latex] or [latex]x \\in B[\/latex] (or both).<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li>The <strong>intersection <\/strong>of two sets contains only the elements that are in both sets.\r\n\r\n\r\n<ul>\r\n\t<li>The intersection is notated [latex]A \\cap B[\/latex]. More formally, [latex]x \\in A \\cap B[\/latex] if [latex]x \\in A[\/latex] and [latex]x \\in B[\/latex].<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li>The <strong>complement<\/strong> of a set [latex]A[\/latex] contains everything that is <em>not<\/em> in the [latex]A[\/latex].\r\n\r\n\r\n<ul>\r\n\t<li>The complement is notated [latex]A'[\/latex], or [latex]A^c[\/latex], or sometimes \u00a0~[latex]A[\/latex].<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li>The <strong>difference<\/strong> of two sets is the list of all the elements that are in one set but not present in the other.\r\n\r\n\r\n<ul>\r\n\t<li>The difference between two sets is notated [latex]A \\setminus B[\/latex]. More formally, [latex]x \\in A \\setminus B[\/latex] if [latex]x \\in A[\/latex] &amp; [latex]x \\notin B [\/latex].<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/div>\r\n<\/section>\r\n<section class=\"textbox proTip\">Notice in the descriptions of the notation introduced above that the\u00a0<strong>complement<\/strong> of a set is denoted [latex]A^{c}[\/latex]. This superscript is\u00a0not an exponent. It is a decoration that denotes\u00a0<strong>the complement of a set<\/strong>.<\/section>\r\n<p>Let\u2019s try applying the set operations.<\/p>\r\n<section class=\"textbox seeExample\">Consider the sets:\r\n\r\n\r\n<ul>\r\n\t<li>[latex]A = \\{\\text{red, green, blue}\\}[\/latex]<\/li>\r\n\t<li>[latex]B = \\{\\text{red, yellow, orange}\\}[\/latex]<\/li>\r\n\t<li>[latex]C = \\{\\text{red, orange, yellow, green, blue, purple}\\}[\/latex]<\/li>\r\n<\/ul>\r\n\r\n\r\nFind the following:\r\n\r\n\r\n<ol>\r\n\t<li>Find [latex]A \\cup B[\/latex]<\/li>\r\n\t<li>Find [latex]A \\cap B[\/latex]<\/li>\r\n\t<li>Find [latex]A^c \\cap C[\/latex]<\/li>\r\n<\/ol>\r\n<p>[reveal-answer q=\"3541\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"3541\"]<\/p>\r\n<ol>\r\n\t<li>The union contains all the elements in either set: [latex]A \\cup B = \\{\\text{red, green, blue, yellow, orange}\\}[\/latex]. Notice we only list red once.<\/li>\r\n\t<li>The intersection contains all the elements in both sets: [latex]A \\cap B = \\{\\text{red}\\}[\/latex].<\/li>\r\n\t<li>Here we\u2019re looking for all the elements that are <em>not<\/em> in set [latex]A[\/latex] and are also in [latex]C[\/latex] . [latex]A^c \\cap C = \\{\\text{orange, yellow, purple}\\}[\/latex]<\/li>\r\n<\/ol>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]2203[\/ohm2_question]<\/section>\r\n<p>Set operations can be grouped together \u2013 for example, [latex]A^c \\cap C[\/latex]. Grouping symbols can be used like they are with arithmetic \u2013 to force an order of operations.<\/p>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]2204[\/ohm2_question]<\/section>\r\n<section class=\"textbox keyTakeaway\">\r\n<div>\r\n<h3>universal set<\/h3>\r\n<p>A <strong>universal set<\/strong> is a set that contains all the elements we are interested in. This would have to be defined by the context.<\/p>\r\n<p>&nbsp;<\/p>\r\n<p>A complement is relative to the universal set, so [latex]A^c[\/latex] contains all the elements in the universal set that are not in [latex]A[\/latex].<\/p>\r\n<\/div>\r\n<\/section>\r\n<section class=\"textbox youChoose\">[videopicker divId=\"tnh-video-picker\" title=\"Sets and Subsets\" label=\"Select Video\"]<br \/>\r\n[videooption displayName=\"Three Exercises in Sets and Subsets\" value=\"\/\/plugin.3playmedia.com\/show?mf=12421101&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=v818mV6jgBQ&amp;video_target=tpm-plugin-u8k6ekka-v818mV6jgBQ\"][videooption displayName=\"Intersection of Sets, Union of Sets and Venn Diagrams\" value=\"\/\/plugin.3playmedia.com\/show?mf=12421102&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=xZELQc11ACY&amp;video_target=tpm-plugin-z8k0cb68-xZELQc11ACY\"] [videooption displayName=\"Sets: Elements Of and Subsets\" value=\"https:\/\/youtu.be\/kt6oWPgBkAM\"]<br \/>\r\n[\/videopicker]\r\n<p><\/p>\r\n<p>You can view the <a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/THREE+EXERCISES+IN+SETS+AND+SUBSETS+-+DISCRETE+MATHEMATICS.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cTHREE EXERCISES IN SETS AND SUBSETS - DISCRETE MATHEMATICS\u201d here (opens in new window).<\/a><\/p>\r\n<p>You can view the <a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Intersection+of+Sets%2C+Union+of+Sets+and+Venn+Diagrams.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cIntersection of Sets, Union of Sets and Venn Diagrams\u201d here (opens in new window).<\/a><\/p>\r\n<p>You can view the <a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Sets_+Elements+Of+and+Subsets.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cSets: Elements Of and Subsets\u201d here (opens in new window).<\/a><\/p>\r\n<\/section>","rendered":"<h2>Set Operations<\/h2>\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<section class=\"textbox keyTakeaway\">\n<div>\n<h3>set operations: union, intersection, complement, and difference<\/h3>\n<ul>\n<li>The <strong>union<\/strong> of two sets contains all the elements contained in either set (or both sets).\n<ul>\n<li>The union is notated [latex]A \\cup B[\/latex].\u00a0More formally, [latex]x \\in A \\cup B[\/latex] if [latex]x \\in A[\/latex] or [latex]x \\in B[\/latex] (or both).<\/li>\n<\/ul>\n<\/li>\n<li>The <strong>intersection <\/strong>of two sets contains only the elements that are in both sets.\n<ul>\n<li>The intersection is notated [latex]A \\cap B[\/latex]. More formally, [latex]x \\in A \\cap B[\/latex] if [latex]x \\in A[\/latex] and [latex]x \\in B[\/latex].<\/li>\n<\/ul>\n<\/li>\n<li>The <strong>complement<\/strong> of a set [latex]A[\/latex] contains everything that is <em>not<\/em> in the [latex]A[\/latex].\n<ul>\n<li>The complement is notated [latex]A'[\/latex], or [latex]A^c[\/latex], or sometimes \u00a0~[latex]A[\/latex].<\/li>\n<\/ul>\n<\/li>\n<li>The <strong>difference<\/strong> of two sets is the list of all the elements that are in one set but not present in the other.\n<ul>\n<li>The difference between two sets is notated [latex]A \\setminus B[\/latex]. More formally, [latex]x \\in A \\setminus B[\/latex] if [latex]x \\in A[\/latex] &amp; [latex]x \\notin B[\/latex].<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<\/section>\n<section class=\"textbox proTip\">Notice in the descriptions of the notation introduced above that the\u00a0<strong>complement<\/strong> of a set is denoted [latex]A^{c}[\/latex]. This superscript is\u00a0not an exponent. It is a decoration that denotes\u00a0<strong>the complement of a set<\/strong>.<\/section>\n<p>Let\u2019s try applying the set operations.<\/p>\n<section class=\"textbox seeExample\">Consider the sets:<\/p>\n<ul>\n<li>[latex]A = \\{\\text{red, green, blue}\\}[\/latex]<\/li>\n<li>[latex]B = \\{\\text{red, yellow, orange}\\}[\/latex]<\/li>\n<li>[latex]C = \\{\\text{red, orange, yellow, green, blue, purple}\\}[\/latex]<\/li>\n<\/ul>\n<p>Find the following:<\/p>\n<ol>\n<li>Find [latex]A \\cup B[\/latex]<\/li>\n<li>Find [latex]A \\cap B[\/latex]<\/li>\n<li>Find [latex]A^c \\cap C[\/latex]<\/li>\n<\/ol>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q3541\">Show Solution<\/button><\/p>\n<div id=\"q3541\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>The union contains all the elements in either set: [latex]A \\cup B = \\{\\text{red, green, blue, yellow, orange}\\}[\/latex]. Notice we only list red once.<\/li>\n<li>The intersection contains all the elements in both sets: [latex]A \\cap B = \\{\\text{red}\\}[\/latex].<\/li>\n<li>Here we\u2019re looking for all the elements that are <em>not<\/em> in set [latex]A[\/latex] and are also in [latex]C[\/latex] . [latex]A^c \\cap C = \\{\\text{orange, yellow, purple}\\}[\/latex]<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm2203\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=2203&theme=lumen&iframe_resize_id=ohm2203&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<p>Set operations can be grouped together \u2013 for example, [latex]A^c \\cap C[\/latex]. Grouping symbols can be used like they are with arithmetic \u2013 to force an order of operations.<\/p>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm2204\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=2204&theme=lumen&iframe_resize_id=ohm2204&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section class=\"textbox keyTakeaway\">\n<div>\n<h3>universal set<\/h3>\n<p>A <strong>universal set<\/strong> is a set that contains all the elements we are interested in. This would have to be defined by the context.<\/p>\n<p>&nbsp;<\/p>\n<p>A complement is relative to the universal set, so [latex]A^c[\/latex] contains all the elements in the universal set that are not in [latex]A[\/latex].<\/p>\n<\/div>\n<\/section>\n<section class=\"textbox youChoose\">\n<div id=\"tnh-video-picker\" class=\"videoPicker\">\n<h3>Sets and Subsets<\/h3>\n<form><label>Select Video:<\/label><select name=\"video\"><option value=\"\/\/plugin.3playmedia.com\/show?mf=12421101&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=v818mV6jgBQ&amp;video_target=tpm-plugin-u8k6ekka-v818mV6jgBQ\">Three Exercises in Sets and Subsets<\/option><option value=\"\/\/plugin.3playmedia.com\/show?mf=12421102&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=xZELQc11ACY&amp;video_target=tpm-plugin-z8k0cb68-xZELQc11ACY\">Intersection of Sets, Union of Sets and Venn Diagrams<\/option><option value=\"https:\/\/www.youtube.com\/embed\/kt6oWPgBkAM\">Sets: Elements Of and Subsets<\/option><\/select><\/form>\n<div class=\"videoContainer threePlay\"><iframe src=\"\/\/plugin.3playmedia.com\/show?mf=12421101&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=v818mV6jgBQ&amp;video_target=tpm-plugin-u8k6ekka-v818mV6jgBQ\" allowfullscreen><\/iframe><\/div>\n<\/p>\n<p>You can view the <a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/THREE+EXERCISES+IN+SETS+AND+SUBSETS+-+DISCRETE+MATHEMATICS.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cTHREE EXERCISES IN SETS AND SUBSETS &#8211; DISCRETE MATHEMATICS\u201d here (opens in new window).<\/a><\/p>\n<p>You can view the <a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Intersection+of+Sets%2C+Union+of+Sets+and+Venn+Diagrams.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cIntersection of Sets, Union of Sets and Venn Diagrams\u201d here (opens in new window).<\/a><\/p>\n<p>You can view the <a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Sets_+Elements+Of+and+Subsets.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cSets: Elements Of and Subsets\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n","protected":false},"author":15,"menu_order":5,"template":"","meta":{"_candela_citation":"[{\"type\":\"original\",\"description\":\"Revision and Adaptation\",\"author\":\"Lumen Learning\",\"organization\":\"\",\"url\":\"\",\"project\":\"\",\"license\":\"pd\",\"license_terms\":\"\"},{\"type\":\"original\",\"description\":\"Question ID 132343\",\"author\":\"\",\"organization\":\"Lumen Learning\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"IMathAS Community License CC-BY + GPL\"},{\"type\":\"cc\",\"description\":\"Math in Society\",\"author\":\"Open Textbook Store, Transition Math Project, and the Open Course Library\",\"organization\":\"\",\"url\":\"http:\/\/www.opentextbookstore.com\/mathinsociety\/\",\"project\":\"\",\"license\":\"cc-by-sa\",\"license_terms\":\"\"},{\"type\":\"copyrighted_video\",\"description\":\"THREE EXERCISES IN SETS AND SUBSETS - 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