{"id":1065,"date":"2021-10-12T17:47:58","date_gmt":"2021-10-12T17:47:58","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/?post_type=chapter&#038;p=1065"},"modified":"2026-03-18T16:38:36","modified_gmt":"2026-03-18T16:38:36","slug":"set-theory-3","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/chapter\/set-theory-3\/","title":{"raw":"1.1.1: Set Theory","rendered":"1.1.1: Set Theory"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h1>Learning Objectives<\/h1>\r\n<ul>\r\n \t<li>Define sets, subsets,\u00a0universal set,\u00a0empty set, and infinite sets<\/li>\r\n \t<li>Determine subsets of a set<\/li>\r\n \t<li>Determine subsets, and complement of a set<\/li>\r\n \t<li>Apply the set operations union, intersection, and complements of sets<\/li>\r\n \t<li>Use Venn diagrams to illustrate set operations<\/li>\r\n<\/ul>\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h1>KEY words<\/h1>\r\n<ul>\r\n \t<li><strong>Subset<\/strong>: A set containing elements of another set<\/li>\r\n \t<li><strong>Empty set<\/strong>: a set containing no elements<\/li>\r\n \t<li><strong>Universal set<\/strong>: a set containing all elements we are interested in<\/li>\r\n \t<li><strong>Infinite se<\/strong>t: a set that contains an infinite number of elements<\/li>\r\n \t<li><strong>Complement<\/strong>: the complement of a set is all elements in the universal set that are not in the original set<\/li>\r\n \t<li><strong>Variable<\/strong>:\u00a0a letter that is used as a stand-in to represent any element<\/li>\r\n<\/ul>\r\n<\/div>\r\nIn mathematics, it is helpful to consider where numbers come from and how they interact with each other. To look at collections of different numbers, we will first define sets.\r\n<h2>Sets<\/h2>\r\nAn art collector owns a collection of paintings, while a music lover might keep a collection of CDs. Any collection of items can form a <em><strong>set<\/strong><\/em>.\r\n<div class=\"textbox shaded\">\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 braces and separated by commas.\r\n\r\n<\/div>\r\nIt is common to name a set, to make it easier to refer to that set later. Capital letters are often used for this purpose. Sets can have a <em><strong>finite<\/strong><\/em> number of elements or an <em><strong>infinite<\/strong><\/em> number of elements and are referred to as finite or infinite sets.\u00a0A 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<div class=\"bcc-box bcc-success\">\r\n<h3>Example<\/h3>\r\nSome examples of sets defined by describing the contents:\r\n<ol>\r\n \t<li>[latex]\\text{C}\\;=\\;\\text{the set of all books written about travel to Chile}[\/latex]. This is a finite set.<\/li>\r\n \t<li>[latex]\\mathbb{Z}=\\text{the set of all integers}[\/latex]. This is an infinite set.<\/li>\r\n<\/ol>\r\nSome examples of sets defined by listing the elements of the set:\r\n<ol>\r\n \t<li>[latex]A\\;= \\;\\{1, 3, 9, 12\\}[\/latex].This is a finite set.<\/li>\r\n \t<li>[latex]\\text{R}\\;=\\;\\{\\text{red, orange, yellow, green, blue, indigo, purple}\\}[\/latex]. This is a finite set.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox\">\r\n<h3>Notation<\/h3>\r\nThe symbol [latex]\\in[\/latex] means \u201cis an element of\u201d.\r\n\r\nA set that contains no elements, [latex]\\{\\;\\}[\/latex], is called the <strong>empty set<\/strong> and is notated [latex]\\varnothing[\/latex].\r\n\r\n<\/div>\r\n<div class=\"bcc-box bcc-success\">\r\n<h3>Example<\/h3>\r\nLet\u00a0[latex]A\\;= \\;\\{1, 2, 3, 4\\}[\/latex]\r\n\r\nTo say that 2 is an element of the set, we write [latex]2\\;\\in\\;A[\/latex]<em>.<\/em>\r\n\r\n<\/div>\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 <em><strong>subset<\/strong><\/em> of the larger set of all Madonna albums.\r\n<div class=\"textbox\">\r\n<h3>Subset<\/h3>\r\nA <strong>subset<\/strong> of a set [latex]A[\/latex]\u00a0is another set that contains only elements from the set [latex]A[\/latex]<em>. <\/em>A subset may contain some elements of set [latex]A[\/latex], no elements of set [latex]A[\/latex]\u00a0(i.e. the empty set, [latex]\\varnothing[\/latex]) or all elements of set [latex]A[\/latex]\u00a0(i.e. [latex]A[\/latex]).\r\n\r\nIf set [latex]B[\/latex]\u00a0is a subset of set [latex]A[\/latex], we write [latex]B\\;\\subseteq\\;A[\/latex].\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 set [latex]B[\/latex]\u00a0is a proper subset of set [latex]A[\/latex], we write [latex]B\\;\\subset\\;A[\/latex].\r\n\r\n<\/div>\r\nThe bottom bar in the subset notation\u00a0 [latex]\\subseteq[\/latex] represents the equality sign, meaning the whole set is a subset of itself. Notice that the proper subset notation\u00a0[latex]\\subset[\/latex] does not have that equals bar so does not include the whole set as a subset of itself.\r\n<div class=\"bcc-box bcc-success\">\r\n<h3>Example<\/h3>\r\nConsider these three sets:\r\n<p style=\"padding-left: 30px;\">[latex]A=\\text{the set of all even numbers}[\/latex]\r\n[latex]B=\\{2,4,6\\}[\/latex]\r\n[latex]C=\\{2,3,4,6\\}[\/latex]<\/p>\r\nHere [latex]B\\:\\subset\\:A[\/latex]\u00a0since every element of [latex]B[\/latex]\u00a0is an even number, so is an element of [latex]A[\/latex].\r\n\r\nMore formally, we could say [latex]B\\:\\subset\\:A[\/latex]\u00a0since if [latex]x\\:\\in\\:B[\/latex], then [latex]x\\:\\in\\:A[\/latex].\r\n\r\nIt is also true that [latex]B\\:\\subset\\:C[\/latex] because every element of [latex]B[\/latex] is an element of [latex]C[\/latex].\r\n\r\n[latex]C[\/latex] is NOT a subset of [latex]A[\/latex], since [latex]C[\/latex] contains an element, 3, that is not contained in [latex]A[\/latex]<em>.<\/em>\r\n\r\n<\/div>\r\n<div class=\"bcc-box bcc-success\">\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\r\nThere 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.\r\n\r\n<\/div>\r\n<h2>Universal Set<\/h2>\r\n<div class=\"textbox\">\r\n<h3>Universal Set<\/h3>\r\nA <strong>universal set<\/strong> is a set that contains all the elements we are interested in. This is defined by context.\r\n\r\n<\/div>\r\n<div class=\"bcc-box bcc-success\">\r\n<h3>Example<\/h3>\r\n<ol>\r\n \t<li>If we are discussing searching for books, the universal set might be all the books in the library.<\/li>\r\n \t<li>If we are grouping our Facebook friends, the universal set would be all of our Facebook friends.<\/li>\r\n \t<li>If we are 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<h2>Union, Intersection, and Complement<\/h2>\r\nIt is common for sets to 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 are 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 <em>either<\/em> set.\u00a0The union is notated [latex]A\\:\\cup\\:B[\/latex].<em>\u00a0<\/em>More formally, [latex]x\\in \\:A\\cup\\:B[\/latex] if [latex]x\\in\\:A[\/latex] or [latex]x\\in\\:B[\/latex].\r\n\r\nThe <strong>intersection <\/strong>of two sets contains only the elements that are in <em>both<\/em> sets.\u00a0The intersection is notated [latex]A\\cap\\,B[\/latex]<em>.\u00a0<\/em>More formally, [latex]x\\:\\in\\,A\\cap\\,B\\:\\text{if}\\:x\\in\\,A\\:\\text{and}\\:x\\in\\,B[\/latex].\r\n\r\nThe <strong>complement<\/strong> of a set [latex]A[\/latex] contains everything that is <em>not<\/em> in the set [latex]A[\/latex].\u00a0The complement is notated [latex]A^{\\prime}[\/latex],\u00a0or [latex]A^c[\/latex]. A complement is relative to the universal set, so [latex]A^{\\prime}[\/latex]\u00a0contains all the elements in the universal set that are not in [latex]A[\/latex].\r\n\r\n<\/div>\r\nNotice that we used a letter, [latex]x[\/latex], to represent any element in a set. This letter is called a\u00a0<em><strong>variable<\/strong><\/em> and is just a stand-in for any element in a given set.\r\n<div class=\"bcc-box bcc-success\">\r\n<h3>Example<\/h3>\r\nSuppose the universal set is [latex]U[\/latex]\u00a0= all whole numbers from 1 to 9. i.e. [latex]U=\\{1,2,3,4,5,6,7,8,9\\}[\/latex]\r\n\r\nIf [latex]A=\\{1, 2, 4\\}[\/latex], then [latex]A^{\\prime}=\\{3,5,6,7,8,9\\}[\/latex].\r\n\r\n<\/div>\r\n<div class=\"bcc-box bcc-success\">\r\n<h3>Example<\/h3>\r\nConsider the sets:\r\n<p style=\"padding-left: 30px;\">[latex]A=\\{\\text{red, green, blue}\\}[\/latex]\r\n[latex]B=\\{\\text{red, yellow, orange}\\}[\/latex]\r\n[latex]C=\\{\\text{red, orange, yellow, green, blue, purple}\\}[\/latex]<\/p>\r\nFind the following:\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^{\\prime}\\cap\\,C[\/latex]<\/li>\r\n<\/ol>\r\n<h4>Answers<\/h4>\r\n<ol>\r\n \t<li>The union contains all the elements in <em>either<\/em> 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 <em>both<\/em> sets: [latex]A\\cap\\,B=\\{\\text{red}\\}[\/latex]. Red is the only color in both sets.<\/li>\r\n \t<li>Here we\u2019re looking for all the elements that are <em>not<\/em> in set [latex]A[\/latex] but are in set [latex]C[\/latex]<em>: \u00a0<\/em>[latex]A^{\\prime}\\cap\\,C=\\{\\text{orange, yellow, purple}\\}[\/latex]<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox shaded\">\r\n<h3>Try It<\/h3>\r\nUsing the sets from the previous example, find [latex]A\\cup\\,C[\/latex]\u00a0and [latex]B^{\\prime}\\cap\\,A[\/latex]\r\n\r\n[reveal-answer q=\"423428\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"423428\"]\r\n\r\n[latex]A\\cup\\,C=\\{\\text{red, green, blue, orange yellow, purple}\\}[\/latex]\r\n\r\n[latex]B^{\\prime}\\cap\\,A=\\{\\text{green, blue}\\}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nSet operations can be grouped together using grouping symbols to force an order of operations, just like in arithmetic. Parentheses are used to tell us which operation to perform first.\r\n<div class=\"bcc-box bcc-success\">\r\n<h3>Example<\/h3>\r\nSuppose [latex]H=\\{\\text{cat, dog, rabbit, mouse}\\}[\/latex], [latex]F=\\{\\text{dog, cow, duck, pig, rabbit}\\}[\/latex], and [latex]W=\\{\\text{duck, rabbit, deer, frog, mouse}\\}[\/latex].\r\n<ol>\r\n \t<li>Find<em>\u00a0<\/em>[latex](H\\cap\\,F)\\cup\\,W[\/latex]<\/li>\r\n \t<li>Find [latex]H\\cap\\,(F\\cup\\,W)[\/latex]<\/li>\r\n \t<li>Find [latex](H\\cap\\,F)^{\\prime}\\cap\\,W[\/latex]<\/li>\r\n<\/ol>\r\n<h4>Solutions<\/h4>\r\n<ol>\r\n \t<li>We start with the intersection since it is in parentheses:\u00a0[latex](H\\cap\\,F)=\\{\\text{dog, rabbit}\\}[\/latex].\u00a0Now we union that result with [latex]W[\/latex]: <em>\u00a0<\/em>[latex](H\\cap\\,F)\\cup\\,W=\\{\\text{dog, duck, rabbit, deer, frog, mouse}\\}[\/latex].<\/li>\r\n \t<li>We start with the union inside parentheses: [latex](F\\cup\\,W)=\\{\\text{dog, cow, duck, pig, rabbit, deer, frog, mouse}\\}[\/latex]\u00a0.\u00a0Now we intersect that result with [latex]H[\/latex]: [latex]H\\cap\\,(F\\cup\\,W)=\\{\\text{dog, rabbit, mouse}\\}[\/latex].<\/li>\r\n \t<li>We start with the intersection inside parentheses: [latex](H\\cap\\,F)=\\{\\text{dog, rabbit}\\}[\/latex].\u00a0Now we want to find the elements of [latex]W[\/latex]\u00a0that are <em>not<\/em> in [latex](H\\cap\\,F)[\/latex]<em>. \u00a0<\/em>[latex](H\\cap\\,F)^{\\prime}\\cap\\,W= \\{\\text{duck, deer, frog, mouse}\\}[\/latex].<\/li>\r\n<\/ol>\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 eighteenth\u00a0century. These illustrations are now called <em><strong>Venn Diagrams<\/strong><\/em>.\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\n<\/div>\r\n<div class=\"bcc-box bcc-success\">\r\n<h3>Example<\/h3>\r\nSuppose the universal set is students in a class. [latex]A[\/latex] = all students with blonde hair; [latex]B[\/latex] = all students with blue eyes; [latex]C[\/latex] = all students at least 6 feet tall.\r\n\r\n&nbsp;\r\n<ol>\r\n \t<li>What does [latex]B^{\\prime}[\/latex] represent?<\/li>\r\n \t<li>What does\u00a0\u00a0[latex]A\\cap\\,B^{\\prime}[\/latex] represent?<\/li>\r\n \t<li>What does\u00a0\u00a0[latex](A\\cup C)^{\\prime}\\cap B[\/latex] represent?Use the Venn to write set interactions to represent each category.<\/li>\r\n \t<li>Students with blonde hair and blue eyes.<\/li>\r\n \t<li>Students with blue eyes that are at least 6 feet tall.<\/li>\r\n \t<li>Students who have blonde hair, blue eyes, and are at least 6 feet tall.<\/li>\r\n \t<li>Students who are under 6 feet tall and have neither blonde hair nor blue eyes.<\/li>\r\n \t<li><span style=\"font-size: 1rem; text-align: initial;\">Students who have neither blonde hair nor blue eyes but are at least 6' tall.<\/span><\/li>\r\n<\/ol>\r\n<img class=\"aligncenter wp-image-1477 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5676\/2021\/10\/07220834\/Class-Ven.png\" alt=\"Venn diagram\" width=\"964\" height=\"692\" \/>\r\n\r\n&nbsp;\r\n\r\n[reveal-answer q=\"HM8872\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"HM8872\"]\r\n<ol>\r\n \t<li>[latex]B^{\\prime}[\/latex] represents students who do not have blue eyes.<\/li>\r\n \t<li>Students who are blonde but do not have blue eyes.<\/li>\r\n \t<li>Students with blue eyes who are neither blonde nor at least 6' tall.<\/li>\r\n \t<li>This is found in the intersection of the blue and yellow circles: \u00a0[latex]A\\cap\\,B[\/latex]<\/li>\r\n \t<li>This is found in the intersection of the blue and green circles: \u00a0[latex]B\\cap\\,C[\/latex]<\/li>\r\n \t<li>This is found in the intersection of the yellow, blue and green circles: \u00a0[latex]A\\cap\\,B\\cap\\,C[\/latex]<\/li>\r\n \t<li>If they are under 6' tall they are not in the green circle. If they do not have blonde hair they are not in the yellow circle. If they do not have blue eyes they are not in the blue circle. Therefore they are outside all 3 circles. \u00a0 [latex](A\\cup\\,B\\cup\\,C)^{\\prime}[\/latex]<\/li>\r\n \t<li>These students are found in the green circle but are neither in the blue nor yellow circles. \u00a0 [latex](A\\cup\\,B)^{\\prime}\\cap\\,C[\/latex]<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>TRY IT<\/strong>\r\n\r\nThe Venn diagram shows the interaction of sports watched on TV by people who answered a survey. [latex]S[\/latex] = {watched soccer}, [latex]F[\/latex] = {watched football}, [latex]B[\/latex] = {watched baseball}, [latex]G[\/latex] = {watched golf}\r\n\r\n<img class=\"aligncenter wp-image-1478 size-large\" src=\"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-content\/uploads\/sites\/5676\/2021\/10\/Sports-Venn1-1024x574.png\" alt=\"A Venn diagram where sets S, B, and F intersects each other and set G only intersects with F and not with S or B.\" width=\"1024\" height=\"574\" \/>\r\n\r\n&nbsp;\r\n\r\nUse set interactions to represent each category.\r\n<ol>\r\n \t<li>People who watch only baseball?<\/li>\r\n \t<li>People who watch golf and football?<\/li>\r\n \t<li>People who watch football and baseball?<\/li>\r\n \t<li>People who watch football, baseball, and soccer?<\/li>\r\n \t<li>People who watch football but neither baseball nor soccer?<\/li>\r\n \t<li>People who do not watch golf?<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"706599\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"706599\"]\r\n<ol>\r\n \t<li>[latex](S\\cup\\,F)^{\\prime}\\cap\\,B[\/latex]<\/li>\r\n \t<li>[latex]G\\cap\\,F[\/latex]<\/li>\r\n \t<li>[latex]F\\cap\\,B[\/latex]<\/li>\r\n \t<li>[latex]F\\cap\\,B\\cap\\,S[\/latex]<\/li>\r\n \t<li>[latex](B\\cup\\,S)^{\\prime}\\cap\\,F[\/latex]<\/li>\r\n \t<li>[latex]G^{\\prime}[\/latex]<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nEven when the elements of the sets are unknown, Venn diagrams can illustrate unions, intersections, and complements.\r\n<div class=\"bcc-box bcc-success\">\r\n<h3>Example<\/h3>\r\nCreate Venn diagrams to illustrate [latex]A\\cup\\,B[\/latex], [latex]A\\cap\\,B[\/latex], and [latex]A^{\\prime}\\cap\\,B[\/latex].\r\n\r\n[latex]A\\cup\\,B[\/latex]\u00a0contains all elements in <em>either<\/em> set.\r\n\r\n<img class=\"alignnone wp-image-166 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1212\/2015\/11\/02174017\/Fig3_1_1.png\" alt=\"Both sets A and B, which intersect each other, are enclosed by a bold red line.\" width=\"312\" height=\"213\" \/>\r\n\r\n[latex]A\\cap\\,B[\/latex] contains only those elements in both sets\u2014in the overlap of the circles.\r\n\r\n<img class=\"alignnone wp-image-167 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1212\/2015\/11\/02174019\/Fig3_1_2.png\" alt=\"The intersection area of sets A and B is enclosed by a bold red line.\" width=\"308\" height=\"209\" \/>\r\n\r\n[latex]A^{\\prime}[\/latex] will contain all elements <em>not<\/em> in the set [latex]A[\/latex]. [latex]A^{\\prime}\\cap\\,B[\/latex]\u00a0will contain the elements in set [latex]B[\/latex]\u00a0that are not in set\u00a0[latex]A[\/latex].\r\n\r\n<img class=\"alignnone wp-image-168 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1212\/2015\/11\/02174020\/Fig3_1_3.png\" alt=\"Set B, without its intersection with set A, is enclosed by a bold red line.\" width=\"309\" height=\"205\" \/>\r\n\r\n<\/div>\r\n<div class=\"bcc-box bcc-success\">\r\n<h3>Example<\/h3>\r\nUse a Venn diagram to illustrate [latex](H\\cap\\,F)^{\\prime}\\cap\\,W[\/latex]\r\n\r\nWe\u2019ll start by identifying everything in the set [latex]H\\cap\\,F[\/latex]\r\n\r\n<img class=\"alignnone wp-image-169 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1212\/2015\/11\/02174021\/Fig3_1_4.png\" alt=\"The intersection of sets H and F, where sets H, F and W intersect each other, is enclosed by a bold red line.\" width=\"303\" height=\"277\" \/>\r\n\r\n&nbsp;\r\n\r\nNow, [latex](H\\cap\\,F)^{\\prime}\\cap\\,W[\/latex] 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-170 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1212\/2015\/11\/02174022\/Fig3_1_5.png\" alt=\"Set W, excluding the intersection region of all three sets H and F and W, is enclosed by a bold red line.\" width=\"306\" height=\"276\" \/>\r\n\r\n<\/div>\r\n<div class=\"bcc-box bcc-success\">\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 wp-image-171 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1212\/2015\/11\/02174023\/Fig3_1_6.png\" alt=\"The intersection of sets H and F, excluding the intersection area of all three sets H and F and W, is enclosed by a bold red line.\" width=\"305\" height=\"269\" \/>\r\n\r\nThe elements in the outlined set <em>are<\/em> in sets [latex]H[\/latex] and [latex]F[\/latex], but are not in set [latex]W[\/latex]. So we represent this set as [latex]H\\cap\\,F\\cap\\,W^{\\prime}[\/latex].\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">\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 wp-image-172 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1212\/2015\/11\/02174025\/Fig3_1_7.png\" alt=\"The intersection of sets A and B, excluding the intersection region with set C and the rest of set C, is enclosed by a bold red line.\" width=\"305\" height=\"275\" \/>\r\n\r\n[reveal-answer q=\"475309\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"475309\"]\r\n\r\n[latex]C[\/latex] is excluded so [latex]C^{\\prime}[\/latex] is intersected with [latex]A[\/latex] and [latex]B[\/latex]: [latex]C^{\\prime}\\cap\\,(A\\cup\\,B)[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>","rendered":"<div class=\"textbox learning-objectives\">\n<h1>Learning Objectives<\/h1>\n<ul>\n<li>Define sets, subsets,\u00a0universal set,\u00a0empty set, and infinite sets<\/li>\n<li>Determine subsets of a set<\/li>\n<li>Determine subsets, and complement of a set<\/li>\n<li>Apply the set operations union, intersection, and complements of sets<\/li>\n<li>Use Venn diagrams to illustrate set operations<\/li>\n<\/ul>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h1>KEY words<\/h1>\n<ul>\n<li><strong>Subset<\/strong>: A set containing elements of another set<\/li>\n<li><strong>Empty set<\/strong>: a set containing no elements<\/li>\n<li><strong>Universal set<\/strong>: a set containing all elements we are interested in<\/li>\n<li><strong>Infinite se<\/strong>t: a set that contains an infinite number of elements<\/li>\n<li><strong>Complement<\/strong>: the complement of a set is all elements in the universal set that are not in the original set<\/li>\n<li><strong>Variable<\/strong>:\u00a0a letter that is used as a stand-in to represent any element<\/li>\n<\/ul>\n<\/div>\n<p>In mathematics, it is helpful to consider where numbers come from and how they interact with each other. To look at collections of different numbers, we will first define sets.<\/p>\n<h2>Sets<\/h2>\n<p>An art collector owns a collection of paintings, while a music lover might keep a collection of CDs. Any collection of items can form a <em><strong>set<\/strong><\/em>.<\/p>\n<div class=\"textbox shaded\">\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 braces and separated by commas.<\/p>\n<\/div>\n<p>It is common to name a set, to make it easier to refer to that set later. Capital letters are often used for this purpose. Sets can have a <em><strong>finite<\/strong><\/em> number of elements or an <em><strong>infinite<\/strong><\/em> number of elements and are referred to as finite or infinite sets.\u00a0A 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<div class=\"bcc-box bcc-success\">\n<h3>Example<\/h3>\n<p>Some examples of sets defined by describing the contents:<\/p>\n<ol>\n<li>[latex]\\text{C}\\;=\\;\\text{the set of all books written about travel to Chile}[\/latex]. This is a finite set.<\/li>\n<li>[latex]\\mathbb{Z}=\\text{the set of all integers}[\/latex]. This is an infinite set.<\/li>\n<\/ol>\n<p>Some examples of sets defined by listing the elements of the set:<\/p>\n<ol>\n<li>[latex]A\\;= \\;\\{1, 3, 9, 12\\}[\/latex].This is a finite set.<\/li>\n<li>[latex]\\text{R}\\;=\\;\\{\\text{red, orange, yellow, green, blue, indigo, purple}\\}[\/latex]. This is a finite set.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox\">\n<h3>Notation<\/h3>\n<p>The symbol [latex]\\in[\/latex] means \u201cis an element of\u201d.<\/p>\n<p>A set that contains no elements, [latex]\\{\\;\\}[\/latex], is called the <strong>empty set<\/strong> and is notated [latex]\\varnothing[\/latex].<\/p>\n<\/div>\n<div class=\"bcc-box bcc-success\">\n<h3>Example<\/h3>\n<p>Let\u00a0[latex]A\\;= \\;\\{1, 2, 3, 4\\}[\/latex]<\/p>\n<p>To say that 2 is an element of the set, we write [latex]2\\;\\in\\;A[\/latex]<em>.<\/em><\/p>\n<\/div>\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 <em><strong>subset<\/strong><\/em> of the larger set of all Madonna albums.<\/p>\n<div class=\"textbox\">\n<h3>Subset<\/h3>\n<p>A <strong>subset<\/strong> of a set [latex]A[\/latex]\u00a0is another set that contains only elements from the set [latex]A[\/latex]<em>. <\/em>A subset may contain some elements of set [latex]A[\/latex], no elements of set [latex]A[\/latex]\u00a0(i.e. the empty set, [latex]\\varnothing[\/latex]) or all elements of set [latex]A[\/latex]\u00a0(i.e. [latex]A[\/latex]).<\/p>\n<p>If set [latex]B[\/latex]\u00a0is a subset of set [latex]A[\/latex], we write [latex]B\\;\\subseteq\\;A[\/latex].<\/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 set [latex]B[\/latex]\u00a0is a proper subset of set [latex]A[\/latex], we write [latex]B\\;\\subset\\;A[\/latex].<\/p>\n<\/div>\n<p>The bottom bar in the subset notation\u00a0 [latex]\\subseteq[\/latex] represents the equality sign, meaning the whole set is a subset of itself. Notice that the proper subset notation\u00a0[latex]\\subset[\/latex] does not have that equals bar so does not include the whole set as a subset of itself.<\/p>\n<div class=\"bcc-box bcc-success\">\n<h3>Example<\/h3>\n<p>Consider these three sets:<\/p>\n<p style=\"padding-left: 30px;\">[latex]A=\\text{the set of all even numbers}[\/latex]<br \/>\n[latex]B=\\{2,4,6\\}[\/latex]<br \/>\n[latex]C=\\{2,3,4,6\\}[\/latex]<\/p>\n<p>Here [latex]B\\:\\subset\\:A[\/latex]\u00a0since every element of [latex]B[\/latex]\u00a0is an even number, so is an element of [latex]A[\/latex].<\/p>\n<p>More formally, we could say [latex]B\\:\\subset\\:A[\/latex]\u00a0since if [latex]x\\:\\in\\:B[\/latex], then [latex]x\\:\\in\\:A[\/latex].<\/p>\n<p>It is also true that [latex]B\\:\\subset\\:C[\/latex] because every element of [latex]B[\/latex] is an element of [latex]C[\/latex].<\/p>\n<p>[latex]C[\/latex] is NOT a subset of [latex]A[\/latex], since [latex]C[\/latex] contains an element, 3, that is not contained in [latex]A[\/latex]<em>.<\/em><\/p>\n<\/div>\n<div class=\"bcc-box bcc-success\">\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<p>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.<\/p>\n<\/div>\n<h2>Universal Set<\/h2>\n<div class=\"textbox\">\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 is defined by context.<\/p>\n<\/div>\n<div class=\"bcc-box bcc-success\">\n<h3>Example<\/h3>\n<ol>\n<li>If we are discussing searching for books, the universal set might be all the books in the library.<\/li>\n<li>If we are grouping our Facebook friends, the universal set would be all of our Facebook friends.<\/li>\n<li>If we are 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<h2>Union, Intersection, and Complement<\/h2>\n<p>It is common for sets to 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 are 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 <em>either<\/em> set.\u00a0The union is notated [latex]A\\:\\cup\\:B[\/latex].<em>\u00a0<\/em>More formally, [latex]x\\in \\:A\\cup\\:B[\/latex] if [latex]x\\in\\:A[\/latex] or [latex]x\\in\\:B[\/latex].<\/p>\n<p>The <strong>intersection <\/strong>of two sets contains only the elements that are in <em>both<\/em> sets.\u00a0The intersection is notated [latex]A\\cap\\,B[\/latex]<em>.\u00a0<\/em>More formally, [latex]x\\:\\in\\,A\\cap\\,B\\:\\text{if}\\:x\\in\\,A\\:\\text{and}\\:x\\in\\,B[\/latex].<\/p>\n<p>The <strong>complement<\/strong> of a set [latex]A[\/latex] contains everything that is <em>not<\/em> in the set [latex]A[\/latex].\u00a0The complement is notated [latex]A^{\\prime}[\/latex],\u00a0or [latex]A^c[\/latex]. A complement is relative to the universal set, so [latex]A^{\\prime}[\/latex]\u00a0contains all the elements in the universal set that are not in [latex]A[\/latex].<\/p>\n<\/div>\n<p>Notice that we used a letter, [latex]x[\/latex], to represent any element in a set. This letter is called a\u00a0<em><strong>variable<\/strong><\/em> and is just a stand-in for any element in a given set.<\/p>\n<div class=\"bcc-box bcc-success\">\n<h3>Example<\/h3>\n<p>Suppose the universal set is [latex]U[\/latex]\u00a0= all whole numbers from 1 to 9. i.e. [latex]U=\\{1,2,3,4,5,6,7,8,9\\}[\/latex]<\/p>\n<p>If [latex]A=\\{1, 2, 4\\}[\/latex], then [latex]A^{\\prime}=\\{3,5,6,7,8,9\\}[\/latex].<\/p>\n<\/div>\n<div class=\"bcc-box bcc-success\">\n<h3>Example<\/h3>\n<p>Consider the sets:<\/p>\n<p style=\"padding-left: 30px;\">[latex]A=\\{\\text{red, green, blue}\\}[\/latex]<br \/>\n[latex]B=\\{\\text{red, yellow, orange}\\}[\/latex]<br \/>\n[latex]C=\\{\\text{red, orange, yellow, green, blue, purple}\\}[\/latex]<\/p>\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^{\\prime}\\cap\\,C[\/latex]<\/li>\n<\/ol>\n<h4>Answers<\/h4>\n<ol>\n<li>The union contains all the elements in <em>either<\/em> 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 <em>both<\/em> sets: [latex]A\\cap\\,B=\\{\\text{red}\\}[\/latex]. Red is the only color in both sets.<\/li>\n<li>Here we\u2019re looking for all the elements that are <em>not<\/em> in set [latex]A[\/latex] but are in set [latex]C[\/latex]<em>: \u00a0<\/em>[latex]A^{\\prime}\\cap\\,C=\\{\\text{orange, yellow, purple}\\}[\/latex]<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox shaded\">\n<h3>Try It<\/h3>\n<p>Using the sets from the previous example, find [latex]A\\cup\\,C[\/latex]\u00a0and [latex]B^{\\prime}\\cap\\,A[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q423428\">Show Answer<\/span><\/p>\n<div id=\"q423428\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]A\\cup\\,C=\\{\\text{red, green, blue, orange yellow, purple}\\}[\/latex]<\/p>\n<p>[latex]B^{\\prime}\\cap\\,A=\\{\\text{green, blue}\\}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>Set operations can be grouped together using grouping symbols to force an order of operations, just like in arithmetic. Parentheses are used to tell us which operation to perform first.<\/p>\n<div class=\"bcc-box bcc-success\">\n<h3>Example<\/h3>\n<p>Suppose [latex]H=\\{\\text{cat, dog, rabbit, mouse}\\}[\/latex], [latex]F=\\{\\text{dog, cow, duck, pig, rabbit}\\}[\/latex], and [latex]W=\\{\\text{duck, rabbit, deer, frog, mouse}\\}[\/latex].<\/p>\n<ol>\n<li>Find<em>\u00a0<\/em>[latex](H\\cap\\,F)\\cup\\,W[\/latex]<\/li>\n<li>Find [latex]H\\cap\\,(F\\cup\\,W)[\/latex]<\/li>\n<li>Find [latex](H\\cap\\,F)^{\\prime}\\cap\\,W[\/latex]<\/li>\n<\/ol>\n<h4>Solutions<\/h4>\n<ol>\n<li>We start with the intersection since it is in parentheses:\u00a0[latex](H\\cap\\,F)=\\{\\text{dog, rabbit}\\}[\/latex].\u00a0Now we union that result with [latex]W[\/latex]: <em>\u00a0<\/em>[latex](H\\cap\\,F)\\cup\\,W=\\{\\text{dog, duck, rabbit, deer, frog, mouse}\\}[\/latex].<\/li>\n<li>We start with the union inside parentheses: [latex](F\\cup\\,W)=\\{\\text{dog, cow, duck, pig, rabbit, deer, frog, mouse}\\}[\/latex]\u00a0.\u00a0Now we intersect that result with [latex]H[\/latex]: [latex]H\\cap\\,(F\\cup\\,W)=\\{\\text{dog, rabbit, mouse}\\}[\/latex].<\/li>\n<li>We start with the intersection inside parentheses: [latex](H\\cap\\,F)=\\{\\text{dog, rabbit}\\}[\/latex].\u00a0Now we want to find the elements of [latex]W[\/latex]\u00a0that are <em>not<\/em> in [latex](H\\cap\\,F)[\/latex]<em>. \u00a0<\/em>[latex](H\\cap\\,F)^{\\prime}\\cap\\,W= \\{\\text{duck, deer, frog, mouse}\\}[\/latex].<\/li>\n<\/ol>\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 eighteenth\u00a0century. These illustrations are now called <em><strong>Venn Diagrams<\/strong><\/em>.<\/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<\/div>\n<div class=\"bcc-box bcc-success\">\n<h3>Example<\/h3>\n<p>Suppose the universal set is students in a class. [latex]A[\/latex] = all students with blonde hair; [latex]B[\/latex] = all students with blue eyes; [latex]C[\/latex] = all students at least 6 feet tall.<\/p>\n<p>&nbsp;<\/p>\n<ol>\n<li>What does [latex]B^{\\prime}[\/latex] represent?<\/li>\n<li>What does\u00a0\u00a0[latex]A\\cap\\,B^{\\prime}[\/latex] represent?<\/li>\n<li>What does\u00a0\u00a0[latex](A\\cup C)^{\\prime}\\cap B[\/latex] represent?Use the Venn to write set interactions to represent each category.<\/li>\n<li>Students with blonde hair and blue eyes.<\/li>\n<li>Students with blue eyes that are at least 6 feet tall.<\/li>\n<li>Students who have blonde hair, blue eyes, and are at least 6 feet tall.<\/li>\n<li>Students who are under 6 feet tall and have neither blonde hair nor blue eyes.<\/li>\n<li><span style=\"font-size: 1rem; text-align: initial;\">Students who have neither blonde hair nor blue eyes but are at least 6&#8242; tall.<\/span><\/li>\n<\/ol>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-1477 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5676\/2021\/10\/07220834\/Class-Ven.png\" alt=\"Venn diagram\" width=\"964\" height=\"692\" \/><\/p>\n<p>&nbsp;<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qHM8872\">Show Answer<\/span><\/p>\n<div id=\"qHM8872\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>[latex]B^{\\prime}[\/latex] represents students who do not have blue eyes.<\/li>\n<li>Students who are blonde but do not have blue eyes.<\/li>\n<li>Students with blue eyes who are neither blonde nor at least 6&#8242; tall.<\/li>\n<li>This is found in the intersection of the blue and yellow circles: \u00a0[latex]A\\cap\\,B[\/latex]<\/li>\n<li>This is found in the intersection of the blue and green circles: \u00a0[latex]B\\cap\\,C[\/latex]<\/li>\n<li>This is found in the intersection of the yellow, blue and green circles: \u00a0[latex]A\\cap\\,B\\cap\\,C[\/latex]<\/li>\n<li>If they are under 6&#8242; tall they are not in the green circle. If they do not have blonde hair they are not in the yellow circle. If they do not have blue eyes they are not in the blue circle. Therefore they are outside all 3 circles. \u00a0 [latex](A\\cup\\,B\\cup\\,C)^{\\prime}[\/latex]<\/li>\n<li>These students are found in the green circle but are neither in the blue nor yellow circles. \u00a0 [latex](A\\cup\\,B)^{\\prime}\\cap\\,C[\/latex]<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox shaded\">\n<p><strong>TRY IT<\/strong><\/p>\n<p>The Venn diagram shows the interaction of sports watched on TV by people who answered a survey. [latex]S[\/latex] = {watched soccer}, [latex]F[\/latex] = {watched football}, [latex]B[\/latex] = {watched baseball}, [latex]G[\/latex] = {watched golf}<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-1478 size-large\" src=\"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-content\/uploads\/sites\/5676\/2021\/10\/Sports-Venn1-1024x574.png\" alt=\"A Venn diagram where sets S, B, and F intersects each other and set G only intersects with F and not with S or B.\" width=\"1024\" height=\"574\" srcset=\"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-content\/uploads\/sites\/5676\/2021\/10\/Sports-Venn1-1024x574.png 1024w, https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-content\/uploads\/sites\/5676\/2021\/10\/Sports-Venn1-300x168.png 300w, https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-content\/uploads\/sites\/5676\/2021\/10\/Sports-Venn1-768x431.png 768w, https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-content\/uploads\/sites\/5676\/2021\/10\/Sports-Venn1-65x36.png 65w, https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-content\/uploads\/sites\/5676\/2021\/10\/Sports-Venn1-225x126.png 225w, https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-content\/uploads\/sites\/5676\/2021\/10\/Sports-Venn1-350x196.png 350w, https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-content\/uploads\/sites\/5676\/2021\/10\/Sports-Venn1.png 1352w\" sizes=\"auto, (max-width: 1024px) 100vw, 1024px\" \/><\/p>\n<p>&nbsp;<\/p>\n<p>Use set interactions to represent each category.<\/p>\n<ol>\n<li>People who watch only baseball?<\/li>\n<li>People who watch golf and football?<\/li>\n<li>People who watch football and baseball?<\/li>\n<li>People who watch football, baseball, and soccer?<\/li>\n<li>People who watch football but neither baseball nor soccer?<\/li>\n<li>People who do not watch golf?<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q706599\">Show Answer<\/span><\/p>\n<div id=\"q706599\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>[latex](S\\cup\\,F)^{\\prime}\\cap\\,B[\/latex]<\/li>\n<li>[latex]G\\cap\\,F[\/latex]<\/li>\n<li>[latex]F\\cap\\,B[\/latex]<\/li>\n<li>[latex]F\\cap\\,B\\cap\\,S[\/latex]<\/li>\n<li>[latex](B\\cup\\,S)^{\\prime}\\cap\\,F[\/latex]<\/li>\n<li>[latex]G^{\\prime}[\/latex]<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<p>Even when the elements of the sets are unknown, Venn diagrams can illustrate unions, intersections, and complements.<\/p>\n<div class=\"bcc-box bcc-success\">\n<h3>Example<\/h3>\n<p>Create Venn diagrams to illustrate [latex]A\\cup\\,B[\/latex], [latex]A\\cap\\,B[\/latex], and [latex]A^{\\prime}\\cap\\,B[\/latex].<\/p>\n<p>[latex]A\\cup\\,B[\/latex]\u00a0contains all elements in <em>either<\/em> set.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-166 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1212\/2015\/11\/02174017\/Fig3_1_1.png\" alt=\"Both sets A and B, which intersect each other, are enclosed by a bold red line.\" width=\"312\" height=\"213\" \/><\/p>\n<p>[latex]A\\cap\\,B[\/latex] contains only those elements in both sets\u2014in the overlap of the circles.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-167 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1212\/2015\/11\/02174019\/Fig3_1_2.png\" alt=\"The intersection area of sets A and B is enclosed by a bold red line.\" width=\"308\" height=\"209\" \/><\/p>\n<p>[latex]A^{\\prime}[\/latex] will contain all elements <em>not<\/em> in the set [latex]A[\/latex]. [latex]A^{\\prime}\\cap\\,B[\/latex]\u00a0will contain the elements in set [latex]B[\/latex]\u00a0that are not in set\u00a0[latex]A[\/latex].<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-168 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1212\/2015\/11\/02174020\/Fig3_1_3.png\" alt=\"Set B, without its intersection with set A, is enclosed by a bold red line.\" width=\"309\" height=\"205\" \/><\/p>\n<\/div>\n<div class=\"bcc-box bcc-success\">\n<h3>Example<\/h3>\n<p>Use a Venn diagram to illustrate [latex](H\\cap\\,F)^{\\prime}\\cap\\,W[\/latex]<\/p>\n<p>We\u2019ll start by identifying everything in the set [latex]H\\cap\\,F[\/latex]<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-169 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1212\/2015\/11\/02174021\/Fig3_1_4.png\" alt=\"The intersection of sets H and F, where sets H, F and W intersect each other, is enclosed by a bold red line.\" width=\"303\" height=\"277\" \/><\/p>\n<p>&nbsp;<\/p>\n<p>Now, [latex](H\\cap\\,F)^{\\prime}\\cap\\,W[\/latex] 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-170 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1212\/2015\/11\/02174022\/Fig3_1_5.png\" alt=\"Set W, excluding the intersection region of all three sets H and F and W, is enclosed by a bold red line.\" width=\"306\" height=\"276\" \/><\/p>\n<\/div>\n<div class=\"bcc-box bcc-success\">\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 wp-image-171 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1212\/2015\/11\/02174023\/Fig3_1_6.png\" alt=\"The intersection of sets H and F, excluding the intersection area of all three sets H and F and W, is enclosed by a bold red line.\" width=\"305\" height=\"269\" \/><\/p>\n<p>The elements in the outlined set <em>are<\/em> in sets [latex]H[\/latex] and [latex]F[\/latex], but are not in set [latex]W[\/latex]. So we represent this set as [latex]H\\cap\\,F\\cap\\,W^{\\prime}[\/latex].<\/p>\n<\/div>\n<div class=\"textbox shaded\">\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 wp-image-172 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1212\/2015\/11\/02174025\/Fig3_1_7.png\" alt=\"The intersection of sets A and B, excluding the intersection region with set C and the rest of set C, is enclosed by a bold red line.\" width=\"305\" height=\"275\" \/><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q475309\">Show Answer<\/span><\/p>\n<div id=\"q475309\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]C[\/latex] is excluded so [latex]C^{\\prime}[\/latex] is intersected with [latex]A[\/latex] and [latex]B[\/latex]: [latex]C^{\\prime}\\cap\\,(A\\cup\\,B)[\/latex]<\/p>\n<\/div>\n<\/div>\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-1065\">\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>Venn diagram examples and Try It. <strong>Authored by<\/strong>: Hazel McKenna. <strong>Provided by<\/strong>: Utah Valley University. <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>Adapted: Lumen Learning. <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>. <strong>License Terms<\/strong>:  CC BY-SA: Attribution-ShareAlike<\/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":160,"menu_order":1,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Adapted: Lumen Learning\",\"author\":\"David Lippman\",\"organization\":\"\",\"url\":\" http:\/\/www.opentextbookstore.com\/mathinsociety\/\",\"project\":\"Math in Society\",\"license\":\"cc-by-sa\",\"license_terms\":\" CC BY-SA: Attribution-ShareAlike\"},{\"type\":\"original\",\"description\":\"Venn diagram examples and Try It\",\"author\":\"Hazel McKenna\",\"organization\":\"Utah Valley University\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"}]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-1065","chapter","type-chapter","status-publish","hentry"],"part":587,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/pressbooks\/v2\/chapters\/1065","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/wp\/v2\/users\/160"}],"version-history":[{"count":57,"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/pressbooks\/v2\/chapters\/1065\/revisions"}],"predecessor-version":[{"id":3169,"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/pressbooks\/v2\/chapters\/1065\/revisions\/3169"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/pressbooks\/v2\/parts\/587"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/pressbooks\/v2\/chapters\/1065\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/wp\/v2\/media?parent=1065"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/pressbooks\/v2\/chapter-type?post=1065"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/wp\/v2\/contributor?post=1065"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/wp\/v2\/license?post=1065"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}