{"id":6778,"date":"2021-12-30T21:27:32","date_gmt":"2021-12-30T21:27:32","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/frontrange-mathforliberalartscorequisite1\/?post_type=chapter&#038;p=6778"},"modified":"2022-01-06T22:31:37","modified_gmt":"2022-01-06T22:31:37","slug":"1-1-language-of-sets","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/frontrange-mathforliberalartscorequisite1\/chapter\/1-1-language-of-sets\/","title":{"raw":"1.1 Language of Sets","rendered":"1.1 Language of Sets"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3 style=\"padding-left: 30px;\">Learning Objectives<\/h3>\r\nThe Basics of Sets\r\n<ul>\r\n \t<li>Elements<\/li>\r\n \t<li>Empty set<\/li>\r\n \t<li>Cardinality<\/li>\r\n \t<li>Forms of writing sets\r\n<ol>\r\n \t<li>Description form<\/li>\r\n \t<li>Roster form<\/li>\r\n \t<li>Set Builder Notation form<\/li>\r\n<\/ol>\r\n<\/li>\r\n<\/ul>\r\n<\/div>\r\nAn art collector might own a collection of paintings, while a music lover might keep a collection of CDs. Any collection of items can form a <strong>set<\/strong>.\r\n<div class=\"textbox examples\">\r\n<h3>recall sets of real numbers<\/h3>\r\nRecall the sets of real numbers you studied previously. Each number contained in a set is an element of the set that contains it. For example, the number [latex]1[\/latex] is an element of the set of counting numbers. The number [latex]\\dfrac{2}{3}[\/latex] is an element of the set of rational numbers. And so on. The same idea applies to any set of distinct objects, as described below.\r\n\r\n<\/div>\r\n<div class=\"textbox\">\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.\r\n\r\nA set is <strong>well defined<\/strong> if its elements are clearly described. There\u2019s no question about what the set\u2019s elements are.\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nSome examples of sets defined by describing the contents:\r\n<ol>\r\n \t<li>The set of all even numbers<\/li>\r\n \t<li>The set of all books written about travel to Chile<\/li>\r\n<\/ol>\r\nSome examples of sets defined by listing the elements of the set:\r\n<ol>\r\n \t<li>{1, 3, 9, 12}<\/li>\r\n \t<li>{red, orange, yellow, green, blue, indigo, purple}<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox\">\r\n<h3>Notation<\/h3>\r\nCommonly, we will use a capital letter to represent a set, to make it easier to refer to that set later.\r\n\r\n<\/div>\r\n<div class=\"textbox\">\r\n<h3>Element<\/h3>\r\nThe objects used to form a set are called its element or its members.\r\n\r\nLet A = {v, w, x, y, z}\r\n\r\nHere \u2018A\u2019 is the name of the set whose elements (members) are v, w, x, y, z.\r\n\r\n<\/div>\r\n<div class=\"textbox\">\r\n<h3>Notation<\/h3>\r\nThe symbol \u2208\u00a0means \u201cis an element of\u201d and the symbol\u00a0\u2209 means \"is not an element of\"\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Examples<\/h3>\r\nIf a set A = {3, 6, 9, 10, 13, 18}. State whether the following statements are \u2018true\u2019 or \u2018false\u2019:\r\n\r\na)\u00a0 \u00a07 \u2208 A\r\n\r\nb)\u00a0 10 \u2209 A\r\n\r\nc)\u00a0 \u00a013 \u2208 A\r\n\r\nd)\u00a0 \u00a0 9, 10 \u2208 A\r\n\r\n&nbsp;\r\n\r\nSOLUTIONS:\r\n\r\na)\u00a0 False, since the element 7 does not belong to the given set A.\r\n\r\nb)\u00a0 False, since the element 10 belongs to the given set A.\r\n\r\nc)\u00a0 True, since the element 13 belongs to the given set A.\r\n\r\nd) True, since the elements 9 and 10 both belong to the given set A.\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox\">\r\n<h3>EMPTy Set<\/h3>\r\nA set that does not contain any elements is called an\u00a0<strong>empty set<\/strong>\u00a0or <strong>null set<\/strong>.\r\n\r\n<\/div>\r\n<div class=\"textbox\">\r\n<h3>Notation<\/h3>\r\nAn empty set is represented as { }, containing no element at all. It is also represented using the symbol \u2205 (read as 'phi').\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Examples<\/h3>\r\nLet's consider the following examples where we need to determine if the given sets are empty sets.\r\n\r\na)\u00a0 A month with 33 days\r\n<p class=\"indent\">\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 Since there are no months with 33 days, we can conclude that this is an empty set.<\/p>\r\nb)\u00a0X = {x | x is a prime number and 14&lt;x&lt;16}\r\n<p class=\"indent\">\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 Let's start by making a list of prime numbers 2, 3, 5, 7, 11, 13, 17\u2026,\r\nWe can see that\u00a0there are no prime numbers between 14 and 16.\u00a0 We can conclude that this is an empty\u00a0set.<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox\">\r\n<h3>Cardinality<\/h3>\r\n<p style=\"orphans: 2;\">The\u00a0<b>cardinality<\/b>\u00a0of a set is basically the size of the set. Cardinality is simply the number of elements in <span style=\"color: #333333;\"><span style=\"white-space: nowrap; background-color: #f6f4f2;\">the set.<\/span><\/span><\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox\">\r\n<h3>Notation<\/h3>\r\nn(A) means the number of members in set A.\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>ExAMPLES<\/h3>\r\nFind the cardinality of the following sets\r\n\r\na)\u00a0 Let <em>A<\/em> = {1, 2, 3, 4, 5, 6},\u00a0 n(A) = 6\r\n\r\nb)\u00a0 Let S be the letters in the word bubble.\u00a0 n(S) = 4.\u00a0 \u00a0 \u00a0 \u00a0The answer is 4 because the set is S = {b, u, l, e} which has 4 elements in it.\r\n\r\nc) Let B =\u00a0\u2205.\u00a0 n(B) = 0.\u00a0 The answer is 0 because the set is empty.\r\n\r\n<\/div>\r\nA 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\r\nSets can be listed in three ways: <strong>Description form, Roster form<\/strong> and <strong>Set Builder Notation form<\/strong>.\r\n\r\nSet builder what? Yeah, we\u2019ll get there in a second. First, let\u2019s go over Roster form. Have you ever played on a\r\nteam? The list of people on the team is the Team Roster.\r\n\r\n<strong>Roster form<\/strong> is just a way of listing the elements.\r\nFor example: If the Set A is \u201cthe set of Natural numbers less than 6,\u201d it would be written in Roster form as:\r\nA={1, 2, 3, 4, 5}\r\n\r\n<strong>Description form\u00a0<\/strong>is simply describing the elements in the set, like, Set A is the \u201cSet of days that end in Y.\u201d\r\nA={ Monday, Tuesday, Wednesday, Thursday, Friday, Saturday, Sunday}\r\n\r\nAnd last but not least, the dreaded <strong>Set Builder Notation form<\/strong>. This is probably the trickiest of the three ways to notate a set, but it\u2019s actually pretty fun once you get the hang of it. It\u2019s kind of like cracking a code. It\u2019s a statement and each part of the statement is telling you something.\r\n\r\nExample:\r\nSet A is the set of all x such that x is an element of the Natural numbers and x is less than 100. What the\u2026? Looks like this: A={x|x\u00a0\u2208 N and x&lt;100} The answer would be A= { 2, 3, 4, 5,..., 99}\r\n\r\nBreak it down like this:\r\nA \u201cA is the set\u201d\r\n{x \u201cOf all x\u201d\r\n|x \u201cSuch that x\u201d\r\n\u2208 N \u201cIs an element of the Natural numbers\u201d\r\nAnd x &lt; \u201cAnd x is less than 100\u201d\r\nSee, that wasn\u2019t so hard.\r\n\r\nHere's another example: Write the following in set builder notation. X is a letter in the word MATH.\r\n\r\nWe'd want to start with braces, then the variable then a vertical line.\r\n\r\n{x|\r\n\r\nNext, we want to say what x is an element of. In this case it is a letter.\r\n\r\n{x|x\u2208English Letters, all letters in MATH}\r\n<p style=\"text-align: center;\"><strong><span style=\"color: #ff0000;\">This is the end of the section. Close this tab and proceed to the corresponding assignment.<\/span><\/strong><\/p>","rendered":"<div class=\"textbox learning-objectives\">\n<h3 style=\"padding-left: 30px;\">Learning Objectives<\/h3>\n<p>The Basics of Sets<\/p>\n<ul>\n<li>Elements<\/li>\n<li>Empty set<\/li>\n<li>Cardinality<\/li>\n<li>Forms of writing sets\n<ol>\n<li>Description form<\/li>\n<li>Roster form<\/li>\n<li>Set Builder Notation form<\/li>\n<\/ol>\n<\/li>\n<\/ul>\n<\/div>\n<p>An art collector might own a collection of paintings, while a music lover might keep a collection of CDs. Any collection of items can form a <strong>set<\/strong>.<\/p>\n<div class=\"textbox examples\">\n<h3>recall sets of real numbers<\/h3>\n<p>Recall the sets of real numbers you studied previously. Each number contained in a set is an element of the set that contains it. For example, the number [latex]1[\/latex] is an element of the set of counting numbers. The number [latex]\\dfrac{2}{3}[\/latex] is an element of the set of rational numbers. And so on. The same idea applies to any set of distinct objects, as described below.<\/p>\n<\/div>\n<div class=\"textbox\">\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.<\/p>\n<p>A set is <strong>well defined<\/strong> if its elements are clearly described. There\u2019s no question about what the set\u2019s elements are.<\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Some examples of sets defined by describing the contents:<\/p>\n<ol>\n<li>The set of all even numbers<\/li>\n<li>The set of all books written about travel to Chile<\/li>\n<\/ol>\n<p>Some examples of sets defined by listing the elements of the set:<\/p>\n<ol>\n<li>{1, 3, 9, 12}<\/li>\n<li>{red, orange, yellow, green, blue, indigo, purple}<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox\">\n<h3>Notation<\/h3>\n<p>Commonly, we will use a capital letter to represent a set, to make it easier to refer to that set later.<\/p>\n<\/div>\n<div class=\"textbox\">\n<h3>Element<\/h3>\n<p>The objects used to form a set are called its element or its members.<\/p>\n<p>Let A = {v, w, x, y, z}<\/p>\n<p>Here \u2018A\u2019 is the name of the set whose elements (members) are v, w, x, y, z.<\/p>\n<\/div>\n<div class=\"textbox\">\n<h3>Notation<\/h3>\n<p>The symbol \u2208\u00a0means \u201cis an element of\u201d and the symbol\u00a0\u2209 means &#8220;is not an element of&#8221;<\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Examples<\/h3>\n<p>If a set A = {3, 6, 9, 10, 13, 18}. State whether the following statements are \u2018true\u2019 or \u2018false\u2019:<\/p>\n<p>a)\u00a0 \u00a07 \u2208 A<\/p>\n<p>b)\u00a0 10 \u2209 A<\/p>\n<p>c)\u00a0 \u00a013 \u2208 A<\/p>\n<p>d)\u00a0 \u00a0 9, 10 \u2208 A<\/p>\n<p>&nbsp;<\/p>\n<p>SOLUTIONS:<\/p>\n<p>a)\u00a0 False, since the element 7 does not belong to the given set A.<\/p>\n<p>b)\u00a0 False, since the element 10 belongs to the given set A.<\/p>\n<p>c)\u00a0 True, since the element 13 belongs to the given set A.<\/p>\n<p>d) True, since the elements 9 and 10 both belong to the given set A.<\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox\">\n<h3>EMPTy Set<\/h3>\n<p>A set that does not contain any elements is called an\u00a0<strong>empty set<\/strong>\u00a0or <strong>null set<\/strong>.<\/p>\n<\/div>\n<div class=\"textbox\">\n<h3>Notation<\/h3>\n<p>An empty set is represented as { }, containing no element at all. It is also represented using the symbol \u2205 (read as &#8216;phi&#8217;).<\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Examples<\/h3>\n<p>Let&#8217;s consider the following examples where we need to determine if the given sets are empty sets.<\/p>\n<p>a)\u00a0 A month with 33 days<\/p>\n<p class=\"indent\">\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 Since there are no months with 33 days, we can conclude that this is an empty set.<\/p>\n<p>b)\u00a0X = {x | x is a prime number and 14&lt;x&lt;16}<\/p>\n<p class=\"indent\">\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 Let&#8217;s start by making a list of prime numbers 2, 3, 5, 7, 11, 13, 17\u2026,<br \/>\nWe can see that\u00a0there are no prime numbers between 14 and 16.\u00a0 We can conclude that this is an empty\u00a0set.<\/p>\n<\/div>\n<div class=\"textbox\">\n<h3>Cardinality<\/h3>\n<p style=\"orphans: 2;\">The\u00a0<b>cardinality<\/b>\u00a0of a set is basically the size of the set. Cardinality is simply the number of elements in <span style=\"color: #333333;\"><span style=\"white-space: nowrap; background-color: #f6f4f2;\">the set.<\/span><\/span><\/p>\n<\/div>\n<div class=\"textbox\">\n<h3>Notation<\/h3>\n<p>n(A) means the number of members in set A.<\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>ExAMPLES<\/h3>\n<p>Find the cardinality of the following sets<\/p>\n<p>a)\u00a0 Let <em>A<\/em> = {1, 2, 3, 4, 5, 6},\u00a0 n(A) = 6<\/p>\n<p>b)\u00a0 Let S be the letters in the word bubble.\u00a0 n(S) = 4.\u00a0 \u00a0 \u00a0 \u00a0The answer is 4 because the set is S = {b, u, l, e} which has 4 elements in it.<\/p>\n<p>c) Let B =\u00a0\u2205.\u00a0 n(B) = 0.\u00a0 The answer is 0 because the set is empty.<\/p>\n<\/div>\n<p>A 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<p>Sets can be listed in three ways: <strong>Description form, Roster form<\/strong> and <strong>Set Builder Notation form<\/strong>.<\/p>\n<p>Set builder what? Yeah, we\u2019ll get there in a second. First, let\u2019s go over Roster form. Have you ever played on a<br \/>\nteam? The list of people on the team is the Team Roster.<\/p>\n<p><strong>Roster form<\/strong> is just a way of listing the elements.<br \/>\nFor example: If the Set A is \u201cthe set of Natural numbers less than 6,\u201d it would be written in Roster form as:<br \/>\nA={1, 2, 3, 4, 5}<\/p>\n<p><strong>Description form\u00a0<\/strong>is simply describing the elements in the set, like, Set A is the \u201cSet of days that end in Y.\u201d<br \/>\nA={ Monday, Tuesday, Wednesday, Thursday, Friday, Saturday, Sunday}<\/p>\n<p>And last but not least, the dreaded <strong>Set Builder Notation form<\/strong>. This is probably the trickiest of the three ways to notate a set, but it\u2019s actually pretty fun once you get the hang of it. It\u2019s kind of like cracking a code. It\u2019s a statement and each part of the statement is telling you something.<\/p>\n<p>Example:<br \/>\nSet A is the set of all x such that x is an element of the Natural numbers and x is less than 100. What the\u2026? Looks like this: A={x|x\u00a0\u2208 N and x&lt;100} The answer would be A= { 2, 3, 4, 5,&#8230;, 99}<\/p>\n<p>Break it down like this:<br \/>\nA \u201cA is the set\u201d<br \/>\n{x \u201cOf all x\u201d<br \/>\n|x \u201cSuch that x\u201d<br \/>\n\u2208 N \u201cIs an element of the Natural numbers\u201d<br \/>\nAnd x &lt; \u201cAnd x is less than 100\u201d<br \/>\nSee, that wasn\u2019t so hard.<\/p>\n<p>Here&#8217;s another example: Write the following in set builder notation. X is a letter in the word MATH.<\/p>\n<p>We&#8217;d want to start with braces, then the variable then a vertical line.<\/p>\n<p>{x|<\/p>\n<p>Next, we want to say what x is an element of. In this case it is a letter.<\/p>\n<p>{x|x\u2208English Letters, all letters in MATH}<\/p>\n<p style=\"text-align: center;\"><strong><span style=\"color: #ff0000;\">This is the end of the section. Close this tab and proceed to the corresponding assignment.<\/span><\/strong><\/p>\n","protected":false},"author":359705,"menu_order":2,"template":"","meta":{"_candela_citation":"[]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-6778","chapter","type-chapter","status-publish","hentry"],"part":159,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/frontrange-mathforliberalartscorequisite1\/wp-json\/pressbooks\/v2\/chapters\/6778","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/frontrange-mathforliberalartscorequisite1\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/frontrange-mathforliberalartscorequisite1\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/frontrange-mathforliberalartscorequisite1\/wp-json\/wp\/v2\/users\/359705"}],"version-history":[{"count":25,"href":"https:\/\/courses.lumenlearning.com\/frontrange-mathforliberalartscorequisite1\/wp-json\/pressbooks\/v2\/chapters\/6778\/revisions"}],"predecessor-version":[{"id":7007,"href":"https:\/\/courses.lumenlearning.com\/frontrange-mathforliberalartscorequisite1\/wp-json\/pressbooks\/v2\/chapters\/6778\/revisions\/7007"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/frontrange-mathforliberalartscorequisite1\/wp-json\/pressbooks\/v2\/parts\/159"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/frontrange-mathforliberalartscorequisite1\/wp-json\/pressbooks\/v2\/chapters\/6778\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/frontrange-mathforliberalartscorequisite1\/wp-json\/wp\/v2\/media?parent=6778"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/frontrange-mathforliberalartscorequisite1\/wp-json\/pressbooks\/v2\/chapter-type?post=6778"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/frontrange-mathforliberalartscorequisite1\/wp-json\/wp\/v2\/contributor?post=6778"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/frontrange-mathforliberalartscorequisite1\/wp-json\/wp\/v2\/license?post=6778"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}