{"id":446,"date":"2016-10-12T21:59:46","date_gmt":"2016-10-12T21:59:46","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/math4libarts\/?post_type=chapter&#038;p=446"},"modified":"2019-05-30T16:36:44","modified_gmt":"2019-05-30T16:36:44","slug":"union-intersection-and-complement","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/waymakermath4libarts\/chapter\/union-intersection-and-complement\/","title":{"raw":"Union, Intersection, and Complement","rendered":"Union, Intersection, and Complement"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Describe memberships of sets, including the empty set, using proper notation, and decide whether given items are members and determine the cardinality of a given set.<\/li>\r\n \t<li>Describe the relations between sets regarding membership, equality, subset, and proper subset, using proper notation.<\/li>\r\n \t<li>Perform the operations of union, intersection, complement, and difference on sets using proper notation.<\/li>\r\n \t<li>Be able to draw and interpret Venn diagrams of set relations and operations and use Venn diagrams to solve problems.<\/li>\r\n \t<li>Recognize when set theory is applicable to real-life situations, solve real-life problems, and communicate real-life problems and solutions to others.<\/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\">\r\n<h3>Union, Intersection, and Complement<\/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\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>, or <em>A<sup>c<\/sup>, or sometimes ~<em>A<\/em>.<\/em>\r\n\r\nA <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.\r\n\r\nA complement is relative to the universal set, so\u00a0<em>A<sup>c<\/sup><\/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&nbsp;\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&nbsp;\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<sup>c<\/sup> <\/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<iframe id=\"mom10\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=110368&amp;theme=oea&amp;iframe_resize_id=mom10\" width=\"100%\" height=\"350\"><\/iframe>\r\n\r\n<\/div>\r\n&nbsp;\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<sup>c<\/sup><\/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<sup>c<\/sup> <\/em>\u22c2<em> C<\/em> = {orange, yellow, purple}<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n<iframe id=\"mom1\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=125865&amp;theme=oea&amp;iframe_resize_id=mom1\" width=\"100%\" height=\"450\"><\/iframe>\r\n\r\n<\/div>\r\nNotice that in the example above, it would be hard to just ask for <em>A<sup>c<\/sup><\/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\nAs we saw earlier with the expression\u00a0<em>A<sup>c<\/sup><\/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\r\n&nbsp;\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>)<em><sup>c<\/sup><\/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)<sup>c<\/sup><\/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>Venn Diagrams<\/h2>\r\nTo visualize the interaction of sets, John Venn in 1880 thought to use overlapping circles, building on a similar idea used by Leonhard Euler in the 18th century. These illustrations now called <strong>Venn Diagrams<\/strong>.\r\n<div class=\"textbox\">\r\n<h3>Venn Diagram<\/h3>\r\nA Venn diagram represents each set by a circle, usually drawn inside of a containing box representing the universal set. Overlapping areas indicate elements common to both sets.\r\n\r\nBasic Venn diagrams can illustrate the interaction of two or three sets.\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nCreate Venn diagrams to illustrate <em>A <\/em>\u22c3<em> B<\/em>, <em>A <\/em>\u22c2<em> B<\/em>, and <em>Ac <\/em>\u22c2<em> B<\/em>\r\n\r\n<em>A <\/em>\u22c3<em> B<\/em> contains all elements in <em>either<\/em> set.\r\n[reveal-answer q=\"252649\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"252649\"]\r\n\r\n<img class=\"alignnone size-full wp-image-449\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/276\/2016\/10\/12220329\/ab1.png\" alt=\"ab1\" width=\"179\" height=\"125\" \/>\r\n\r\n<em>A <\/em>\u22c2<em> B<\/em> contains only those elements in both sets \u2013 in the overlap of the circles.\r\n\r\n<img class=\"alignnone size-full wp-image-450\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/276\/2016\/10\/12220349\/ab2.png\" alt=\"ab2\" width=\"180\" height=\"125\" \/>\r\n\r\n<em>Ac <\/em>will contain all elements <em>not<\/em> in the set A. <em>A<sup>c <\/sup><\/em>\u22c2<em> B<\/em> will contain the elements in set <em>B<\/em> that are not in set <em>A<\/em>.\r\n\r\n<img class=\"alignnone size-full wp-image-451\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/276\/2016\/10\/12220403\/ab3.png\" alt=\"ab3\" width=\"180\" height=\"125\" \/>\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nUse a Venn diagram to illustrate (<em>H <\/em>\u22c2<em> F<\/em>)<sup><em>c<\/em><\/sup> \u22c2<em> W\r\n<\/em>[reveal-answer q=\"311976\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"311976\"]\r\n\r\nWe\u2019ll start by identifying everything in the set <em>H <\/em>\u22c2<em> F<\/em>\r\n\r\n<img class=\"alignnone wp-image-452 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/276\/2016\/10\/12220617\/hfw1.png\" alt=\"A Venn diagram depicting three overlapping circles, labeled H, F, and W, respectively. The area where H and F overlap is outlined in red.\" width=\"180\" height=\"152\" \/>\r\n\r\nNow, (<em>H <\/em>\u22c2<em> F<\/em>)<em>c<\/em> \u22c2<em> W<\/em> will contain everything <em>not<\/em> in the set identified above that is also in set <em>W<\/em>.\r\n\r\n<img class=\"alignnone wp-image-453 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/276\/2016\/10\/12220632\/hfw2.png\" alt=\"A Venn diagram depicting three overlapping circles, labeled H, F, and W, respectively. The area of circle W where it does not overlap with H and F at the same time is outlined in red.\" width=\"180\" height=\"156\" \/>\r\n\r\n&nbsp;\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nhttps:\/\/youtu.be\/CPeeOUldZ6M\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nCreate an expression to represent the outlined part of the Venn diagram shown.\r\n\r\n<img class=\"alignnone size-full wp-image-454\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/276\/2016\/10\/12220943\/hfw3.png\" alt=\"hfw3\" width=\"180\" height=\"157\" \/>\r\n[reveal-answer q=\"828282\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"828282\"]\r\n\r\nThe elements in the outlined set <em>are<\/em> in sets <em>H<\/em> and <em>F<\/em>, but are not in set <em>W<\/em>. So we could represent this set as <em>H <\/em>\u22c2<em> F<\/em> \u22c2<em> W<sup>c<\/sup><\/em>\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\nCreate an expression to represent the outlined portion of the Venn diagram shown.\r\n\r\n<img class=\"alignnone size-full wp-image-455\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/276\/2016\/10\/12221621\/abc1.png\" alt=\"abc1\" width=\"180\" height=\"157\" \/>\r\n<iframe id=\"mom2\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=6699&amp;theme=oea&amp;iframe_resize_id=mom2\" width=\"100%\" height=\"200\"><\/iframe>\r\n\r\n<\/div>\r\nhttps:\/\/youtu.be\/VfC8rhLdYYg","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Describe memberships of sets, including the empty set, using proper notation, and decide whether given items are members and determine the cardinality of a given set.<\/li>\n<li>Describe the relations between sets regarding membership, equality, subset, and proper subset, using proper notation.<\/li>\n<li>Perform the operations of union, intersection, complement, and difference on sets using proper notation.<\/li>\n<li>Be able to draw and interpret Venn diagrams of set relations and operations and use Venn diagrams to solve problems.<\/li>\n<li>Recognize when set theory is applicable to real-life situations, solve real-life problems, and communicate real-life problems and solutions to others.<\/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\">\n<h3>Union, Intersection, and Complement<\/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>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>, or <em>A<sup>c<\/sup>, or sometimes ~<em>A<\/em>.<\/em><\/p>\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>A complement is relative to the universal set, so\u00a0<em>A<sup>c<\/sup><\/em><em>\u00a0<\/em>contains all the elements in the universal set that are not in <em>A<\/em>.<\/p>\n<\/div>\n<p>&nbsp;<\/p>\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<p>&nbsp;<\/p>\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<sup>c<\/sup> <\/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=\"mom10\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=110368&amp;theme=oea&amp;iframe_resize_id=mom10\" width=\"100%\" height=\"350\"><\/iframe><\/p>\n<\/div>\n<p>&nbsp;<\/p>\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<sup>c<\/sup><\/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<sup>c<\/sup> <\/em>\u22c2<em> C<\/em> = {orange, yellow, purple}<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"mom1\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=125865&amp;theme=oea&amp;iframe_resize_id=mom1\" width=\"100%\" height=\"450\"><\/iframe><\/p>\n<\/div>\n<p>Notice that in the example above, it would be hard to just ask for <em>A<sup>c<\/sup><\/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<p>As we saw earlier with the expression\u00a0<em>A<sup>c<\/sup><\/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<p>&nbsp;<\/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>)<em><sup>c<\/sup><\/em> \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)<sup>c<\/sup><\/em>\u00a0\u22c2<em> W<\/em> = {duck, deer, frog, mouse}<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<h2>Venn Diagrams<\/h2>\n<p>To visualize the interaction of sets, John Venn in 1880 thought to use overlapping circles, building on a similar idea used by Leonhard Euler in the 18th century. These illustrations now called <strong>Venn Diagrams<\/strong>.<\/p>\n<div class=\"textbox\">\n<h3>Venn Diagram<\/h3>\n<p>A Venn diagram represents each set by a circle, usually drawn inside of a containing box representing the universal set. Overlapping areas indicate elements common to both sets.<\/p>\n<p>Basic Venn diagrams can illustrate the interaction of two or three sets.<\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Create Venn diagrams to illustrate <em>A <\/em>\u22c3<em> B<\/em>, <em>A <\/em>\u22c2<em> B<\/em>, and <em>Ac <\/em>\u22c2<em> B<\/em><\/p>\n<p><em>A <\/em>\u22c3<em> B<\/em> contains all elements in <em>either<\/em> set.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q252649\">Show Solution<\/span><\/p>\n<div id=\"q252649\" class=\"hidden-answer\" style=\"display: none\">\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-449\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/276\/2016\/10\/12220329\/ab1.png\" alt=\"ab1\" width=\"179\" height=\"125\" \/><\/p>\n<p><em>A <\/em>\u22c2<em> B<\/em> contains only those elements in both sets \u2013 in the overlap of the circles.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-450\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/276\/2016\/10\/12220349\/ab2.png\" alt=\"ab2\" width=\"180\" height=\"125\" \/><\/p>\n<p><em>Ac <\/em>will contain all elements <em>not<\/em> in the set A. <em>A<sup>c <\/sup><\/em>\u22c2<em> B<\/em> will contain the elements in set <em>B<\/em> that are not in set <em>A<\/em>.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-451\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/276\/2016\/10\/12220403\/ab3.png\" alt=\"ab3\" width=\"180\" height=\"125\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Use a Venn diagram to illustrate (<em>H <\/em>\u22c2<em> F<\/em>)<sup><em>c<\/em><\/sup> \u22c2<em> W<br \/>\n<\/em><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q311976\">Show Solution<\/span><\/p>\n<div id=\"q311976\" class=\"hidden-answer\" style=\"display: none\">\n<p>We\u2019ll start by identifying everything in the set <em>H <\/em>\u22c2<em> F<\/em><\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-452 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/276\/2016\/10\/12220617\/hfw1.png\" alt=\"A Venn diagram depicting three overlapping circles, labeled H, F, and W, respectively. The area where H and F overlap is outlined in red.\" width=\"180\" height=\"152\" \/><\/p>\n<p>Now, (<em>H <\/em>\u22c2<em> F<\/em>)<em>c<\/em> \u22c2<em> W<\/em> will contain everything <em>not<\/em> in the set identified above that is also in set <em>W<\/em>.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-453 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/276\/2016\/10\/12220632\/hfw2.png\" alt=\"A Venn diagram depicting three overlapping circles, labeled H, F, and W, respectively. The area of circle W where it does not overlap with H and F at the same time is outlined in red.\" width=\"180\" height=\"156\" \/><\/p>\n<p>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Sets: drawing a Venn diagram\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/CPeeOUldZ6M?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Create an expression to represent the outlined part of the Venn diagram shown.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-454\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/276\/2016\/10\/12220943\/hfw3.png\" alt=\"hfw3\" width=\"180\" height=\"157\" \/><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q828282\">Show Solution<\/span><\/p>\n<div id=\"q828282\" class=\"hidden-answer\" style=\"display: none\">\n<p>The elements in the outlined set <em>are<\/em> in sets <em>H<\/em> and <em>F<\/em>, but are not in set <em>W<\/em>. So we could represent this set as <em>H <\/em>\u22c2<em> F<\/em> \u22c2<em> W<sup>c<\/sup><\/em><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Create an expression to represent the outlined portion of the Venn diagram shown.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-455\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/276\/2016\/10\/12221621\/abc1.png\" alt=\"abc1\" width=\"180\" height=\"157\" \/><br \/>\n<iframe loading=\"lazy\" id=\"mom2\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=6699&amp;theme=oea&amp;iframe_resize_id=mom2\" width=\"100%\" height=\"200\"><\/iframe><\/p>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"oembed-2\" title=\"Sets: writing an expression for a Venn diagram region\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/VfC8rhLdYYg?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/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-446\">\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\/about\/pdm\">Public Domain: No Known Copyright<\/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>. <strong>License Terms<\/strong>: IMathAS Community License CC-BY + GPL<\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Math in Society. <strong>Authored by<\/strong>: Open Textbook Store, Transition Math Project, and the Open Course Library. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/www.opentextbookstore.com\/mathinsociety\/\">http:\/\/www.opentextbookstore.com\/mathinsociety\/<\/a>. <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>Sets: drawing a Venn diagram. <strong>Authored by<\/strong>: David Lippman. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/CPeeOUldZ6M?list=PL7138FAEC01D6F3F3\">https:\/\/youtu.be\/CPeeOUldZ6M?list=PL7138FAEC01D6F3F3<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Sets: drawing a Venn diagram. <strong>Authored by<\/strong>: David Lippman. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/CPeeOUldZ6M\">https:\/\/youtu.be\/CPeeOUldZ6M<\/a>. <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 6699. <strong>Authored by<\/strong>: Morales,Lawrence. <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 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 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