{"id":29,"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-learn-it-page-3\/"},"modified":"2025-08-29T21:00:59","modified_gmt":"2025-08-29T21:00:59","slug":"set-theory-basics-learn-it-3","status":"web-only","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/chapter\/set-theory-basics-learn-it-3\/","title":{"raw":"Set Theory Basics: Learn It 3","rendered":"Set Theory Basics: Learn It 3"},"content":{"raw":"<h2>Venn Diagrams<\/h2>\r\n<p>To visualize the interaction of sets, John Venn in 1880 thought to use overlapping circles, building on a similar idea used by Leonhard Euler in the 18th century. These illustrations now called <strong>Venn diagrams<\/strong>.<\/p>\r\n<section class=\"textbox keyTakeaway\">\r\n<div>\r\n<h3>Venn diagram<\/h3>\r\n<p>A <strong>Venn diagram<\/strong> represents each set by a circle, usually drawn inside of a containing box representing the universal set. Overlapping areas indicate elements common to both sets. Basic Venn diagrams can illustrate the interaction of two or three sets.<\/p>\r\n<\/div>\r\n<\/section>\r\n<p>Venn diagrams can be used to illustrate the union, intersection, and complements of sets.<\/p>\r\n<center>\r\n[caption id=\"attachment_6427\" align=\"aligncenter\" width=\"500\"]<img class=\"wp-image-6427\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/18\/2023\/01\/10162953\/Set-theory-basics-1-1-284x300.png\" alt=\"Three Venn diagrams are shown. The first is for the union of set A and set B. The entire outside surface of the Venn diagram is outlined in red since A \u22c3 B contains all elements in either set. The second is for the intersection of set A and set B. The overlap of the circles is outlined in red since A \u22c2 B contains only those elements in both sets. The third is for the intersection of the compliment of set A and set B. Only the portion of the Venn diagram containing set B is outlined in red since A\u1d9c will contain all elements not in the set A. A\u1d9c \u22c2 B will contain the elements in set B that are not in set A.\" width=\"500\" height=\"528\" \/> \u00a0Figure 1. These Venn diagrams show the union (upper left), intersection (upper right), and compliment (second row) of sets[\/caption]\r\n<\/center>\r\n<section class=\"textbox seeExample\">Use a Venn diagram to illustrate [latex](H \\cap F)^{c} \\cap W[\/latex].<br \/>\r\n[reveal-answer q=\"62178\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"62178\"]We\u2019ll start by identifying everything in the set [latex]H \\cap F[\/latex].<center><img class=\"wp-image-452 size-full aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/276\/2016\/10\/12220617\/hfw1.png\" alt=\"A Venn diagram depicting three overlapping circles, labeled H, F, and W, respectively. The area where H and F overlap is outlined in red.\" width=\"180\" height=\"152\" \/><\/center>Now, [latex](H \\cap F)^{c} \\cap W[\/latex] will contain everything <em>not<\/em> in the set identified above that is also in set [latex]W[\/latex].<center><img class=\"wp-image-453 size-full aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/276\/2016\/10\/12220632\/hfw2.png\" alt=\"A Venn diagram depicting three overlapping circles, labeled H, F, and W, respectively. The area of circle W where it does not overlap with H and F at the same time is outlined in red.\" width=\"180\" height=\"156\" \/><\/center><br \/>\r\n[\/hidden-answer]<\/section>\r\n<section class=\"textbox seeExample\">Create an expression to represent the outlined part of the Venn diagram shown.<br \/>\r\n<center><img class=\"aligncenter wp-image-6197 size-medium\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/18\/2023\/01\/07161716\/Set-theory-basics-2-300x300.png\" alt=\"A Venn diagram with the elements shared by set H and set F highlighted.\" width=\"300\" height=\"300\" \/><\/center><br \/>\r\n[reveal-answer q=\"621789\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"621789\"]The elements in the outlined set <em>are<\/em> in sets [latex]H[\/latex] and [latex]F[\/latex], but are not in set [latex]W[\/latex]. So we could represent this set as [latex]H \\cap F \\cap W^c[\/latex].[\/hidden-answer]<\/section>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]2254[\/ohm2_question]<\/section>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]2255[\/ohm2_question]<\/section>","rendered":"<h2>Venn Diagrams<\/h2>\n<p>To visualize the interaction of sets, John Venn in 1880 thought to use overlapping circles, building on a similar idea used by Leonhard Euler in the 18th century. These illustrations now called <strong>Venn diagrams<\/strong>.<\/p>\n<section class=\"textbox keyTakeaway\">\n<div>\n<h3>Venn diagram<\/h3>\n<p>A <strong>Venn diagram<\/strong> represents each set by a circle, usually drawn inside of a containing box representing the universal set. Overlapping areas indicate elements common to both sets. Basic Venn diagrams can illustrate the interaction of two or three sets.<\/p>\n<\/div>\n<\/section>\n<p>Venn diagrams can be used to illustrate the union, intersection, and complements of sets.<\/p>\n<div style=\"text-align: center;\">\n<figure id=\"attachment_6427\" aria-describedby=\"caption-attachment-6427\" style=\"width: 500px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-6427\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/18\/2023\/01\/10162953\/Set-theory-basics-1-1-284x300.png\" alt=\"Three Venn diagrams are shown. The first is for the union of set A and set B. The entire outside surface of the Venn diagram is outlined in red since A \u22c3 B contains all elements in either set. The second is for the intersection of set A and set B. The overlap of the circles is outlined in red since A \u22c2 B contains only those elements in both sets. The third is for the intersection of the compliment of set A and set B. Only the portion of the Venn diagram containing set B is outlined in red since A\u1d9c will contain all elements not in the set A. A\u1d9c \u22c2 B will contain the elements in set B that are not in set A.\" width=\"500\" height=\"528\" srcset=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/18\/2023\/01\/10162953\/Set-theory-basics-1-1-284x300.png 284w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/18\/2023\/01\/10162953\/Set-theory-basics-1-1-768x811.png 768w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/18\/2023\/01\/10162953\/Set-theory-basics-1-1-65x69.png 65w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/18\/2023\/01\/10162953\/Set-theory-basics-1-1-225x238.png 225w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/18\/2023\/01\/10162953\/Set-theory-basics-1-1-350x370.png 350w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/18\/2023\/01\/10162953\/Set-theory-basics-1-1.png 930w\" sizes=\"(max-width: 500px) 100vw, 500px\" \/><figcaption id=\"caption-attachment-6427\" class=\"wp-caption-text\">\u00a0Figure 1. These Venn diagrams show the union (upper left), intersection (upper right), and compliment (second row) of sets<\/figcaption><\/figure>\n<\/div>\n<section class=\"textbox seeExample\">Use a Venn diagram to illustrate [latex](H \\cap F)^{c} \\cap W[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q62178\">Show Solution<\/button><\/p>\n<div id=\"q62178\" class=\"hidden-answer\" style=\"display: none\">We\u2019ll start by identifying everything in the set [latex]H \\cap F[\/latex].<\/p>\n<div style=\"text-align: center;\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-452 size-full aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/276\/2016\/10\/12220617\/hfw1.png\" alt=\"A Venn diagram depicting three overlapping circles, labeled H, F, and W, respectively. The area where H and F overlap is outlined in red.\" width=\"180\" height=\"152\" \/><\/div>\n<p>Now, [latex](H \\cap F)^{c} \\cap W[\/latex] will contain everything <em>not<\/em> in the set identified above that is also in set [latex]W[\/latex].<\/p>\n<div style=\"text-align: center;\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-453 size-full aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/276\/2016\/10\/12220632\/hfw2.png\" alt=\"A Venn diagram depicting three overlapping circles, labeled H, F, and W, respectively. The area of circle W where it does not overlap with H and F at the same time is outlined in red.\" width=\"180\" height=\"156\" \/><\/div>\n<p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox seeExample\">Create an expression to represent the outlined part of the Venn diagram shown.<\/p>\n<div style=\"text-align: center;\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-6197 size-medium\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/18\/2023\/01\/07161716\/Set-theory-basics-2-300x300.png\" alt=\"A Venn diagram with the elements shared by set H and set F highlighted.\" width=\"300\" height=\"300\" srcset=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/18\/2023\/01\/07161716\/Set-theory-basics-2-300x300.png 300w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/18\/2023\/01\/07161716\/Set-theory-basics-2-150x150.png 150w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/18\/2023\/01\/07161716\/Set-theory-basics-2-768x768.png 768w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/18\/2023\/01\/07161716\/Set-theory-basics-2-65x65.png 65w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/18\/2023\/01\/07161716\/Set-theory-basics-2-225x225.png 225w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/18\/2023\/01\/07161716\/Set-theory-basics-2-350x350.png 350w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/18\/2023\/01\/07161716\/Set-theory-basics-2.png 900w\" sizes=\"(max-width: 300px) 100vw, 300px\" \/><\/div>\n<div class=\"wp-nocaption \"><\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q621789\">Show Solution<\/button><\/p>\n<div id=\"q621789\" class=\"hidden-answer\" style=\"display: none\">The elements in the outlined set <em>are<\/em> in sets [latex]H[\/latex] and [latex]F[\/latex], but are not in set [latex]W[\/latex]. So we could represent this set as [latex]H \\cap F \\cap W^c[\/latex].<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm2254\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=2254&theme=lumen&iframe_resize_id=ohm2254&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/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","protected":false},"author":15,"menu_order":6,"template":"","meta":{"_candela_citation":"[{\"type\":\"original\",\"description\":\"Revision and Adaptation\",\"author\":\"\",\"organization\":\"Lumen Learning\",\"url\":\"\",\"project\":\"\",\"license\":\"pd\",\"license_terms\":\"\"},{\"type\":\"original\",\"description\":\"Question ID 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