{"id":3276,"date":"2023-05-23T18:07:59","date_gmt":"2023-05-23T18:07:59","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/?post_type=chapter&#038;p=3276"},"modified":"2024-10-18T20:50:11","modified_gmt":"2024-10-18T20:50:11","slug":"logic-basics-learn-it-1","status":"web-only","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/chapter\/logic-basics-learn-it-1\/","title":{"raw":"Logic Basics: Learn It 1","rendered":"Logic Basics: Learn It 1"},"content":{"raw":"<section class=\"textbox learningGoals\">\r\n<ul>\r\n\t<li>Combine sets using Boolean logic and proper notation<\/li>\r\n\t<li>Create and interpret expressions using statements and conditionals<\/li>\r\n\t<li>Construct and analyze truth tables for complex statements or conditionals<\/li>\r\n\t<li>Determine the logical equivalence between two statements<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Logic<\/h2>\r\n<p><strong>Logic <\/strong>is a systematic way of thinking that allows us to deduce new information\u00a0from old information and to parse the meanings of sentences.\u00a0You use logic informally in everyday life and certainly also in doing mathematics. Logic is, basically, the study of valid reasoning.<\/p>\r\n<p>For example, suppose you are working with a certain circle, call it\u00a0\u201cCircle [latex]X[\/latex],\u201d and you have available the following two pieces of information.<\/p>\r\n<ol>\r\n\t<li>Circle [latex]X[\/latex] has radius equal to [latex]3[\/latex].<\/li>\r\n\t<li>If any circle has radius [latex]r[\/latex], then its area is [latex]\\pi{r}^{2}[\/latex]\u00a0square units.<\/li>\r\n<\/ol>\r\n<p>You have no trouble putting these two facts together to get:<\/p>\r\n<ol start=\"3\">\r\n\t<li>Circle [latex]X[\/latex] has area [latex]9\\pi[\/latex] square units.<\/li>\r\n<\/ol>\r\n<p>You are using logic to combine existing information to\u00a0produce new information. Since a major objective in mathematics is to\u00a0deduce new information, logic must play a fundamental role. This chapter\u00a0is intended to give you a sufficient mastery of logic.<\/p>\r\n<h3>Boolean Logic<\/h3>\r\n<p>We can often classify items as belonging to sets. If you went the library to search for a book and they asked you to express your search using unions, intersections, and complements of sets, that would feel a little strange. Instead, we typically using words like \u201cand,\u201d \u201cor,\" and \u201cnot\u201d to connect our keywords together to form a search. These words, which form the basis of <strong>Boolean logic<\/strong>, are directly related to set operations with the same terminology.<\/p>\r\n<section class=\"textbox keyTakeaway\">\r\n<div>\r\n<h3>Boolean logic<\/h3>\r\n<p>Boolean logic combines multiple statements that are either true or false into an expression that is either true or false.<\/p>\r\n<ul>\r\n\t<li>In connection to sets, a boolean search is true if the element in question is part of the set being searched.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<\/section>\r\n<p>Suppose <em>M<\/em> is the set of all mystery books, and <em>C<\/em> is the set of all comedy books. If we search for \u201cmystery\u201d, we are looking for all the books that are an element of the set <em>M<\/em>; the search is true for books that are in the set.<\/p>\r\n<p>When we search for \u201cmystery <em>and<\/em> comedy\u201d we are looking for a book that is an element of both sets, in the <strong>[pb_glossary id=\"13320\"]intersection[\/pb_glossary]<\/strong>. If we were to search for \u201cmystery<em> or<\/em> comedy\u201d we are looking for a book that is a mystery, a comedy, or both, which is the <strong>[pb_glossary id=\"13319\"]union [\/pb_glossary] <\/strong>of the sets. If we searched for \u201c<em>not <\/em>comedy\u201d we are looking for any book in the library that is not a comedy, the <strong>[pb_glossary id=\"13321\"]complement[\/pb_glossary] <\/strong>of the set <em>C<\/em>.<\/p>\r\n<section class=\"textbox connectIt\">\r\n<p><strong>Connection to set operations:<\/strong><\/p>\r\n<center>[latex]\\begin{array}{r@{\\hfill}l}<br \/>\r\nA \\text{ and } B &amp;&amp; \\text{elements in the intersection } A \\cap B \\\\<br \/>\r\nA \\text{ or } B &amp;&amp; \\text{elements in the union } A \\cup B \\\\<br \/>\r\n\\text{ Not } A &amp;&amp; \\text{elements in the complement } A^c \\\\<br \/>\r\n\\end{array}[\/latex]<\/center><\/section>\r\n<section class=\"textbox proTip\">Notice here that <em>or<\/em> is not exclusive. This is a difference between the Boolean logic use of the word and common everyday use. When your significant other asks \u201cdo you want to go to the park or the movies?\u201d they usually are proposing an exclusive choice \u2013 one option or the other, but not both. In Boolean logic, the <em>or<\/em> is not exclusive \u2013 more like being asked at a restaurant \u201cwould you like fries or a drink with that?\u201d Answering \u201cboth, please\u201d is an acceptable answer.<\/section>\r\n<section class=\"textbox example\">Suppose we are searching a library database for Mexican universities. Express a reasonable search using Boolean logic.<br \/>\r\n[reveal-answer q=\"912486\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"912486\"]We could start with the search \u201cMexico <em>and <\/em>university\u201d, but would be likely to find results for the U.S. state New Mexico. To account for this, we could revise our search to read: Mexico <em>and<\/em> university <em>not <\/em>\u201cNew Mexico.\u201d[\/hidden-answer]<\/section>\r\n<section class=\"textbox questionHelp\">In most internet search engines, it is not necessary to include the word <em>and<\/em>; the search engine assumes that if you provide two keywords you are looking for both. In Google\u2019s search, the keyword <em>or<\/em> has be capitalized as OR, and a negative sign in front of a word is used to indicate <em>not<\/em>. Quotes around a phrase indicate that the entire phrase should be looked for. The search from the previous example on Google could be written:\r\n\r\n<p style=\"text-align: center;\">Mexico university -\u201cNew Mexico\u201d<\/p>\r\n<\/section>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]7053[\/ohm2_question]<\/section>\r\n<h3>Which Comes First?<\/h3>\r\n<p>Sometimes statements made in English can be ambiguous. For this reason, Boolean logic uses parentheses to show precedent, just like in algebraic order of operations.<\/p>\r\n<p>The English phrase \u201cGo to the store and buy me eggs and bagels or cereal\u201d is ambiguous; it is not clear whether the requestors is asking for eggs always along with either bagels or cereal, or whether they\u2019re asking for either the combination of eggs and bagels, or just cereal.<\/p>\r\n<p>For this reason, using parentheses clarifies the intent:<\/p>\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td>Eggs and (bagels or cereal) means<\/td>\r\n<td>Option 1: Eggs and bagels, Option 2: Eggs and cereal<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>(Eggs and bagels) or cereal means<\/td>\r\n<td>\u00a0Option 1: Eggs and bagels, Option 2: Cereal<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<section class=\"textbox example\">Describe the numbers that meet the condition: odd number and less than [latex]20[\/latex] and greater than [latex]0 [\/latex] and (multiple of [latex]3[\/latex] or multiple of [latex]5[\/latex]).<br \/>\r\n[reveal-answer q=\"877489\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"877489\"]The first three conditions limit us to the set [latex]\\{1, 3, 5, 7, 9, 11, 13, 15, 17, 19\\}[\/latex]. The last grouped conditions tell us to find elements of this set that are also either a multiple of [latex]3[\/latex] or a multiple of [latex]5[\/latex]. This leaves us with the set [latex]\\{3, 5, 9, 15\\}[\/latex]. Notice that we would have gotten a very different result if we had written (odd number and less than [latex]20[\/latex] and greater than [latex]0[\/latex] and multiple of [latex]3[\/latex]) or multiple of [latex]5[\/latex]. The first grouped set of conditions would give [latex]\\{3, 9, 15\\}[\/latex]. When combined with the last condition, though, this set expands without limits:\r\n\r\n<p style=\"text-align: center;\">[latex]\\{3, 5, 9, 15, 20, 25, 30, 35, 40, 45, \u2026\\}[\/latex]<\/p>\r\n<p>Be aware that when a string of conditions is written without grouping symbols, it is often interpreted from the left to right, resulting in the latter interpretation.<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<h2>Quantified Statements<\/h2>\r\n<p>Words that describe an entire set, such as \u201call\u201d, \u201cevery\u201d, or \u201cnone\u201d, are called <strong>universal quantifiers<\/strong> because that set could be considered a universal set. In contrast, words or phrases such as \u201csome\u201d, \u201cone\u201d, or \u201cat least one\u201d are called <strong>existential quantifiers<\/strong> because they describe the existence of at least one element in a set.<\/p>\r\n<section class=\"textbox keyTakeaway\">\r\n<div>\r\n<h3>quantifiers<\/h3>\r\n<ul>\r\n\t<li>An <strong>universal quantifier<\/strong> states that an entire set of things share a characteristic.\r\n\r\n<ul>\r\n\t<li><strong>Notation<\/strong>: The universal quantifier is typically symbolized by [latex]\\forall[\/latex] (an inverted letter 'A'), which stands for \"for all\" or \"for every.\"<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li>An <strong>existential quantifier<\/strong> states that a set contains at least one element.\r\n\r\n<ul>\r\n\t<li><strong>Notation<\/strong>: The existential quantifier is symbolized by [latex]\\exists[\/latex] (a backwards letter 'E'), which stands for \"there exists.\"<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/div>\r\n<\/section>\r\n<section class=\"textbox example\">Suppose your friend says \u201cEverybody cheats on their taxes.\u201d What is the minimum amount of evidence you would need to prove your friend wrong?\r\n\r\n<p>To show that it is not true that everybody cheats on their taxes, all you need is <span style=\"text-decoration: underline;\">one<\/span> person who does not cheat on their taxes. It would be perfectly fine to produce more people who do not cheat, but one counterexample is all you need.<\/p>\r\n\r\nIt is important to note that you do <span style=\"text-decoration: underline;\">not<\/span> need to show that absolutely nobody cheats on their taxes.<\/section>\r\n<p>Something interesting happens when we <strong>negate <\/strong>\u2013 or state the opposite of \u2013 a quantified statement. When we negate a statement with a universal quantifier, we get a statement with an existential quantifier, and vice-versa.<\/p>\r\n<section class=\"textbox keyTakeaway\">\r\n<div>\r\n<h3>negating a quantified statement<\/h3>\r\n<p>The negation of \u201call [latex]A[\/latex] are [latex]B[\/latex]\u201d is \u201cat least one [latex]A[\/latex] is not [latex]B[\/latex] .\u201d<\/p>\r\n<p>&nbsp;<\/p>\r\n<p>The negation of \u201cno [latex]A[\/latex] are [latex]B[\/latex]\u201d is \u201cat least one [latex]A[\/latex] is [latex]B[\/latex] .\u201d<\/p>\r\n<p>&nbsp;<\/p>\r\n<p>The negation of \u201cat least one [latex]A[\/latex] is [latex]B[\/latex]\u201d is \u201cno [latex]A[\/latex] are [latex]B[\/latex] .\u201d<\/p>\r\n<p>&nbsp;<\/p>\r\n<p>The negation of \u201cat least one [latex]A[\/latex] is not [latex]B[\/latex]\u201d is \u201call [latex]A[\/latex] are [latex]B[\/latex] .\u201d<\/p>\r\n<\/div>\r\n<\/section>\r\n<section class=\"textbox example\">Write the negation of this statement: \u201cSomebody brought a flashlight.\u201d[reveal-answer q=\"877481\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"877481\"]<br \/>\r\nThe negation is \u201cNobody brought a flashlight.\u201d<br \/>\r\n[\/hidden-answer]<\/section>","rendered":"<section class=\"textbox learningGoals\">\n<ul>\n<li>Combine sets using Boolean logic and proper notation<\/li>\n<li>Create and interpret expressions using statements and conditionals<\/li>\n<li>Construct and analyze truth tables for complex statements or conditionals<\/li>\n<li>Determine the logical equivalence between two statements<\/li>\n<\/ul>\n<\/section>\n<h2>Logic<\/h2>\n<p><strong>Logic <\/strong>is a systematic way of thinking that allows us to deduce new information\u00a0from old information and to parse the meanings of sentences.\u00a0You use logic informally in everyday life and certainly also in doing mathematics. Logic is, basically, the study of valid reasoning.<\/p>\n<p>For example, suppose you are working with a certain circle, call it\u00a0\u201cCircle [latex]X[\/latex],\u201d and you have available the following two pieces of information.<\/p>\n<ol>\n<li>Circle [latex]X[\/latex] has radius equal to [latex]3[\/latex].<\/li>\n<li>If any circle has radius [latex]r[\/latex], then its area is [latex]\\pi{r}^{2}[\/latex]\u00a0square units.<\/li>\n<\/ol>\n<p>You have no trouble putting these two facts together to get:<\/p>\n<ol start=\"3\">\n<li>Circle [latex]X[\/latex] has area [latex]9\\pi[\/latex] square units.<\/li>\n<\/ol>\n<p>You are using logic to combine existing information to\u00a0produce new information. Since a major objective in mathematics is to\u00a0deduce new information, logic must play a fundamental role. This chapter\u00a0is intended to give you a sufficient mastery of logic.<\/p>\n<h3>Boolean Logic<\/h3>\n<p>We can often classify items as belonging to sets. If you went the library to search for a book and they asked you to express your search using unions, intersections, and complements of sets, that would feel a little strange. Instead, we typically using words like \u201cand,\u201d \u201cor,&#8221; and \u201cnot\u201d to connect our keywords together to form a search. These words, which form the basis of <strong>Boolean logic<\/strong>, are directly related to set operations with the same terminology.<\/p>\n<section class=\"textbox keyTakeaway\">\n<div>\n<h3>Boolean logic<\/h3>\n<p>Boolean logic combines multiple statements that are either true or false into an expression that is either true or false.<\/p>\n<ul>\n<li>In connection to sets, a boolean search is true if the element in question is part of the set being searched.<\/li>\n<\/ul>\n<\/div>\n<\/section>\n<p>Suppose <em>M<\/em> is the set of all mystery books, and <em>C<\/em> is the set of all comedy books. If we search for \u201cmystery\u201d, we are looking for all the books that are an element of the set <em>M<\/em>; the search is true for books that are in the set.<\/p>\n<p>When we search for \u201cmystery <em>and<\/em> comedy\u201d we are looking for a book that is an element of both sets, in the <strong><a class=\"glossary-term\" aria-haspopup=\"dialog\" aria-describedby=\"definition\" href=\"#term_3276_13320\">intersection<\/a><\/strong>. If we were to search for \u201cmystery<em> or<\/em> comedy\u201d we are looking for a book that is a mystery, a comedy, or both, which is the <strong><a class=\"glossary-term\" aria-haspopup=\"dialog\" aria-describedby=\"definition\" href=\"#term_3276_13319\">union <\/a> <\/strong>of the sets. If we searched for \u201c<em>not <\/em>comedy\u201d we are looking for any book in the library that is not a comedy, the <strong><a class=\"glossary-term\" aria-haspopup=\"dialog\" aria-describedby=\"definition\" href=\"#term_3276_13321\">complement<\/a> <\/strong>of the set <em>C<\/em>.<\/p>\n<section class=\"textbox connectIt\">\n<p><strong>Connection to set operations:<\/strong><\/p>\n<div style=\"text-align: center;\">[latex]\\begin{array}{r@{\\hfill}l}<br \/>  A \\text{ and } B && \\text{elements in the intersection } A \\cap B \\\\<br \/>  A \\text{ or } B && \\text{elements in the union } A \\cup B \\\\<br \/>  \\text{ Not } A && \\text{elements in the complement } A^c \\\\<br \/>  \\end{array}[\/latex]<\/div>\n<\/section>\n<section class=\"textbox proTip\">Notice here that <em>or<\/em> is not exclusive. This is a difference between the Boolean logic use of the word and common everyday use. When your significant other asks \u201cdo you want to go to the park or the movies?\u201d they usually are proposing an exclusive choice \u2013 one option or the other, but not both. In Boolean logic, the <em>or<\/em> is not exclusive \u2013 more like being asked at a restaurant \u201cwould you like fries or a drink with that?\u201d Answering \u201cboth, please\u201d is an acceptable answer.<\/section>\n<section class=\"textbox example\">Suppose we are searching a library database for Mexican universities. Express a reasonable search using Boolean logic.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q912486\">Show Solution<\/button><\/p>\n<div id=\"q912486\" class=\"hidden-answer\" style=\"display: none\">We could start with the search \u201cMexico <em>and <\/em>university\u201d, but would be likely to find results for the U.S. state New Mexico. To account for this, we could revise our search to read: Mexico <em>and<\/em> university <em>not <\/em>\u201cNew Mexico.\u201d<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox questionHelp\">In most internet search engines, it is not necessary to include the word <em>and<\/em>; the search engine assumes that if you provide two keywords you are looking for both. In Google\u2019s search, the keyword <em>or<\/em> has be capitalized as OR, and a negative sign in front of a word is used to indicate <em>not<\/em>. Quotes around a phrase indicate that the entire phrase should be looked for. The search from the previous example on Google could be written:<\/p>\n<p style=\"text-align: center;\">Mexico university -\u201cNew Mexico\u201d<\/p>\n<\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm7053\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=7053&theme=lumen&iframe_resize_id=ohm7053&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<h3>Which Comes First?<\/h3>\n<p>Sometimes statements made in English can be ambiguous. For this reason, Boolean logic uses parentheses to show precedent, just like in algebraic order of operations.<\/p>\n<p>The English phrase \u201cGo to the store and buy me eggs and bagels or cereal\u201d is ambiguous; it is not clear whether the requestors is asking for eggs always along with either bagels or cereal, or whether they\u2019re asking for either the combination of eggs and bagels, or just cereal.<\/p>\n<p>For this reason, using parentheses clarifies the intent:<\/p>\n<table>\n<tbody>\n<tr>\n<td>Eggs and (bagels or cereal) means<\/td>\n<td>Option 1: Eggs and bagels, Option 2: Eggs and cereal<\/td>\n<\/tr>\n<tr>\n<td>(Eggs and bagels) or cereal means<\/td>\n<td>\u00a0Option 1: Eggs and bagels, Option 2: Cereal<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<section class=\"textbox example\">Describe the numbers that meet the condition: odd number and less than [latex]20[\/latex] and greater than [latex]0[\/latex] and (multiple of [latex]3[\/latex] or multiple of [latex]5[\/latex]).<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q877489\">Show Solution<\/button><\/p>\n<div id=\"q877489\" class=\"hidden-answer\" style=\"display: none\">The first three conditions limit us to the set [latex]\\{1, 3, 5, 7, 9, 11, 13, 15, 17, 19\\}[\/latex]. The last grouped conditions tell us to find elements of this set that are also either a multiple of [latex]3[\/latex] or a multiple of [latex]5[\/latex]. This leaves us with the set [latex]\\{3, 5, 9, 15\\}[\/latex]. Notice that we would have gotten a very different result if we had written (odd number and less than [latex]20[\/latex] and greater than [latex]0[\/latex] and multiple of [latex]3[\/latex]) or multiple of [latex]5[\/latex]. The first grouped set of conditions would give [latex]\\{3, 9, 15\\}[\/latex]. When combined with the last condition, though, this set expands without limits:<\/p>\n<p style=\"text-align: center;\">[latex]\\{3, 5, 9, 15, 20, 25, 30, 35, 40, 45, \u2026\\}[\/latex]<\/p>\n<p>Be aware that when a string of conditions is written without grouping symbols, it is often interpreted from the left to right, resulting in the latter interpretation.<\/p>\n<\/div>\n<\/div>\n<\/section>\n<h2>Quantified Statements<\/h2>\n<p>Words that describe an entire set, such as \u201call\u201d, \u201cevery\u201d, or \u201cnone\u201d, are called <strong>universal quantifiers<\/strong> because that set could be considered a universal set. In contrast, words or phrases such as \u201csome\u201d, \u201cone\u201d, or \u201cat least one\u201d are called <strong>existential quantifiers<\/strong> because they describe the existence of at least one element in a set.<\/p>\n<section class=\"textbox keyTakeaway\">\n<div>\n<h3>quantifiers<\/h3>\n<ul>\n<li>An <strong>universal quantifier<\/strong> states that an entire set of things share a characteristic.\n<ul>\n<li><strong>Notation<\/strong>: The universal quantifier is typically symbolized by [latex]\\forall[\/latex] (an inverted letter &#8216;A&#8217;), which stands for &#8220;for all&#8221; or &#8220;for every.&#8221;<\/li>\n<\/ul>\n<\/li>\n<li>An <strong>existential quantifier<\/strong> states that a set contains at least one element.\n<ul>\n<li><strong>Notation<\/strong>: The existential quantifier is symbolized by [latex]\\exists[\/latex] (a backwards letter &#8216;E&#8217;), which stands for &#8220;there exists.&#8221;<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<\/section>\n<section class=\"textbox example\">Suppose your friend says \u201cEverybody cheats on their taxes.\u201d What is the minimum amount of evidence you would need to prove your friend wrong?<\/p>\n<p>To show that it is not true that everybody cheats on their taxes, all you need is <span style=\"text-decoration: underline;\">one<\/span> person who does not cheat on their taxes. It would be perfectly fine to produce more people who do not cheat, but one counterexample is all you need.<\/p>\n<p>It is important to note that you do <span style=\"text-decoration: underline;\">not<\/span> need to show that absolutely nobody cheats on their taxes.<\/section>\n<p>Something interesting happens when we <strong>negate <\/strong>\u2013 or state the opposite of \u2013 a quantified statement. When we negate a statement with a universal quantifier, we get a statement with an existential quantifier, and vice-versa.<\/p>\n<section class=\"textbox keyTakeaway\">\n<div>\n<h3>negating a quantified statement<\/h3>\n<p>The negation of \u201call [latex]A[\/latex] are [latex]B[\/latex]\u201d is \u201cat least one [latex]A[\/latex] is not [latex]B[\/latex] .\u201d<\/p>\n<p>&nbsp;<\/p>\n<p>The negation of \u201cno [latex]A[\/latex] are [latex]B[\/latex]\u201d is \u201cat least one [latex]A[\/latex] is [latex]B[\/latex] .\u201d<\/p>\n<p>&nbsp;<\/p>\n<p>The negation of \u201cat least one [latex]A[\/latex] is [latex]B[\/latex]\u201d is \u201cno [latex]A[\/latex] are [latex]B[\/latex] .\u201d<\/p>\n<p>&nbsp;<\/p>\n<p>The negation of \u201cat least one [latex]A[\/latex] is not [latex]B[\/latex]\u201d is \u201call [latex]A[\/latex] are [latex]B[\/latex] .\u201d<\/p>\n<\/div>\n<\/section>\n<section class=\"textbox example\">Write the negation of this statement: \u201cSomebody brought a flashlight.\u201d<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q877481\">Show Solution<\/button><\/p>\n<div id=\"q877481\" class=\"hidden-answer\" style=\"display: none\">\nThe negation is \u201cNobody brought a flashlight.\u201d\n<\/div>\n<\/div>\n<\/section>\n<div class=\"glossary\"><span class=\"screen-reader-text\" id=\"definition\">definition<\/span><template id=\"term_3276_13320\"><div class=\"glossary__definition\" role=\"dialog\" data-id=\"term_3276_13320\"><div tabindex=\"-1\"><p>The intersection of two sets contains only the elements that are in both sets.<\/p>\n<\/div><button><span aria-hidden=\"true\">&times;<\/span><span class=\"screen-reader-text\">Close definition<\/span><\/button><\/div><\/template><template id=\"term_3276_13319\"><div class=\"glossary__definition\" role=\"dialog\" data-id=\"term_3276_13319\"><div tabindex=\"-1\"><p>The union of two sets contains all the elements contained in either set (or both sets).<\/p>\n<\/div><button><span aria-hidden=\"true\">&times;<\/span><span class=\"screen-reader-text\">Close definition<\/span><\/button><\/div><\/template><template id=\"term_3276_13321\"><div class=\"glossary__definition\" role=\"dialog\" data-id=\"term_3276_13321\"><div tabindex=\"-1\"><p>The complement of a set contains everything that is not in the set.<\/p>\n<\/div><button><span aria-hidden=\"true\">&times;<\/span><span class=\"screen-reader-text\">Close definition<\/span><\/button><\/div><\/template><\/div>","protected":false},"author":15,"menu_order":11,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":24,"module-header":"learn_it","content_attributions":[],"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\/3276"}],"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":40,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters\/3276\/revisions"}],"predecessor-version":[{"id":15058,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters\/3276\/revisions\/15058"}],"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\/3276\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/media?parent=3276"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapter-type?post=3276"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/contributor?post=3276"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/license?post=3276"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}