{"id":488,"date":"2025-10-09T15:14:17","date_gmt":"2025-10-09T15:14:17","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/calculus3\/?post_type=chapter&#038;p=488"},"modified":"2026-05-22T18:22:07","modified_gmt":"2026-05-22T18:22:07","slug":"foundations-of-vectors-and-3d-space-background-youll-need-2","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/calculus3\/chapter\/foundations-of-vectors-and-3d-space-background-youll-need-2\/","title":{"raw":"Foundations of Vectors and 3D Space: Background You'll Need 2","rendered":"Foundations of Vectors and 3D Space: Background You&#8217;ll Need 2"},"content":{"raw":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li><span data-sheets-root=\"1\">Understand set notation and basic set theory concepts<\/span><\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Applying Set and Interval Notation<\/h2>\r\nSet notation and interval notation are crucial tools in mathematics for describing ranges of values precisely and concisely. These notations assist in expressing solutions to equations, defining function domains and ranges, and setting up integration and summation intervals.\r\n\r\n<strong>Set Notation<\/strong> generally refers to a way of describing sets with a list of elements or a rule that the elements must follow, often expressed in the form [latex] \\{x | \\text{ property of } x\\}[\/latex]. This can include descriptions of sets that are not continuous, such as [latex]\\{1, 2, 3, 4\\}[\/latex] or [latex]\\{x | x \\text{ is an even integer}\\}[\/latex].\r\n\r\n<strong>Interval Notation<\/strong> focuses on describing continuous intervals on the real number line, often using parentheses and brackets, such as [latex](a, b)[\/latex] or [latex][c, d)[\/latex]. In interval notation:\r\n<ul>\r\n \t<li>We use <strong>parentheses ( )<\/strong> for intervals to indicate that the endpoint is not included, known as an open interval.<\/li>\r\n \t<li>We use <strong>brackets [ ]<\/strong> for intervals to indicate that the endpoint is included, known as a closed interval.<\/li>\r\n \t<li>The <strong>union symbol<\/strong> [latex]\\cup[\/latex] is used to combine disjoint sets or intervals that are part of the domain or range but do not directly connect.<\/li>\r\n \t<li><strong>Infinity (\u221e)<\/strong> is always accompanied by a parenthesis because infinity is not a number but rather a concept of endlessness.<\/li>\r\n<\/ul>\r\n<section class=\"textbox keyTakeaway\">\r\n<h3>set and interval notation<\/h3>\r\n<ul>\r\n \t<li><strong>Set Notation<\/strong>: Uses curly braces [latex]\\{\\}[\/latex] to list elements explicitly or to describe them with conditions.<\/li>\r\n \t<li><strong>Interval Notation<\/strong>: Efficient for describing continuous ranges, using brackets [latex][][\/latex]for closed intervals and parentheses [latex]( )[\/latex] for open intervals.<\/li>\r\n<\/ul>\r\n<\/section><section class=\"textbox example\">Some examples of set and interval notation being used are:\r\n<ol>\r\n \t<li><strong>Describing a Domain<\/strong>: The domain of [latex]f(x)=\\sqrt{x\u22123}[\/latex] is all [latex]x[\/latex] such that [latex]x\u22123\u22650[\/latex]. In interval notation, this is [latex][3,\u221e)[\/latex].<\/li>\r\n \t<li><strong>Solution Sets<\/strong>: For the inequality [latex]x^2\u22124&lt;12[\/latex], solve to find [latex]x&lt;4[\/latex] and [latex]x&gt;\u22124[\/latex], described as [latex](\u22124,4)[\/latex].<\/li>\r\n \t<li><strong>Defining Function Ranges<\/strong>: If a function [latex]f[\/latex] maps real numbers to their squares, the range can be set as [latex][0,\u221e)[\/latex], representing all non-negative real numbers.<\/li>\r\n<\/ol>\r\n<\/section><section class=\"textbox proTip\">Always determine whether the interval should include the endpoints based on the conditions given.\r\n\r\n<\/section><section class=\"textbox example\">Write the following sets using interval notation.\r\n<ol>\r\n \t<li>The set of all real numbers greater than [latex]5[\/latex]<\/li>\r\n \t<li>[latex] {x\u2223x\u2264\u22122}[\/latex]<\/li>\r\n \t<li>[latex]{x\u2223x\u2264\u22123 \\text{ or } x\u22653}[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"519465\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"519465\"]\r\n<ol>\r\n \t<li>This set includes every real number greater than [latex]5[\/latex], but not [latex]5[\/latex] itself. In interval notation, you represent this as [latex](5,\u221e)[\/latex].<\/li>\r\n \t<li>The set describes all real numbers less than or equal to [latex]-2[\/latex]. In interval notation, this is written as [latex](\u2212\u221e,\u22122][\/latex].<\/li>\r\n \t<li>This set includes all real numbers less than or equal to [latex]-3[\/latex] and all real numbers greater than or equal to [latex]3[\/latex]. In interval notation, these two conditions are represented as two separate intervals combined using the union symbol. Thus, the notation for this set is [latex](\u2212\u221e,\u22123]\u222a[3,\u221e)[\/latex].[\/hidden-answer]<\/li>\r\n<\/ol>\r\n<\/section><section class=\"textbox tryIt\">[ohm_question hide_question_numbers=1]287041[\/ohm_question]\r\n\r\n<\/section>\r\n<h2>Set Operations<\/h2>\r\nCommonly, sets interact. For example, you and a new roommate decide to have a house party, and you both invite your circle of friends. At this party, two sets are being combined, though it might turn out that there are some friends that were in both sets.\r\n\r\n<section class=\"textbox keyTakeaway\">\r\n<div>\r\n<h3>set operations: union, intersection, complement, and difference<\/h3>\r\n<ul>\r\n \t<li>The <strong>union<\/strong> of two sets contains all the elements contained in either set (or both sets).\r\n<ul>\r\n \t<li>The union is notated [latex]A \\cup B[\/latex].\u00a0More formally, [latex]x \\in A \\cup B[\/latex] if [latex]x \\in A[\/latex] or [latex]x \\in B[\/latex] (or both).<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li>The <strong>intersection <\/strong>of two sets contains only the elements that are in both sets.\r\n<ul>\r\n \t<li>The 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].<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li>The <strong>complement<\/strong> of a set [latex]A[\/latex] contains everything that is <em>not<\/em> in the [latex]A[\/latex].\r\n<ul>\r\n \t<li>The complement is notated [latex]A'[\/latex], or [latex]A^c[\/latex], or sometimes \u00a0~[latex]A[\/latex].<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li>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.\r\n<ul>\r\n \t<li>The difference between two sets is notated [latex]A \\setminus B[\/latex]. More formally, [latex]x \\in A \\setminus B[\/latex] if [latex]x \\in A[\/latex] &amp; [latex]x \\notin B [\/latex].<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/div>\r\n<\/section><section class=\"textbox proTip\">Notice in the descriptions of the notation introduced above that the\u00a0<strong>complement<\/strong> of a set is denoted [latex]A^{c}[\/latex]. This superscript is\u00a0not an exponent. It is a decoration that denotes\u00a0<strong>the complement of a set<\/strong>.<\/section>Let\u2019s try applying the set operations.\r\n\r\n<section class=\"textbox seeExample\">Consider the sets:\r\n<ul>\r\n \t<li>[latex]A = \\{\\text{red, green, blue}\\}[\/latex]<\/li>\r\n \t<li>[latex]B = \\{\\text{red, yellow, orange}\\}[\/latex]<\/li>\r\n \t<li>[latex]C = \\{\\text{red, orange, yellow, green, blue, purple}\\}[\/latex]<\/li>\r\n<\/ul>\r\nFind the following:\r\n<ol>\r\n \t<li>Find [latex]A \\cup B[\/latex]<\/li>\r\n \t<li>Find [latex]A \\cap B[\/latex]<\/li>\r\n \t<li>Find [latex]A^c \\cap C[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"3541\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"3541\"]\r\n<ol>\r\n \t<li>The union contains all the elements in either set: [latex]A \\cup B = \\{\\text{red, green, blue, yellow, orange}\\}[\/latex]. Notice we only list red once.<\/li>\r\n \t<li>The intersection contains all the elements in both sets: [latex]A \\cap B = \\{\\text{red}\\}[\/latex].<\/li>\r\n \t<li>Here we\u2019re looking for all the elements that are <em>not<\/em> in set [latex]A[\/latex] and are also in [latex]C[\/latex] . [latex]A^c \\cap C = \\{\\text{orange, yellow, purple}\\}[\/latex]<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]2203[\/ohm2_question]<\/section>Set operations can be grouped together \u2013 for example, [latex]A^c \\cap C[\/latex]. Grouping symbols can be used like they are with arithmetic \u2013 to force an order of operations.\r\n\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]2204[\/ohm2_question]<\/section>","rendered":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li><span data-sheets-root=\"1\">Understand set notation and basic set theory concepts<\/span><\/li>\n<\/ul>\n<\/section>\n<h2>Applying Set and Interval Notation<\/h2>\n<p>Set notation and interval notation are crucial tools in mathematics for describing ranges of values precisely and concisely. These notations assist in expressing solutions to equations, defining function domains and ranges, and setting up integration and summation intervals.<\/p>\n<p><strong>Set Notation<\/strong> generally refers to a way of describing sets with a list of elements or a rule that the elements must follow, often expressed in the form [latex]\\{x | \\text{ property of } x\\}[\/latex]. This can include descriptions of sets that are not continuous, such as [latex]\\{1, 2, 3, 4\\}[\/latex] or [latex]\\{x | x \\text{ is an even integer}\\}[\/latex].<\/p>\n<p><strong>Interval Notation<\/strong> focuses on describing continuous intervals on the real number line, often using parentheses and brackets, such as [latex](a, b)[\/latex] or [latex][c, d)[\/latex]. In interval notation:<\/p>\n<ul>\n<li>We use <strong>parentheses ( )<\/strong> for intervals to indicate that the endpoint is not included, known as an open interval.<\/li>\n<li>We use <strong>brackets [ ]<\/strong> for intervals to indicate that the endpoint is included, known as a closed interval.<\/li>\n<li>The <strong>union symbol<\/strong> [latex]\\cup[\/latex] is used to combine disjoint sets or intervals that are part of the domain or range but do not directly connect.<\/li>\n<li><strong>Infinity (\u221e)<\/strong> is always accompanied by a parenthesis because infinity is not a number but rather a concept of endlessness.<\/li>\n<\/ul>\n<section class=\"textbox keyTakeaway\">\n<h3>set and interval notation<\/h3>\n<ul>\n<li><strong>Set Notation<\/strong>: Uses curly braces [latex]\\{\\}[\/latex] to list elements explicitly or to describe them with conditions.<\/li>\n<li><strong>Interval Notation<\/strong>: Efficient for describing continuous ranges, using brackets [latex][][\/latex]for closed intervals and parentheses [latex]( )[\/latex] for open intervals.<\/li>\n<\/ul>\n<\/section>\n<section class=\"textbox example\">Some examples of set and interval notation being used are:<\/p>\n<ol>\n<li><strong>Describing a Domain<\/strong>: The domain of [latex]f(x)=\\sqrt{x\u22123}[\/latex] is all [latex]x[\/latex] such that [latex]x\u22123\u22650[\/latex]. In interval notation, this is [latex][3,\u221e)[\/latex].<\/li>\n<li><strong>Solution Sets<\/strong>: For the inequality [latex]x^2\u22124<12[\/latex], solve to find [latex]x<4[\/latex] and [latex]x>\u22124[\/latex], described as [latex](\u22124,4)[\/latex].<\/li>\n<li><strong>Defining Function Ranges<\/strong>: If a function [latex]f[\/latex] maps real numbers to their squares, the range can be set as [latex][0,\u221e)[\/latex], representing all non-negative real numbers.<\/li>\n<\/ol>\n<\/section>\n<section class=\"textbox proTip\">Always determine whether the interval should include the endpoints based on the conditions given.<\/p>\n<\/section>\n<section class=\"textbox example\">Write the following sets using interval notation.<\/p>\n<ol>\n<li>The set of all real numbers greater than [latex]5[\/latex]<\/li>\n<li>[latex]{x\u2223x\u2264\u22122}[\/latex]<\/li>\n<li>[latex]{x\u2223x\u2264\u22123 \\text{ or } x\u22653}[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q519465\">Show Answer<\/button><\/p>\n<div id=\"q519465\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>This set includes every real number greater than [latex]5[\/latex], but not [latex]5[\/latex] itself. In interval notation, you represent this as [latex](5,\u221e)[\/latex].<\/li>\n<li>The set describes all real numbers less than or equal to [latex]-2[\/latex]. In interval notation, this is written as [latex](\u2212\u221e,\u22122][\/latex].<\/li>\n<li>This set includes all real numbers less than or equal to [latex]-3[\/latex] and all real numbers greater than or equal to [latex]3[\/latex]. In interval notation, these two conditions are represented as two separate intervals combined using the union symbol. Thus, the notation for this set is [latex](\u2212\u221e,\u22123]\u222a[3,\u221e)[\/latex].<\/div>\n<\/div>\n<\/li>\n<\/ol>\n<\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm287041\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=287041&theme=lumen&iframe_resize_id=ohm287041&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/section>\n<h2>Set Operations<\/h2>\n<p>Commonly, sets interact. For example, you and a new roommate decide to have a house party, and you both invite your circle of friends. At this party, two sets are being combined, though it might turn out that there are some friends that were in both sets.<\/p>\n<section class=\"textbox keyTakeaway\">\n<div>\n<h3>set operations: union, intersection, complement, and difference<\/h3>\n<ul>\n<li>The <strong>union<\/strong> of two sets contains all the elements contained in either set (or both sets).\n<ul>\n<li>The union is notated [latex]A \\cup B[\/latex].\u00a0More formally, [latex]x \\in A \\cup B[\/latex] if [latex]x \\in A[\/latex] or [latex]x \\in B[\/latex] (or both).<\/li>\n<\/ul>\n<\/li>\n<li>The <strong>intersection <\/strong>of two sets contains only the elements that are in both sets.\n<ul>\n<li>The 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].<\/li>\n<\/ul>\n<\/li>\n<li>The <strong>complement<\/strong> of a set [latex]A[\/latex] contains everything that is <em>not<\/em> in the [latex]A[\/latex].\n<ul>\n<li>The complement is notated [latex]A'[\/latex], or [latex]A^c[\/latex], or sometimes \u00a0~[latex]A[\/latex].<\/li>\n<\/ul>\n<\/li>\n<li>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.\n<ul>\n<li>The difference between two sets is notated [latex]A \\setminus B[\/latex]. More formally, [latex]x \\in A \\setminus B[\/latex] if [latex]x \\in A[\/latex] &amp; [latex]x \\notin B[\/latex].<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<\/section>\n<section class=\"textbox proTip\">Notice in the descriptions of the notation introduced above that the\u00a0<strong>complement<\/strong> of a set is denoted [latex]A^{c}[\/latex]. This superscript is\u00a0not an exponent. It is a decoration that denotes\u00a0<strong>the complement of a set<\/strong>.<\/section>\n<p>Let\u2019s try applying the set operations.<\/p>\n<section class=\"textbox seeExample\">Consider the sets:<\/p>\n<ul>\n<li>[latex]A = \\{\\text{red, green, blue}\\}[\/latex]<\/li>\n<li>[latex]B = \\{\\text{red, yellow, orange}\\}[\/latex]<\/li>\n<li>[latex]C = \\{\\text{red, orange, yellow, green, blue, purple}\\}[\/latex]<\/li>\n<\/ul>\n<p>Find the following:<\/p>\n<ol>\n<li>Find [latex]A \\cup B[\/latex]<\/li>\n<li>Find [latex]A \\cap B[\/latex]<\/li>\n<li>Find [latex]A^c \\cap C[\/latex]<\/li>\n<\/ol>\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\">\n<ol>\n<li>The union contains all the elements in either set: [latex]A \\cup B = \\{\\text{red, green, blue, yellow, orange}\\}[\/latex]. Notice we only list red once.<\/li>\n<li>The intersection contains all the elements in both sets: [latex]A \\cap B = \\{\\text{red}\\}[\/latex].<\/li>\n<li>Here we\u2019re looking for all the elements that are <em>not<\/em> in set [latex]A[\/latex] and are also in [latex]C[\/latex] . [latex]A^c \\cap C = \\{\\text{orange, yellow, purple}\\}[\/latex]<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm2203\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=2203&theme=lumen&iframe_resize_id=ohm2203&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<p>Set operations can be grouped together \u2013 for example, [latex]A^c \\cap C[\/latex]. Grouping symbols can be used like they are with arithmetic \u2013 to force an order of operations.<\/p>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm2204\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=2204&theme=lumen&iframe_resize_id=ohm2204&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n","protected":false},"author":15,"menu_order":3,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-488","chapter","type-chapter","status-publish","hentry"],"part":420,"module-header":"- Select Header -","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\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/488","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/users\/15"}],"version-history":[{"count":4,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/488\/revisions"}],"predecessor-version":[{"id":2869,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/488\/revisions\/2869"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/parts\/420"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/488\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/media?parent=488"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapter-type?post=488"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/contributor?post=488"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/license?post=488"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}