{"id":32,"date":"2023-01-25T16:33:54","date_gmt":"2023-01-25T16:33:54","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/chapter\/set-theory-basics-dig-deeper-page-1\/"},"modified":"2024-10-18T20:50:11","modified_gmt":"2024-10-18T20:50:11","slug":"set-theory-basics-fresh-take","status":"web-only","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/chapter\/set-theory-basics-fresh-take\/","title":{"raw":"Set Theory Basics: Fresh Take","rendered":"Set Theory Basics: Fresh Take"},"content":{"raw":"<section class=\"textbox learningGoals\">\r\n<ul>\r\n\t<li>Understand the concepts of sets, including empty sets, subsets, and proper subsets, and use correct set notation<\/li>\r\n\t<li>Describe and perform set operations (union, intersection, complement, and difference) using proper set notation<\/li>\r\n\t<li>Create and interpret Venn diagrams to represent and analyze set relationships and operations<\/li>\r\n\t<li>Apply the concepts of sets, subsets, and cardinality properties to solve real-life problems<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Sets<\/h2>\r\n<div class=\"textbox shaded\">\r\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\r\n<p>A <strong>set<\/strong> is a collection of unique objects or elements. These elements can be anything: numbers, people, other sets, etc. For example, the set [latex]A[\/latex] could be the set of all positive integers, while the set B could be the set of all colors in the rainbow.<\/p>\r\n<p>The <strong>empty set<\/strong>, also known as the null set, is a special set that contains no elements. For example, the set of all dogs with six legs is an empty set, since no such dogs exist.<\/p>\r\n<p><strong>Notation<\/strong><br \/>\r\nIn set theory, we often use curly brackets - [latex]\\{\\}[\/latex] - to denote a set. An example of a set is [latex]A = \\{a, b, c, d\\}[\/latex].<\/p>\r\n<p>Commonly, we will use a variable to represent a set, to make it easier to refer to that set later. We usually denote sets with capital letters such as [latex]A[\/latex], [latex]B[\/latex], [latex]C[\/latex], etc.<\/p>\r\n<p>The symbol [latex]\\in[\/latex]\u00a0means \u201cis an element of\u201d.<\/p>\r\n<p>An empty set is denoted by the symbols is notated by [latex]\\emptyset[\/latex] or [latex]\\{ \\}[\/latex].<\/p>\r\n<\/div>\r\n<section class=\"textbox watchIt\"><iframe src=\"\/\/plugin.3playmedia.com\/show?mf=10205707&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=fes92vSBTg4&amp;video_target=tpm-plugin-1fy67j04-fes92vSBTg4\" width=\"800px\" height=\"450px\" frameborder=\"0\" marginwidth=\"0px\" marginheight=\"0px\"><\/iframe><br \/>\r\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Sets+Basics+-+Introduction+%7C+Don't+Memorise.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cSets Basics - Introduction | Don't Memorise\u201d here (opens in new window).<\/a><\/p>\r\n<\/section>\r\n<section class=\"textbox watchIt\"><iframe src=\"\/\/plugin.3playmedia.com\/show?mf=10205708&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=LumU80IN748&amp;video_target=tpm-plugin-2tflvwyb-LumU80IN748\" width=\"800px\" height=\"450px\" frameborder=\"0\" marginwidth=\"0px\" marginheight=\"0px\"><\/iframe><br \/>\r\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Sets+-+Listing+Method+%7C+Don't+Memorise.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cSets - Listing Method | Don't Memorise\u201d here (opens in new window).<\/a><\/p>\r\n<\/section>\r\n<h2>Subsets<\/h2>\r\n<div class=\"textbox shaded\" style=\"text-align: left;\">\r\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\r\n<p>A set [latex]A[\/latex] is said to be a <strong>subset<\/strong> of another set [latex]B[\/latex] if every element in [latex]A[\/latex] is also in [latex]B[\/latex]. We denote this by [latex]A \\subseteq B[\/latex]. For instance, if [latex]B = \\{\\text{red}, \\text{green}, \\text{blue}, \\text{yellow}\\}[\/latex] and [latex]A = \\{\\text{red}, \\text{blue}, \\text{yellow}\\}[\/latex], then [latex]A[\/latex] is a subset of [latex]B[\/latex] because every color in [latex]A[\/latex] is also in [latex]B[\/latex].<\/p>\r\n<p>A <strong>proper subset<\/strong> is similar to a subset, but with one key difference: if [latex]A[\/latex] is a proper subset of [latex]B[\/latex], then there is at least one element in [latex]B[\/latex] that is not in [latex]A[\/latex]. We denote this by [latex]A \\subset B[\/latex]. If [latex]B = \\{\\text{red}, \\text{green}, \\text{blue}, \\text{yellow}\\}[\/latex] and [latex]A = \\{\\text{red}, \\text{blue}\\}[\/latex], then [latex]A[\/latex] is a proper subset of [latex]B[\/latex].<\/p>\r\n<p>To calculate the number of subsets and proper subsets in a set you must use the following formulas:<\/p>\r\n<p>If you have [latex]n[\/latex] elements in your set:<\/p>\r\n<ul>\r\n\t<li>Number of subsets: [latex]2^{n}[\/latex]<\/li>\r\n\t<li>Number of proper subsets: [latex]2^{n}\u22121[\/latex]<\/li>\r\n<\/ul>\r\n<\/div>\r\n<section class=\"textbox example\">Consider these three sets:<br \/>\r\n<br \/>\r\n[latex]A[\/latex] = the set of all even numbers<br \/>\r\n[latex]B = \\{2, 4, 6\\}[\/latex]<br \/>\r\n[latex]C = \\{2, 3, 4, 6\\}[\/latex]<br \/>\r\n<br \/>\r\nHere [latex]B \\subset A[\/latex] since every element of [latex]B[\/latex] is also an even number, so is an element of [latex]A[\/latex]. More formally, we could say [latex]B \\subset A[\/latex] since if [latex]x \\in B[\/latex], then [latex]x \\in A[\/latex]. It is also true that [latex]B \\subset C[\/latex]. [latex]C[\/latex] is not a subset of [latex]A[\/latex], since [latex]C[\/latex] contains an element, [latex]3[\/latex], that is not contained in [latex]A[\/latex].<\/section>\r\n<section class=\"textbox seeExample\">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?<br \/>\r\n[reveal-answer q=\"857946\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"857946\"]There are many possible answers here. One would be the set of plays by Shakespeare. This is also a subset of the set of all plays ever written. It is also a subset of all British literature.[\/hidden-answer]<\/section>\r\n<section class=\"textbox seeExample\">Consider the\u00a0set\u00a0[latex]A = \\{1, 3, 5\\} [\/latex]. Which of the following sets is [latex]A [\/latex] a subset of?<br \/>\r\n<br \/>\r\n[latex]X = \\{1, 3, 7, 5\\} [\/latex]<br \/>\r\n[latex]Y = \\{1, 3 \\} [\/latex]<br \/>\r\n[latex]Z = \\{1, m, n, 3, 5\\}[\/latex]<br \/>\r\n[reveal-answer q=\"3546\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"3546\"] [latex] X [\/latex] and [latex] Z [\/latex] [\/hidden-answer]<\/section>\r\n<section class=\"textbox seeExample\">Using the set: [latex]\\text{Soda} = \\{\\text{Sprite}, \\text{Dr. Pepper}, \\text{Diet Coke}, \\text{Mountain Dew}, \\text{Cherry Coke}\\}[\/latex], calculate the number of subsets and proper subsets.<br \/>\r\n[reveal-answer q=\"3541\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"3541\"]For this set, there are [latex]5[\/latex] elements which means [latex]n = 5[\/latex]. To determine the number of subsets we must use [latex]2^n[\/latex]:\r\n\r\n\r\n<div style=\"text-align: center;\">[latex]\\begin{array}{ccc}\\hfill 2^5 = 2^5 = 32\\\\ \\end{array}[\/latex]<br \/>\r\n<br \/>\r\n<\/div>\r\n<p>To determine the number of proper subsets we must use [latex]2^n - 1[\/latex]:<\/p>\r\n<div style=\"text-align: center;\">[latex]\\begin{array}{ccc}\\hfill 2^n - 1 = 2^5 - 1 = 32 - 1 = 31\\\\ \\end{array}[\/latex]<br \/>\r\n<br \/>\r\n<\/div>\r\n<p>There are [latex]32[\/latex] subsets and [latex]31[\/latex] proper subsets.<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<section class=\"textbox watchIt\"><iframe src=\"\/\/plugin.3playmedia.com\/show?mf=10371002&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=_9Wvu-R04go&amp;video_target=tpm-plugin-jv0rfpbh-_9Wvu-R04go\" width=\"800px\" height=\"450px\" frameborder=\"0\" marginwidth=\"0px\" marginheight=\"0px\"><\/iframe><br \/>\r\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/What+is+a+Subset%3F+%7C+Don't+Memorise.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cWhat is a Subset? | Don't Memorise\u201d here (opens in new window).<\/a><\/p>\r\n<\/section>\r\n<section class=\"textbox watchIt\"><iframe title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/xotLg-oLboY\" width=\"560\" height=\"315\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><br \/>\r\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Subsets%2C+Proper+Subsets+and+Supersets+_+Don't+Memorise.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cSubsets, Proper Subsets and Supersets | Don't Memorise\u201d here (opens in new window).<\/a><\/p>\r\n<\/section>\r\n<section class=\"textbox watchIt\"><iframe src=\"\/\/plugin.3playmedia.com\/show?mf=10205709&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=BhFgcf0VSYc&amp;video_target=tpm-plugin-kh2dl4wg-BhFgcf0VSYc\" width=\"800px\" height=\"450px\" frameborder=\"0\" marginwidth=\"0px\" marginheight=\"0px\"><\/iframe><br \/>\r\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/What+is+a+Null+Set+%7C+Is+Null+Set+a+Subset+of+Every+Set%3F+%7C+Don't+Memorise.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cWhat is a Null Set | Is Null Set a Subset of Every Set? | Don't Memorise\u201d here (opens in new window).<\/a><\/p>\r\n<\/section>\r\n<h2>Set Operations: Union, Intersection, Complement, and Difference<\/h2>\r\n<p>A big part of understanding proper notation in sets and set operations is being able to read and translate the symbols used. There are a lot of different symbols used that make things confusing. We can use the following table as a reminder of what each symbol means:<\/p>\r\n<table>\r\n<tbody>\r\n<tr>\r\n<th>Symbol<\/th>\r\n<th>Symbol Meaning<\/th>\r\n<th>Translation<\/th>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]\\in[\/latex]<\/td>\r\n<td>\u201cis an element of\u201d or \u201cis an element in\u201d<\/td>\r\n<td>[latex]1 \\in P[\/latex] reads as: [latex]1[\/latex] is an element of the set [latex]P[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]\\subseteq[\/latex]<\/td>\r\n<td>Subset<\/td>\r\n<td>[latex]C \\subseteq R[\/latex] reads as: set [latex]C[\/latex] is a subset of set [latex]R[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]\\subset[\/latex]<\/td>\r\n<td>Proper Subset<\/td>\r\n<td>[latex]C \\subset R[\/latex] reads as: set [latex]C[\/latex] is a proper subset of set [latex]R[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]\\cup[\/latex]<\/td>\r\n<td>Union of two sets<\/td>\r\n<td>[latex]C \\cup R[\/latex] reads as: the union of sets [latex]C[\/latex] and [latex]R[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]\\cap[\/latex]<\/td>\r\n<td>Intersection of two sets<\/td>\r\n<td>[latex]C \\cap R[\/latex] reads as: the intersection of sets [latex]C[\/latex] and [latex]R[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]R'[\/latex], or [latex]R^c[\/latex], or ~[latex]R[\/latex]<\/td>\r\n<td>Complement of a set<\/td>\r\n<td>[latex]R'[\/latex] reads as: the complement of set [latex]R[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]\\setminus[\/latex]<\/td>\r\n<td>Difference of two sets<\/td>\r\n<td>[latex]C \\setminus R[\/latex] reads as: the difference between sets [latex]C[\/latex] and [latex]R[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p>&nbsp;<\/p>\r\n<p>Now that we can see what each symbol translates to let\u2019s break down each of the formal notations of the set operations to better understand what they are stating.<\/p>\r\n<h3>Union<\/h3>\r\n<div class=\"textbox shaded\" style=\"text-align: left;\">\r\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\r\n<p>The <strong>union<\/strong> of two sets contains all the elements contained in either set (or both sets) and is notated [latex]A \\cup B[\/latex].\u00a0The formal notation states, [latex]x \\in A \\cup B[\/latex] if [latex]x \\in A[\/latex] or [latex]x \\in B[\/latex] (or both).<\/p>\r\n<p>If we break this down and translate the symbols into words it is easier to understand.<\/p>\r\n<ul>\r\n\t<li>[latex]x \\in A \\cup B[\/latex] reads as \u201c[latex]x[\/latex] is an element of the union of sets [latex]A[\/latex] and [latex]B[\/latex] \u201c<\/li>\r\n\t<li>[latex]x \\in A[\/latex] reads as \u201c[latex]x[\/latex] is an element of set [latex]A[\/latex] \u201c<\/li>\r\n\t<li>[latex]x \\in B[\/latex] reads as \u201c[latex]x[\/latex] is an element of set [latex]B[\/latex] \u201c<\/li>\r\n<\/ul>\r\n<p>If we add all the parts we translated into words back into the original formal definition it reads as \u201c[latex]x[\/latex] is an element of the union of sets [latex]A[\/latex] and [latex]B[\/latex] if [latex]x[\/latex] is an element of set [latex]A[\/latex] or [latex]x[\/latex] is an element of set [latex]B[\/latex] (or both).\u201d<\/p>\r\n<\/div>\r\n<h3>Intersection<\/h3>\r\n<div class=\"textbox shaded\" style=\"text-align: left;\">\r\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\r\n<p>The <strong>intersection <\/strong>of two sets contains only the elements that are in both sets.\u00a0The 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].<\/p>\r\n<p>If we break this down and translate the symbols into words it is easier to understand.<\/p>\r\n<ul>\r\n\t<li>[latex]x \\in A \\cap B[\/latex] reads as \u201c[latex]x[\/latex] is an element of the intersection of sets [latex]A[\/latex] and [latex]B[\/latex] \u201c<\/li>\r\n\t<li>[latex]x \\in A[\/latex] reads as \u201c[latex]x[\/latex] is an element of set [latex]A[\/latex] \u201c<\/li>\r\n\t<li>[latex]x \\in B[\/latex] reads as \u201c[latex]x[\/latex] is an element of set [latex]B[\/latex] \u201c<\/li>\r\n<\/ul>\r\n<p>If we add all the parts we translated into words back into the original formal definition it reads as \u201c[latex]x[\/latex] is an element of the intersection of sets [latex]A[\/latex] and [latex]B[\/latex] if [latex]x[\/latex] is an element of set [latex]A[\/latex] and [latex]x[\/latex] is an element of set [latex]B[\/latex]. \u201c<\/p>\r\n<\/div>\r\n<section class=\"textbox proTip\">\r\n<h3>intersection and union symbols<\/h3>\r\n<p>The intersection [latex]\\cup[\/latex] and union [latex]\\cap[\/latex] symbols look a little like letters in the alphabet. In fact, that\u2019s a trick for remembering them.<\/p>\r\n<p>The union symbol looks like a capital U, for <em>union<\/em>.<\/p>\r\n<p>The intersection symbol looks a little like a big lower-case n, for <em>in-tersect.<\/em><\/p>\r\n<\/section>\r\n<h3>Difference<\/h3>\r\n<div class=\"textbox shaded\" style=\"text-align: left;\">\r\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\r\n<p>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. The difference between two sets is notated [latex]A \\setminus B[\/latex].<\/p>\r\n<p>More formally, [latex]x \\in A \\setminus B[\/latex] if [latex]x \\in A[\/latex] &amp; [latex]x \\notin B [\/latex].<\/p>\r\n<p>If we break this down and translate the symbols into words it is easier to understand.<\/p>\r\n<ul>\r\n\t<li>[latex]x \\in A \\setminus B [\/latex] reads as \u201c[latex]x[\/latex] is an element of the difference of sets [latex]A[\/latex] and [latex]B[\/latex] \u201c<\/li>\r\n\t<li>[latex]x \\in A[\/latex] reads as \u201c[latex]x[\/latex] is an element of set [latex]A[\/latex] \u201c<\/li>\r\n\t<li>[latex]x \\notin B[\/latex] reads as \u201c[latex]x[\/latex] is not an element of set [latex]B[\/latex] \u201c<\/li>\r\n<\/ul>\r\n<p>If we add all the parts we translated into words back into the original formal definition it reads as \u201c[latex]x[\/latex] is an element of the difference of sets [latex]A[\/latex] and [latex]B[\/latex] if [latex]x[\/latex] is an element of set [latex]A[\/latex] and [latex]x[\/latex] is not an element of set [latex]B[\/latex] .\u201c<\/p>\r\n<\/div>\r\n<h3>Universal Set<\/h3>\r\n<div class=\"textbox shaded\" style=\"text-align: left;\">\r\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\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<\/div>\r\n<section class=\"textbox seeExample\">\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<\/section>\r\n<h3>Complement<\/h3>\r\n<div class=\"textbox shaded\" style=\"text-align: left;\">\r\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\r\n<p>A <strong>complement<\/strong> is relative to the universal set. The <strong>complement<\/strong> of a set [latex]A[\/latex] contains everything that is <em>not<\/em> in the universal set [latex]A[\/latex].\u00a0 The complement is notated [latex]A'[\/latex], or [latex]A^c[\/latex], or sometimes \u00a0~[latex]A[\/latex].<\/p>\r\n<\/div>\r\n<section class=\"textbox seeExample\">Suppose the universal set is <em>U<\/em> = all whole numbers from [latex]1[\/latex] to [latex]9[\/latex]. If [latex]A = {1, 2, 4}[\/latex], then [latex]A^{c} = \\{3, 5, 6, 7, 8, 9\\}[\/latex].<\/section>\r\n<section class=\"textbox seeExample\">Consider the sets:<br \/>\r\n<br \/>\r\n[latex]A = \\{\\text{red, green, blue}\\}[\/latex]<br \/>\r\n[latex]B = \\{\\text{red, yellow, orange}\\}[\/latex]<br \/>\r\n[latex]C = \\{\\text{red, orange, yellow, green, blue, purple}\\}[\/latex] Find the following:\r\n\r\n\r\n<ol>\r\n\t<li>Find [latex]A \\cup C[\/latex]<\/li>\r\n\t<li>Find [latex]B^c \\cap A[\/latex]<\/li>\r\n<\/ol>\r\n<p>[reveal-answer q=\"3551\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"3551\"]<\/p>\r\n<ol>\r\n\t<li style=\"text-align: left;\">The union contains all the elements in either set:\r\n<center>[latex]A \\cup C=\\{\\text { red, orange, yellow, green, blue purple }\\}[\/latex]<\/center><p><\/p><\/li>\r\n\t<li>Here we\u2019re looking for all the elements that are <em>not<\/em> in set [latex]B[\/latex] and are also in [latex]A[\/latex] .<center>[latex]B^{c} \\cap A=\\{\\text { green, blue }\\}[\/latex]<\/center><\/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]2223[\/ohm2_question]<\/section>\r\n<section class=\"textbox watchIt\"><iframe src=\"\/\/plugin.3playmedia.com\/show?mf=10205710&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=fLMN0wtiz-4&amp;video_target=tpm-plugin-oesmkpqb-fLMN0wtiz-4\" width=\"800px\" height=\"450px\" frameborder=\"0\" marginwidth=\"0px\" marginheight=\"0px\"><\/iframe><br \/>\r\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Sets_UnionIntersectionComplement.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cSets: Union, Intersection, Complement\u201d here (opens in new window).<\/a><\/p>\r\n<\/section>\r\n<p>Any of the set operations can be grouped together or stand alone to make statements about different sets. When symbols are grouped together they form an order of operations. Don\u2019t forget, like in arithmetic, when applying the order of operations parenthesis indicates the operation you do first.<\/p>\r\n<section class=\"textbox seeExample\">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].\r\n\r\n\r\n<ol>\r\n\t<li>Find [latex](H \\cap F) \\cup W[\/latex]<\/li>\r\n\t<li>Find [latex](F \\cup W) \\cap H[\/latex]<\/li>\r\n\t<li>Find [latex](H \\cap F)^c \\cap W[\/latex]<\/li>\r\n<\/ol>\r\n<p>[reveal-answer q=\"3552\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"3552\"]<\/p>\r\n<ol>\r\n\t<li>We start with the intersection: [latex](H \\cap F) = \\{\\text{dog, rabbit}\\}[\/latex]. Now we union that result with [latex]W[\/latex]: [latex](H \\cap F) \\cup W = \\{\\text{dog, duck, rabbit, deer, frog, mouse}\\}[\/latex].\r\n<p>&nbsp;<\/p><\/li>\r\n\t<li>We start with the union: [latex](F \\cup W) = \\{\\text{dog, cow, rabbit, duck, pig, deer, frog, mouse}\\}[\/latex]. Now we intersect that result with [latex]H[\/latex]: [latex](F \\cup W) \\cap H = \\{\\text{dog, rabbit, mouse}\\}[\/latex].\r\n<p>&nbsp;<\/p>\r\n<\/li>\r\n\t<li>We start with the intersection: Find [latex](H \\cap F) = \\{\\text{dog, rabbit}\\}[\/latex].\u00a0 Now we want to find the elements of [latex]W[\/latex] that are <em>not<\/em> in [latex](H \\cap F)[\/latex]. [latex](H \\cap F)^c \\cap W = \\{\\text{duck, deer, frog, mouse}\\}[\/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]2224[\/ohm2_question]<\/section>\r\n<h2>Venn Diagrams<\/h2>\r\n<div class=\"textbox shaded\">\r\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\r\n<p>Venn diagrams are a helpful tool for visualizing the relationships between sets and performing set operations. A Venn diagram uses overlapping circles or other shapes to illustrate the logical relationships between two or more sets of items. Each circle represents a set, with the items of the set represented by points within the circle. Here's how you can create and interpret Venn diagrams for various set relationships and operations:<\/p>\r\n<ul>\r\n\t<li><strong>Representing Sets<\/strong>: In a Venn diagram, each set is represented by a circle. For example, if you have two sets A and B, you can represent them with two circles. The universal set is often represented by a rectangle encompassing all the circles.<\/li>\r\n\t<li><strong>Union<\/strong>: The union of two sets A and B (A \u222a B) is represented by the area covered by both circles A and B, including the overlap.<\/li>\r\n\t<li><strong>Intersection<\/strong>: The intersection of two sets A and B (A \u2229 B) is represented by the area where the two circles overlap.<\/li>\r\n\t<li><strong>Difference<\/strong>: The difference of two sets A and B (A - B) is represented by the area that is in circle A but not in circle B.<\/li>\r\n\t<li><strong>Complement<\/strong>: The complement of a set A (A') is represented by the area that is outside circle A but within the universal set.<\/li>\r\n\t<li><strong>Empty Set<\/strong>: An empty set or null set, which contains no elements, can be represented by a circle with nothing in it. If the set is a subset of another set, it can be represented by a circle within the larger set, but with no elements.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<section class=\"textbox example\">Create an expression to represent the outlined portion of the Venn diagram shown:<br \/>\r\n<center><img class=\"size-full wp-image-4218 aligncenter\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/18\/2023\/01\/07151939\/clipboard_ee6f4b6ece2cde3e7320d09311c24e49b.png\" alt=\"A Venn diagram of three sets A B and C are shown overlapping. The region highlighted includes anything in A or anything in B, excluding anything also in C\" width=\"274\" height=\"257\" \/><\/center>[reveal-answer q=\"2178\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"2178\"]<br \/>\r\n[latex]A \\cup B \\cap C^{c}[\/latex]<br \/>\r\n[\/hidden-answer]<\/section>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]2255[\/ohm2_question]<\/section>\r\n<section class=\"textbox watchIt\"><iframe title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/CPeeOUldZ6M\" width=\"560\" height=\"315\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><br \/>\r\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Sets_+drawing+a+Venn+diagram.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cSets: drawing a Venn diagram\u201d here (opens in new window).<\/a><\/p>\r\n<\/section>\r\n<section class=\"textbox watchIt\"><iframe src=\"\/\/plugin.3playmedia.com\/show?mf=10205711&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=32iNIZJ2dI4&amp;video_target=tpm-plugin-a9364ct3-32iNIZJ2dI4\" width=\"800px\" height=\"450px\" frameborder=\"0\" marginwidth=\"0px\" marginheight=\"0px\"><\/iframe><br \/>\r\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Venn+Diagrams.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cVenn Diagrams\u201d here (opens in new window).<\/a><\/p>\r\n<\/section>\r\n<h2>Cardinality<\/h2>\r\n<p><strong>Cardinality<\/strong> is a fundamental concept in set theory and refers to the number of elements in a set. It is a measure of the \"size\" of a set. Remember, cardinality is just a fancy way of saying how many elements are in a set. The cardinality of a set is denoted by [latex]{\\lvert}A{\\rvert}[\/latex], where [latex]A[\/latex] is the set. For example, if you have a set [latex]A = \\{1, 2, 3, 4, 5\\}[\/latex], the cardinality of set [latex]A[\/latex] is [latex]5[\/latex], written as [latex]{\\lvert}A{\\rvert} = 5[\/latex].<\/p>\r\n<section class=\"textbox example\">What is the cardinality of [latex]P[\/latex] = the set of English names for the months of the year?<br \/>\r\n[reveal-answer q=\"3178\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"3178\"]The cardinality of this set is [latex]12[\/latex], since there are [latex]12[\/latex] months in the year.[\/hidden-answer]<\/section>\r\n<h3>Cardinality Properties<\/h3>\r\n<ol>\r\n\t<li><strong>Union of Sets<\/strong>: The cardinality of the union of two sets A and B can be found using the formula:\r\n\r\n\r\n<p>&nbsp;<\/p>\r\n<center>[latex]n(A \\cup B) = n(A) + n(B) \u2013 n(A \\cap B)[\/latex]<\/center>\r\n<p>&nbsp;<\/p>\r\n<\/li>\r\n\t<li><strong>Complement of a Set<\/strong>: The cardinality of a set A can also be found by subtracting the number of elements not in A (the complement of A, represented as [latex]\\bar{A}[\/latex]) from the total number of elements in the universal set U. This is expressed in the formula:<br \/>\r\n<center>[latex]n(A) = n(U) \u2013 n(A)[\/latex]<\/center><\/li>\r\n<\/ol>\r\n<p><em>Notice that the first property can also be written in an equivalent form by solving for the cardinality of the intersection:<\/em><\/p>\r\n<p style=\"text-align: center;\">[latex]n(A \\cap B) = n(A) + n(B) \u2013 n(A \\cup B)[\/latex]<\/p>\r\n<p>It might be easier to understand the cardinality properties if we break them down into words.<\/p>\r\n<table style=\"border-collapse: collapse; width: 100%;\" border=\"1\">\r\n<thead><\/thead>\r\n<tbody>\r\n<tr>\r\n<td style=\"width: 35%;\">Union<\/td>\r\n<td style=\"width: 20%;\">[latex]\\begin{array}{lc}<br \/>\r\nn(A \\cup B) = \\\\<br \/>\r\nn(A) + n(B) - n(A \\cap B)<br \/>\r\n\\end{array}[\/latex]<\/td>\r\n<td style=\"width: 60%;\"><em>the cardinality of the union of set [latex]A[\/latex] with set [latex]B[\/latex] will consists of the cardinality of [latex]A[\/latex] together with the cardinality of [latex]B[\/latex], after deducting the cardinality of their intersection.\u00a0<\/em><\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 35%;\">Intersection<\/td>\r\n<td style=\"width: 20%;\">[latex]\\begin{array}{lc}<br \/>\r\nn(A \\cap B) = \\\\<br \/>\r\nn(A) + n(B) - n(A \\cap B)<br \/>\r\n\\end{array}[\/latex]<\/td>\r\n<td style=\"width: 60%;\"><em>the cardinality of the intersection of set [latex]A[\/latex] with set [latex]B[\/latex] will consists of the cardinality of [latex]A[\/latex] together with the cardinality of [latex]B[\/latex], after deducting the cardinality of their union.\u00a0<\/em><\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 35%;\">Complement<\/td>\r\n<td style=\"width: 20%;\">[latex]n(A) = n(U) - n(A)[\/latex]<\/td>\r\n<td style=\"width: 60%;\"><em>the cardinality of the complement of [latex]A[\/latex] will consist of the cardinality of the universal set less the cardinality of [latex]A[\/latex].\u00a0<\/em>In other words, it\u2019s the cardinality of all the elements that are not in [latex]A[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p>&nbsp;<\/p>\r\n<p>Let\u2019s try some more examples dealing with the cardinality properties.<\/p>\r\n<section class=\"textbox seeExample\">One hundred fifty people were surveyed and asked if they believed in UFOs, ghosts, and Bigfoot.\r\n\r\n\r\n<div style=\"float: left; width: 50%;\">\r\n<ul>\r\n\t<li>[latex]43[\/latex] believed in UFOs<\/li>\r\n\t<li>[latex]25[\/latex] believed in Bigfoot<\/li>\r\n\t<li>[latex]8[\/latex] believed in ghosts and Bigfoot<\/li>\r\n\t<li>[latex]2[\/latex] believed in all three<\/li>\r\n<\/ul>\r\n<\/div>\r\n<div style=\"float: right; width: 50%;\">\r\n<ul>\r\n\t<li>[latex]44[\/latex] believed in ghosts<\/li>\r\n\t<li>[latex]10[\/latex] believed in UFOs and ghosts<\/li>\r\n\t<li>[latex]5[\/latex] believed in UFOs and Bigfoot<\/li>\r\n<\/ul>\r\n<\/div>\r\n<p>&nbsp;<\/p>\r\n<p>How many people surveyed believed in at least one of these things?<\/p>\r\n\r\n\r\n[reveal-answer q=\"8483\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"8483\"]\r\n\r\n\r\n<p>Let\u2019s start off by drawing a Venn diagram of the information we were given.<\/p>\r\n<p>It is easiest to start with the intersection of all three circles and we work our way out. We know that [latex]2[\/latex] people believe in all three \u2013 that goes in our center most region. We can then use the information about people who believe in ghosts and Bigfoot, UFOs and ghosts, and UFOS and Bigfoot to fill in the next layer of regions. For example, since [latex]10[\/latex] people believe in UFOs and Ghosts, and [latex]2[\/latex] believe in all three, that leaves [latex]8[\/latex] that believe in only UFOs and Ghosts. So we would put [latex]8[\/latex] in the region that overlaps UFOs and ghosts. We can then use the total number who believe in ghosts, Bigfoot, and UFO\u2019s to work our way out and fill in all the regions.<\/p>\r\n<p>Here is what the final Venn diagram should look like:<\/p>\r\n<center><img class=\"alignnone wp-image-460 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/276\/2016\/10\/12222338\/ufoghostsbigfoot.png\" alt=\"A Venn diagram depicting three overlapping circles, labeled UFOs, Ghosts, and Bigfoot. It shows that 30 of the people surveyed only believe in UFOs, 28 only believe in Ghosts, 14 only believe in Bigfoot, 8 believe in Ghosts and UFOs, but not Bigfoot, 6 believe in Ghosts and Bigfoot, but not UFOs, 3 believe in UFOs and Bigfoot, but not Ghosts, 2 believe in all 3, and 59 don't believe in any of the three.\" width=\"183\" height=\"152\" \/><\/center>\r\n<p>&nbsp;<\/p>\r\n<p>Once we have completed our diagram, we can add up all the regions, getting [latex]91[\/latex] people in the union of all three sets. This means [latex]91[\/latex] people in at least one thing. This leaves [latex]150 \u2013 91 = 59[\/latex] who believe in none.<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]2251[\/ohm2_question]<\/section>\r\n<section class=\"textbox watchIt\"><iframe src=\"\/\/plugin.3playmedia.com\/show?mf=10205712&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=TWr5VX22HAE&amp;video_target=tpm-plugin-t3r5jlq0-TWr5VX22HAE\" width=\"800px\" height=\"450px\" frameborder=\"0\" marginwidth=\"0px\" marginheight=\"0px\"><\/iframe><br \/>\r\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Properties+of+the+cardinality+of+the+set.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cProperties of the cardinality of the set\u201d here (opens in new window).<\/a><\/p>\r\n<\/section>","rendered":"<section class=\"textbox learningGoals\">\n<ul>\n<li>Understand the concepts of sets, including empty sets, subsets, and proper subsets, and use correct set notation<\/li>\n<li>Describe and perform set operations (union, intersection, complement, and difference) using proper set notation<\/li>\n<li>Create and interpret Venn diagrams to represent and analyze set relationships and operations<\/li>\n<li>Apply the concepts of sets, subsets, and cardinality properties to solve real-life problems<\/li>\n<\/ul>\n<\/section>\n<h2>Sets<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\n<p>A <strong>set<\/strong> is a collection of unique objects or elements. These elements can be anything: numbers, people, other sets, etc. For example, the set [latex]A[\/latex] could be the set of all positive integers, while the set B could be the set of all colors in the rainbow.<\/p>\n<p>The <strong>empty set<\/strong>, also known as the null set, is a special set that contains no elements. For example, the set of all dogs with six legs is an empty set, since no such dogs exist.<\/p>\n<p><strong>Notation<\/strong><br \/>\nIn set theory, we often use curly brackets &#8211; [latex]\\{\\}[\/latex] &#8211; to denote a set. An example of a set is [latex]A = \\{a, b, c, d\\}[\/latex].<\/p>\n<p>Commonly, we will use a variable to represent a set, to make it easier to refer to that set later. We usually denote sets with capital letters such as [latex]A[\/latex], [latex]B[\/latex], [latex]C[\/latex], etc.<\/p>\n<p>The symbol [latex]\\in[\/latex]\u00a0means \u201cis an element of\u201d.<\/p>\n<p>An empty set is denoted by the symbols is notated by [latex]\\emptyset[\/latex] or [latex]\\{ \\}[\/latex].<\/p>\n<\/div>\n<section class=\"textbox watchIt\"><iframe loading=\"lazy\" src=\"\/\/plugin.3playmedia.com\/show?mf=10205707&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=fes92vSBTg4&amp;video_target=tpm-plugin-1fy67j04-fes92vSBTg4\" width=\"800px\" height=\"450px\" frameborder=\"0\" marginwidth=\"0px\" marginheight=\"0px\"><\/iframe><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Sets+Basics+-+Introduction+%7C+Don't+Memorise.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cSets Basics &#8211; Introduction | Don&#8217;t Memorise\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<section class=\"textbox watchIt\"><iframe loading=\"lazy\" src=\"\/\/plugin.3playmedia.com\/show?mf=10205708&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=LumU80IN748&amp;video_target=tpm-plugin-2tflvwyb-LumU80IN748\" width=\"800px\" height=\"450px\" frameborder=\"0\" marginwidth=\"0px\" marginheight=\"0px\"><\/iframe><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Sets+-+Listing+Method+%7C+Don't+Memorise.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cSets &#8211; Listing Method | Don&#8217;t Memorise\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<h2>Subsets<\/h2>\n<div class=\"textbox shaded\" style=\"text-align: left;\">\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\n<p>A set [latex]A[\/latex] is said to be a <strong>subset<\/strong> of another set [latex]B[\/latex] if every element in [latex]A[\/latex] is also in [latex]B[\/latex]. We denote this by [latex]A \\subseteq B[\/latex]. For instance, if [latex]B = \\{\\text{red}, \\text{green}, \\text{blue}, \\text{yellow}\\}[\/latex] and [latex]A = \\{\\text{red}, \\text{blue}, \\text{yellow}\\}[\/latex], then [latex]A[\/latex] is a subset of [latex]B[\/latex] because every color in [latex]A[\/latex] is also in [latex]B[\/latex].<\/p>\n<p>A <strong>proper subset<\/strong> is similar to a subset, but with one key difference: if [latex]A[\/latex] is a proper subset of [latex]B[\/latex], then there is at least one element in [latex]B[\/latex] that is not in [latex]A[\/latex]. We denote this by [latex]A \\subset B[\/latex]. If [latex]B = \\{\\text{red}, \\text{green}, \\text{blue}, \\text{yellow}\\}[\/latex] and [latex]A = \\{\\text{red}, \\text{blue}\\}[\/latex], then [latex]A[\/latex] is a proper subset of [latex]B[\/latex].<\/p>\n<p>To calculate the number of subsets and proper subsets in a set you must use the following formulas:<\/p>\n<p>If you have [latex]n[\/latex] elements in your set:<\/p>\n<ul>\n<li>Number of subsets: [latex]2^{n}[\/latex]<\/li>\n<li>Number of proper subsets: [latex]2^{n}\u22121[\/latex]<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\">Consider these three sets:<\/p>\n<p>[latex]A[\/latex] = the set of all even numbers<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] since every element of [latex]B[\/latex] is also an even number, so is an element of [latex]A[\/latex]. More formally, we could say [latex]B \\subset A[\/latex] since if [latex]x \\in B[\/latex], then [latex]x \\in A[\/latex]. It is also true that [latex]B \\subset C[\/latex]. [latex]C[\/latex] is not a subset of [latex]A[\/latex], since [latex]C[\/latex] contains an element, [latex]3[\/latex], that is not contained in [latex]A[\/latex].<\/section>\n<section class=\"textbox seeExample\">Suppose a set contains the plays \u201cMuch Ado About Nothing,\u201d \u201cMacBeth,\u201d and \u201cA Midsummer\u2019s Night Dream.\u201d What is a larger set this might be a subset of?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q857946\">Show Solution<\/button><\/p>\n<div id=\"q857946\" class=\"hidden-answer\" style=\"display: none\">There are many possible answers here. One would be the set of plays by Shakespeare. This is also a subset of the set of all plays ever written. It is also a subset of all British literature.<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox seeExample\">Consider the\u00a0set\u00a0[latex]A = \\{1, 3, 5\\}[\/latex]. Which of the following sets is [latex]A[\/latex] a subset of?<\/p>\n<p>[latex]X = \\{1, 3, 7, 5\\}[\/latex]<br \/>\n[latex]Y = \\{1, 3 \\}[\/latex]<br \/>\n[latex]Z = \\{1, m, n, 3, 5\\}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q3546\">Show Solution<\/button><\/p>\n<div id=\"q3546\" class=\"hidden-answer\" style=\"display: none\"> [latex]X[\/latex] and [latex]Z[\/latex] <\/div>\n<\/div>\n<\/section>\n<section class=\"textbox seeExample\">Using the set: [latex]\\text{Soda} = \\{\\text{Sprite}, \\text{Dr. Pepper}, \\text{Diet Coke}, \\text{Mountain Dew}, \\text{Cherry Coke}\\}[\/latex], calculate the number of subsets and proper subsets.<\/p>\n<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\">For this set, there are [latex]5[\/latex] elements which means [latex]n = 5[\/latex]. To determine the number of subsets we must use [latex]2^n[\/latex]:<\/p>\n<div style=\"text-align: center;\">[latex]\\begin{array}{ccc}\\hfill 2^5 = 2^5 = 32\\\\ \\end{array}[\/latex]<\/p>\n<\/div>\n<p>To determine the number of proper subsets we must use [latex]2^n - 1[\/latex]:<\/p>\n<div style=\"text-align: center;\">[latex]\\begin{array}{ccc}\\hfill 2^n - 1 = 2^5 - 1 = 32 - 1 = 31\\\\ \\end{array}[\/latex]<\/p>\n<\/div>\n<p>There are [latex]32[\/latex] subsets and [latex]31[\/latex] proper subsets.<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox watchIt\"><iframe loading=\"lazy\" src=\"\/\/plugin.3playmedia.com\/show?mf=10371002&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=_9Wvu-R04go&amp;video_target=tpm-plugin-jv0rfpbh-_9Wvu-R04go\" width=\"800px\" height=\"450px\" frameborder=\"0\" marginwidth=\"0px\" marginheight=\"0px\"><\/iframe><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/What+is+a+Subset%3F+%7C+Don't+Memorise.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cWhat is a Subset? | Don&#8217;t Memorise\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<section class=\"textbox watchIt\"><iframe loading=\"lazy\" title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/xotLg-oLboY\" width=\"560\" height=\"315\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Subsets%2C+Proper+Subsets+and+Supersets+_+Don't+Memorise.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cSubsets, Proper Subsets and Supersets | Don&#8217;t Memorise\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<section class=\"textbox watchIt\"><iframe loading=\"lazy\" src=\"\/\/plugin.3playmedia.com\/show?mf=10205709&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=BhFgcf0VSYc&amp;video_target=tpm-plugin-kh2dl4wg-BhFgcf0VSYc\" width=\"800px\" height=\"450px\" frameborder=\"0\" marginwidth=\"0px\" marginheight=\"0px\"><\/iframe><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/What+is+a+Null+Set+%7C+Is+Null+Set+a+Subset+of+Every+Set%3F+%7C+Don't+Memorise.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cWhat is a Null Set | Is Null Set a Subset of Every Set? | Don&#8217;t Memorise\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<h2>Set Operations: Union, Intersection, Complement, and Difference<\/h2>\n<p>A big part of understanding proper notation in sets and set operations is being able to read and translate the symbols used. There are a lot of different symbols used that make things confusing. We can use the following table as a reminder of what each symbol means:<\/p>\n<table>\n<tbody>\n<tr>\n<th>Symbol<\/th>\n<th>Symbol Meaning<\/th>\n<th>Translation<\/th>\n<\/tr>\n<tr>\n<td>[latex]\\in[\/latex]<\/td>\n<td>\u201cis an element of\u201d or \u201cis an element in\u201d<\/td>\n<td>[latex]1 \\in P[\/latex] reads as: [latex]1[\/latex] is an element of the set [latex]P[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]\\subseteq[\/latex]<\/td>\n<td>Subset<\/td>\n<td>[latex]C \\subseteq R[\/latex] reads as: set [latex]C[\/latex] is a subset of set [latex]R[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]\\subset[\/latex]<\/td>\n<td>Proper Subset<\/td>\n<td>[latex]C \\subset R[\/latex] reads as: set [latex]C[\/latex] is a proper subset of set [latex]R[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]\\cup[\/latex]<\/td>\n<td>Union of two sets<\/td>\n<td>[latex]C \\cup R[\/latex] reads as: the union of sets [latex]C[\/latex] and [latex]R[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]\\cap[\/latex]<\/td>\n<td>Intersection of two sets<\/td>\n<td>[latex]C \\cap R[\/latex] reads as: the intersection of sets [latex]C[\/latex] and [latex]R[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]R'[\/latex], or [latex]R^c[\/latex], or ~[latex]R[\/latex]<\/td>\n<td>Complement of a set<\/td>\n<td>[latex]R'[\/latex] reads as: the complement of set [latex]R[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]\\setminus[\/latex]<\/td>\n<td>Difference of two sets<\/td>\n<td>[latex]C \\setminus R[\/latex] reads as: the difference between sets [latex]C[\/latex] and [latex]R[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>&nbsp;<\/p>\n<p>Now that we can see what each symbol translates to let\u2019s break down each of the formal notations of the set operations to better understand what they are stating.<\/p>\n<h3>Union<\/h3>\n<div class=\"textbox shaded\" style=\"text-align: left;\">\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\n<p>The <strong>union<\/strong> of two sets contains all the elements contained in either set (or both sets) and is notated [latex]A \\cup B[\/latex].\u00a0The formal notation states, [latex]x \\in A \\cup B[\/latex] if [latex]x \\in A[\/latex] or [latex]x \\in B[\/latex] (or both).<\/p>\n<p>If we break this down and translate the symbols into words it is easier to understand.<\/p>\n<ul>\n<li>[latex]x \\in A \\cup B[\/latex] reads as \u201c[latex]x[\/latex] is an element of the union of sets [latex]A[\/latex] and [latex]B[\/latex] \u201c<\/li>\n<li>[latex]x \\in A[\/latex] reads as \u201c[latex]x[\/latex] is an element of set [latex]A[\/latex] \u201c<\/li>\n<li>[latex]x \\in B[\/latex] reads as \u201c[latex]x[\/latex] is an element of set [latex]B[\/latex] \u201c<\/li>\n<\/ul>\n<p>If we add all the parts we translated into words back into the original formal definition it reads as \u201c[latex]x[\/latex] is an element of the union of sets [latex]A[\/latex] and [latex]B[\/latex] if [latex]x[\/latex] is an element of set [latex]A[\/latex] or [latex]x[\/latex] is an element of set [latex]B[\/latex] (or both).\u201d<\/p>\n<\/div>\n<h3>Intersection<\/h3>\n<div class=\"textbox shaded\" style=\"text-align: left;\">\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\n<p>The <strong>intersection <\/strong>of two sets contains only the elements that are in both sets.\u00a0The 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].<\/p>\n<p>If we break this down and translate the symbols into words it is easier to understand.<\/p>\n<ul>\n<li>[latex]x \\in A \\cap B[\/latex] reads as \u201c[latex]x[\/latex] is an element of the intersection of sets [latex]A[\/latex] and [latex]B[\/latex] \u201c<\/li>\n<li>[latex]x \\in A[\/latex] reads as \u201c[latex]x[\/latex] is an element of set [latex]A[\/latex] \u201c<\/li>\n<li>[latex]x \\in B[\/latex] reads as \u201c[latex]x[\/latex] is an element of set [latex]B[\/latex] \u201c<\/li>\n<\/ul>\n<p>If we add all the parts we translated into words back into the original formal definition it reads as \u201c[latex]x[\/latex] is an element of the intersection of sets [latex]A[\/latex] and [latex]B[\/latex] if [latex]x[\/latex] is an element of set [latex]A[\/latex] and [latex]x[\/latex] is an element of set [latex]B[\/latex]. \u201c<\/p>\n<\/div>\n<section class=\"textbox proTip\">\n<h3>intersection and union symbols<\/h3>\n<p>The intersection [latex]\\cup[\/latex] and union [latex]\\cap[\/latex] symbols look a little like letters in the alphabet. In fact, that\u2019s a trick for remembering them.<\/p>\n<p>The union symbol looks like a capital U, for <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<\/section>\n<h3>Difference<\/h3>\n<div class=\"textbox shaded\" style=\"text-align: left;\">\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\n<p>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. The difference between two sets is notated [latex]A \\setminus B[\/latex].<\/p>\n<p>More formally, [latex]x \\in A \\setminus B[\/latex] if [latex]x \\in A[\/latex] &amp; [latex]x \\notin B[\/latex].<\/p>\n<p>If we break this down and translate the symbols into words it is easier to understand.<\/p>\n<ul>\n<li>[latex]x \\in A \\setminus B[\/latex] reads as \u201c[latex]x[\/latex] is an element of the difference of sets [latex]A[\/latex] and [latex]B[\/latex] \u201c<\/li>\n<li>[latex]x \\in A[\/latex] reads as \u201c[latex]x[\/latex] is an element of set [latex]A[\/latex] \u201c<\/li>\n<li>[latex]x \\notin B[\/latex] reads as \u201c[latex]x[\/latex] is not an element of set [latex]B[\/latex] \u201c<\/li>\n<\/ul>\n<p>If we add all the parts we translated into words back into the original formal definition it reads as \u201c[latex]x[\/latex] is an element of the difference of sets [latex]A[\/latex] and [latex]B[\/latex] if [latex]x[\/latex] is an element of set [latex]A[\/latex] and [latex]x[\/latex] is not an element of set [latex]B[\/latex] .\u201c<\/p>\n<\/div>\n<h3>Universal Set<\/h3>\n<div class=\"textbox shaded\" style=\"text-align: left;\">\n<p><strong>The Main Idea\u00a0<\/strong><\/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<\/div>\n<section class=\"textbox seeExample\">\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<\/section>\n<h3>Complement<\/h3>\n<div class=\"textbox shaded\" style=\"text-align: left;\">\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\n<p>A <strong>complement<\/strong> is relative to the universal set. The <strong>complement<\/strong> of a set [latex]A[\/latex] contains everything that is <em>not<\/em> in the universal set [latex]A[\/latex].\u00a0 The complement is notated [latex]A'[\/latex], or [latex]A^c[\/latex], or sometimes \u00a0~[latex]A[\/latex].<\/p>\n<\/div>\n<section class=\"textbox seeExample\">Suppose the universal set is <em>U<\/em> = all whole numbers from [latex]1[\/latex] to [latex]9[\/latex]. If [latex]A = {1, 2, 4}[\/latex], then [latex]A^{c} = \\{3, 5, 6, 7, 8, 9\\}[\/latex].<\/section>\n<section class=\"textbox seeExample\">Consider the sets:<\/p>\n<p>[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] Find the following:<\/p>\n<ol>\n<li>Find [latex]A \\cup C[\/latex]<\/li>\n<li>Find [latex]B^c \\cap A[\/latex]<\/li>\n<\/ol>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q3551\">Show Solution<\/button><\/p>\n<div id=\"q3551\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li style=\"text-align: left;\">The union contains all the elements in either set:\n<div style=\"text-align: center;\">[latex]A \\cup C=\\{\\text { red, orange, yellow, green, blue purple }\\}[\/latex]<\/div>\n<\/p>\n<\/li>\n<li>Here we\u2019re looking for all the elements that are <em>not<\/em> in set [latex]B[\/latex] and are also in [latex]A[\/latex] .\n<div style=\"text-align: center;\">[latex]B^{c} \\cap A=\\{\\text { green, blue }\\}[\/latex]<\/div>\n<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm2223\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=2223&theme=lumen&iframe_resize_id=ohm2223&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section class=\"textbox watchIt\"><iframe loading=\"lazy\" src=\"\/\/plugin.3playmedia.com\/show?mf=10205710&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=fLMN0wtiz-4&amp;video_target=tpm-plugin-oesmkpqb-fLMN0wtiz-4\" width=\"800px\" height=\"450px\" frameborder=\"0\" marginwidth=\"0px\" marginheight=\"0px\"><\/iframe><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Sets_UnionIntersectionComplement.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cSets: Union, Intersection, Complement\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<p>Any of the set operations can be grouped together or stand alone to make statements about different sets. When symbols are grouped together they form an order of operations. Don\u2019t forget, like in arithmetic, when applying the order of operations parenthesis indicates the operation you do first.<\/p>\n<section class=\"textbox seeExample\">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 [latex](H \\cap F) \\cup W[\/latex]<\/li>\n<li>Find [latex](F \\cup W) \\cap H[\/latex]<\/li>\n<li>Find [latex](H \\cap F)^c \\cap W[\/latex]<\/li>\n<\/ol>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q3552\">Show Solution<\/button><\/p>\n<div id=\"q3552\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>We start with the intersection: [latex](H \\cap F) = \\{\\text{dog, rabbit}\\}[\/latex]. Now we union that result with [latex]W[\/latex]: [latex](H \\cap F) \\cup W = \\{\\text{dog, duck, rabbit, deer, frog, mouse}\\}[\/latex].\n<p>&nbsp;<\/p>\n<\/li>\n<li>We start with the union: [latex](F \\cup W) = \\{\\text{dog, cow, rabbit, duck, pig, deer, frog, mouse}\\}[\/latex]. Now we intersect that result with [latex]H[\/latex]: [latex](F \\cup W) \\cap H = \\{\\text{dog, rabbit, mouse}\\}[\/latex].\n<p>&nbsp;<\/p>\n<\/li>\n<li>We start with the intersection: Find [latex](H \\cap F) = \\{\\text{dog, rabbit}\\}[\/latex].\u00a0 Now we want to find the elements of [latex]W[\/latex] that are <em>not<\/em> in [latex](H \\cap F)[\/latex]. [latex](H \\cap F)^c \\cap W = \\{\\text{duck, deer, frog, mouse}\\}[\/latex]<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm2224\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=2224&theme=lumen&iframe_resize_id=ohm2224&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<h2>Venn Diagrams<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\n<p>Venn diagrams are a helpful tool for visualizing the relationships between sets and performing set operations. A Venn diagram uses overlapping circles or other shapes to illustrate the logical relationships between two or more sets of items. Each circle represents a set, with the items of the set represented by points within the circle. Here&#8217;s how you can create and interpret Venn diagrams for various set relationships and operations:<\/p>\n<ul>\n<li><strong>Representing Sets<\/strong>: In a Venn diagram, each set is represented by a circle. For example, if you have two sets A and B, you can represent them with two circles. The universal set is often represented by a rectangle encompassing all the circles.<\/li>\n<li><strong>Union<\/strong>: The union of two sets A and B (A \u222a B) is represented by the area covered by both circles A and B, including the overlap.<\/li>\n<li><strong>Intersection<\/strong>: The intersection of two sets A and B (A \u2229 B) is represented by the area where the two circles overlap.<\/li>\n<li><strong>Difference<\/strong>: The difference of two sets A and B (A &#8211; B) is represented by the area that is in circle A but not in circle B.<\/li>\n<li><strong>Complement<\/strong>: The complement of a set A (A&#8217;) is represented by the area that is outside circle A but within the universal set.<\/li>\n<li><strong>Empty Set<\/strong>: An empty set or null set, which contains no elements, can be represented by a circle with nothing in it. If the set is a subset of another set, it can be represented by a circle within the larger set, but with no elements.<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\">Create an expression to represent the outlined portion of the Venn diagram shown:<\/p>\n<div style=\"text-align: center;\"><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-4218 aligncenter\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/18\/2023\/01\/07151939\/clipboard_ee6f4b6ece2cde3e7320d09311c24e49b.png\" alt=\"A Venn diagram of three sets A B and C are shown overlapping. The region highlighted includes anything in A or anything in B, excluding anything also in C\" width=\"274\" height=\"257\" srcset=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/18\/2023\/01\/07151939\/clipboard_ee6f4b6ece2cde3e7320d09311c24e49b.png 274w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/18\/2023\/01\/07151939\/clipboard_ee6f4b6ece2cde3e7320d09311c24e49b-65x61.png 65w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/18\/2023\/01\/07151939\/clipboard_ee6f4b6ece2cde3e7320d09311c24e49b-225x211.png 225w\" sizes=\"(max-width: 274px) 100vw, 274px\" \/><\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q2178\">Show Solution<\/button><\/p>\n<div id=\"q2178\" class=\"hidden-answer\" style=\"display: none\">\n[latex]A \\cup B \\cap C^{c}[\/latex]\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm2255\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=2255&theme=lumen&iframe_resize_id=ohm2255&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section class=\"textbox watchIt\"><iframe loading=\"lazy\" title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/CPeeOUldZ6M\" width=\"560\" height=\"315\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Sets_+drawing+a+Venn+diagram.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cSets: drawing a Venn diagram\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<section class=\"textbox watchIt\"><iframe loading=\"lazy\" src=\"\/\/plugin.3playmedia.com\/show?mf=10205711&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=32iNIZJ2dI4&amp;video_target=tpm-plugin-a9364ct3-32iNIZJ2dI4\" width=\"800px\" height=\"450px\" frameborder=\"0\" marginwidth=\"0px\" marginheight=\"0px\"><\/iframe><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Venn+Diagrams.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cVenn Diagrams\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<h2>Cardinality<\/h2>\n<p><strong>Cardinality<\/strong> is a fundamental concept in set theory and refers to the number of elements in a set. It is a measure of the &#8220;size&#8221; of a set. Remember, cardinality is just a fancy way of saying how many elements are in a set. The cardinality of a set is denoted by [latex]{\\lvert}A{\\rvert}[\/latex], where [latex]A[\/latex] is the set. For example, if you have a set [latex]A = \\{1, 2, 3, 4, 5\\}[\/latex], the cardinality of set [latex]A[\/latex] is [latex]5[\/latex], written as [latex]{\\lvert}A{\\rvert} = 5[\/latex].<\/p>\n<section class=\"textbox example\">What is the cardinality of [latex]P[\/latex] = the set of English names for the months of the year?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q3178\">Show Solution<\/button><\/p>\n<div id=\"q3178\" class=\"hidden-answer\" style=\"display: none\">The cardinality of this set is [latex]12[\/latex], since there are [latex]12[\/latex] months in the year.<\/div>\n<\/div>\n<\/section>\n<h3>Cardinality Properties<\/h3>\n<ol>\n<li><strong>Union of Sets<\/strong>: The cardinality of the union of two sets A and B can be found using the formula:\n<p>&nbsp;<\/p>\n<div style=\"text-align: center;\">[latex]n(A \\cup B) = n(A) + n(B) \u2013 n(A \\cap B)[\/latex]<\/div>\n<p>&nbsp;<\/p>\n<\/li>\n<li><strong>Complement of a Set<\/strong>: The cardinality of a set A can also be found by subtracting the number of elements not in A (the complement of A, represented as [latex]\\bar{A}[\/latex]) from the total number of elements in the universal set U. This is expressed in the formula:\n<div style=\"text-align: center;\">[latex]n(A) = n(U) \u2013 n(A)[\/latex]<\/div>\n<\/li>\n<\/ol>\n<p><em>Notice that the first property can also be written in an equivalent form by solving for the cardinality of the intersection:<\/em><\/p>\n<p style=\"text-align: center;\">[latex]n(A \\cap B) = n(A) + n(B) \u2013 n(A \\cup B)[\/latex]<\/p>\n<p>It might be easier to understand the cardinality properties if we break them down into words.<\/p>\n<table style=\"border-collapse: collapse; width: 100%;\">\n<thead><\/thead>\n<tbody>\n<tr>\n<td style=\"width: 35%;\">Union<\/td>\n<td style=\"width: 20%;\">[latex]\\begin{array}{lc}<br \/>  n(A \\cup B) = \\\\<br \/>  n(A) + n(B) - n(A \\cap B)<br \/>  \\end{array}[\/latex]<\/td>\n<td style=\"width: 60%;\"><em>the cardinality of the union of set [latex]A[\/latex] with set [latex]B[\/latex] will consists of the cardinality of [latex]A[\/latex] together with the cardinality of [latex]B[\/latex], after deducting the cardinality of their intersection.\u00a0<\/em><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 35%;\">Intersection<\/td>\n<td style=\"width: 20%;\">[latex]\\begin{array}{lc}<br \/>  n(A \\cap B) = \\\\<br \/>  n(A) + n(B) - n(A \\cap B)<br \/>  \\end{array}[\/latex]<\/td>\n<td style=\"width: 60%;\"><em>the cardinality of the intersection of set [latex]A[\/latex] with set [latex]B[\/latex] will consists of the cardinality of [latex]A[\/latex] together with the cardinality of [latex]B[\/latex], after deducting the cardinality of their union.\u00a0<\/em><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 35%;\">Complement<\/td>\n<td style=\"width: 20%;\">[latex]n(A) = n(U) - n(A)[\/latex]<\/td>\n<td style=\"width: 60%;\"><em>the cardinality of the complement of [latex]A[\/latex] will consist of the cardinality of the universal set less the cardinality of [latex]A[\/latex].\u00a0<\/em>In other words, it\u2019s the cardinality of all the elements that are not in [latex]A[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>&nbsp;<\/p>\n<p>Let\u2019s try some more examples dealing with the cardinality properties.<\/p>\n<section class=\"textbox seeExample\">One hundred fifty people were surveyed and asked if they believed in UFOs, ghosts, and Bigfoot.<\/p>\n<div style=\"float: left; width: 50%;\">\n<ul>\n<li>[latex]43[\/latex] believed in UFOs<\/li>\n<li>[latex]25[\/latex] believed in Bigfoot<\/li>\n<li>[latex]8[\/latex] believed in ghosts and Bigfoot<\/li>\n<li>[latex]2[\/latex] believed in all three<\/li>\n<\/ul>\n<\/div>\n<div style=\"float: right; width: 50%;\">\n<ul>\n<li>[latex]44[\/latex] believed in ghosts<\/li>\n<li>[latex]10[\/latex] believed in UFOs and ghosts<\/li>\n<li>[latex]5[\/latex] believed in UFOs and Bigfoot<\/li>\n<\/ul>\n<\/div>\n<p>&nbsp;<\/p>\n<p>How many people surveyed believed in at least one of these things?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q8483\">Show Solution<\/button><\/p>\n<div id=\"q8483\" class=\"hidden-answer\" style=\"display: none\">\n<p>Let\u2019s start off by drawing a Venn diagram of the information we were given.<\/p>\n<p>It is easiest to start with the intersection of all three circles and we work our way out. We know that [latex]2[\/latex] people believe in all three \u2013 that goes in our center most region. We can then use the information about people who believe in ghosts and Bigfoot, UFOs and ghosts, and UFOS and Bigfoot to fill in the next layer of regions. For example, since [latex]10[\/latex] people believe in UFOs and Ghosts, and [latex]2[\/latex] believe in all three, that leaves [latex]8[\/latex] that believe in only UFOs and Ghosts. So we would put [latex]8[\/latex] in the region that overlaps UFOs and ghosts. We can then use the total number who believe in ghosts, Bigfoot, and UFO\u2019s to work our way out and fill in all the regions.<\/p>\n<p>Here is what the final Venn diagram should look like:<\/p>\n<div style=\"text-align: center;\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-460 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/276\/2016\/10\/12222338\/ufoghostsbigfoot.png\" alt=\"A Venn diagram depicting three overlapping circles, labeled UFOs, Ghosts, and Bigfoot. It shows that 30 of the people surveyed only believe in UFOs, 28 only believe in Ghosts, 14 only believe in Bigfoot, 8 believe in Ghosts and UFOs, but not Bigfoot, 6 believe in Ghosts and Bigfoot, but not UFOs, 3 believe in UFOs and Bigfoot, but not Ghosts, 2 believe in all 3, and 59 don't believe in any of the three.\" width=\"183\" height=\"152\" \/><\/div>\n<p>&nbsp;<\/p>\n<p>Once we have completed our diagram, we can add up all the regions, getting [latex]91[\/latex] people in the union of all three sets. This means [latex]91[\/latex] people in at least one thing. This leaves [latex]150 \u2013 91 = 59[\/latex] who believe in none.<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm2251\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=2251&theme=lumen&iframe_resize_id=ohm2251&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section class=\"textbox watchIt\"><iframe loading=\"lazy\" src=\"\/\/plugin.3playmedia.com\/show?mf=10205712&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=TWr5VX22HAE&amp;video_target=tpm-plugin-t3r5jlq0-TWr5VX22HAE\" width=\"800px\" height=\"450px\" frameborder=\"0\" marginwidth=\"0px\" marginheight=\"0px\"><\/iframe><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Properties+of+the+cardinality+of+the+set.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cProperties of the cardinality of the set\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n","protected":false},"author":15,"menu_order":10,"template":"","meta":{"_candela_citation":"[{\"type\":\"original\",\"description\":\"Revision and Adaptation\",\"author\":\"\",\"organization\":\"Lumen Learning\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"Sets\",\"author\":\"David Lippman\",\"organization\":\"\",\"url\":\"http:\/\/www.opentextbookstore.com\/mathinsociety\/\",\"project\":\"Math in Society\",\"license\":\"cc-by-sa\",\"license_terms\":\"\"}]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":24,"module-header":"fresh_take","content_attributions":[{"type":"original","description":"Revision and Adaptation","author":"","organization":"Lumen Learning","url":"","project":"","license":"cc-by","license_terms":""},{"type":"cc","description":"Sets","author":"David Lippman","organization":"","url":"http:\/\/www.opentextbookstore.com\/mathinsociety\/","project":"Math in Society","license":"cc-by-sa","license_terms":""}],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters\/32"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/users\/15"}],"version-history":[{"count":140,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters\/32\/revisions"}],"predecessor-version":[{"id":15056,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters\/32\/revisions\/15056"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/parts\/24"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters\/32\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/media?parent=32"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapter-type?post=32"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/contributor?post=32"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/license?post=32"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}