{"id":31416,"date":"2023-07-04T15:08:00","date_gmt":"2023-07-04T15:08:00","guid":{"rendered":"https:\/\/www.splashlearn.com\/math-vocabulary\/?page_id=31416"},"modified":"2023-07-05T03:19:46","modified_gmt":"2023-07-05T03:19:46","slug":"union-of-sets-definition-formula-examples-facts-faqs","status":"publish","type":"post","link":"https:\/\/www.splashlearn.com\/math-vocabulary\/union-of-sets","title":{"rendered":"Union of Sets &#8211; Definition, Formula, Examples, Facts, FAQs"},"content":{"rendered":"<div class=\"aioseo-breadcrumbs\"><span class=\"aioseo-breadcrumb\">\n\tMath-Vocabulary\n<\/span><\/div>\n\n<div class=\"ub_table-of-contents\" data-showtext=\"show\" data-hidetext=\"hide\" data-scrolltype=\"auto\" id=\"ub_table-of-contents-d1f9867d-08bf-4c84-8dde-8b73c5180096\" data-initiallyhideonmobile=\"false\"\n                    data-initiallyshow=\"true\"><div class=\"ub_table-of-contents-extra-container\"><div class=\"ub_table-of-contents-container ub_table-of-contents-1-column \"><ul><li><a href=https:\/\/www.splashlearn.com\/math-vocabulary\/union-of-sets#0-what-is-the-union-of-sets>What Is the Union of Sets?<\/a><\/li><li><a href=https:\/\/www.splashlearn.com\/math-vocabulary\/union-of-sets#5-number-of-elements-in-a-union-b-formula>Number of Elements in A Union B Formula<\/a><\/li><li><a href=https:\/\/www.splashlearn.com\/math-vocabulary\/union-of-sets#6-properties-of-union-of-sets>Properties of Union of Sets<\/a><\/li><li><a href=https:\/\/www.splashlearn.com\/math-vocabulary\/union-of-sets#9-solved-examples-of-union-of-sets>Solved Examples of Union of Sets<\/a><\/li><li><a href=https:\/\/www.splashlearn.com\/math-vocabulary\/union-of-sets#10-practice-problems-on-union-of-sets>Practice Problems on Union of Sets<\/a><\/li><li><a href=https:\/\/www.splashlearn.com\/math-vocabulary\/union-of-sets#11-frequently-asked-questions-about-union-of-sets>Frequently Asked Questions about Union of Sets<\/a><\/li><\/ul><\/div><\/div><\/div>\n\n\n<h2 class=\"wp-block-heading\" id=\"0-what-is-the-union-of-sets\">What Is the Union of Sets?<\/h2>\n\n\n\n<p><strong>The union of a collection of sets is a set containing all the elements of individual sets. The union of two sets A and B is a new set denoted by <\/strong>$A\\cup B$<strong>, which contains all the elements of the set A and all the elements of set B (without repetition).<\/strong><\/p>\n\n\n\n<p><strong>Example: <\/strong>Find $A\\cup B$ if&nbsp; $A = \\left\\{1, 2, 4, 6\\right\\}$ and $B = \\left\\{2, 3, 6, 7, 8, 9\\right\\}$.<\/p>\n\n\n\n<p>The union of sets A and B has all the numbers of individual sets without repetition.&nbsp;<\/p>\n\n\n\n<p>$A \\cup B = \\left\\{1, 2, 3, 4, 6, 7, 8, 9\\right\\}$.&nbsp;<\/p>\n\n\n\n<div id=\"recommended-games-container-id\" class=\"recommended-games-container\"><h4 class=\"recommended-games-container-headline\">Recommended Games<\/h4><div class=\"recommended-games-container-slides\"><div class=\"game-card-container-outer\">\r\n\t<a href=\"https:\/\/www.splashlearn.com\/s\/math-games\/compare-the-sets-of-objects\" data-vars-ga-category=\"splashlearn_vocab\" data-vars-ga-action=\"games_recommendations\" data-vars-ga-label=\"post_widget\">\r\n\t\t<div class=\"game-card-container-inner-block\">\r\n\t\t\t<img decoding=\"async\" class=\"game-card-container-inner-img\" src=\"https:\/\/cdn.splashmath.com\/curriculum_uploads\/images\/playables\/count_comp_comp_grps_1_pt.png\" alt=\"Compare the Sets of Objects Game\">\r\n\t\t<\/div>\r\n\t\r\n\t\t<div class=\"game-card-container-inner\">\r\n\t\t\t<div class=\"game-card-container-inner-name\">Compare the Sets of Objects Game<\/div>\r\n\t\t\t<span class=\"game-card-container-inner-cta\">Play<\/span>\r\n\t\t<\/div>\r\n\t<\/a>\r\n<\/div><div class=\"game-card-container-outer\">\r\n\t<a href=\"https:\/\/www.splashlearn.com\/s\/math-games\/make-the-sets-of-objects-equal\" data-vars-ga-category=\"splashlearn_vocab\" data-vars-ga-action=\"games_recommendations\" data-vars-ga-label=\"post_widget\">\r\n\t\t<div class=\"game-card-container-inner-block\">\r\n\t\t\t<img decoding=\"async\" class=\"game-card-container-inner-img\" src=\"https:\/\/cdn.splashmath.com\/curriculum_uploads\/images\/playables\/count_comp_eql_grps_pt.png\" alt=\"Make the Sets of Objects Equal Game\">\r\n\t\t<\/div>\r\n\t\r\n\t\t<div class=\"game-card-container-inner\">\r\n\t\t\t<div class=\"game-card-container-inner-name\">Make the Sets of Objects Equal Game<\/div>\r\n\t\t\t<span class=\"game-card-container-inner-cta\">Play<\/span>\r\n\t\t<\/div>\r\n\t<\/a>\r\n<\/div><\/div><p class=\"recommended-games-container-desc\"><a href=\"https:\/\/www.splashlearn.com\/games\">More Games<\/a><\/p><button class=\"scroll-right-arrow\"><\/button><\/div><script type=\"text\/javascript\">\n        document.addEventListener(\"DOMContentLoaded\", function() {\n            const container = document.querySelector(\"#recommended-games-container-id\");\n            const slidesContainer = container.querySelector(\".recommended-games-container-slides\");\n            const cards = slidesContainer.querySelectorAll(\".game-card-container-outer\");\n            const scrollRightArrow = container.querySelector(\".scroll-right-arrow\");\n\n            function adjustContainerStyles() {\n                const numCards = cards.length;\n\n                if (numCards === 1) {\n                    container.style.maxWidth = \"30%\";\n                    container.style.textAlign = \"center\";\n                } else if (numCards === 2) {\n                    container.style.maxWidth = \"50%\";\n                    container.style.textAlign = \"center\";\n                } else if (numCards === 3) {\n                    container.style.maxWidth = \"75%\";\n                    container.style.textAlign = \"center\";\n                } else {\n                    container.style.maxWidth = \"\";\n                    container.style.textAlign = \"\";\n                }\n            }\n\n            function checkScrollPosition() {\n                const maxScrollLeft = slidesContainer.scrollWidth - slidesContainer.clientWidth;\n                if ((slidesContainer.scrollLeft + 10) >= maxScrollLeft) {\n                    scrollRightArrow.style.display = \"none\"; \/\/ Hide the arrow if fully scrolled\n                } else {\n                    scrollRightArrow.style.display = \"block\"; \/\/ Show the arrow if not fully scrolled\n                }\n            }\n\n            scrollRightArrow.addEventListener(\"click\", function() {\n                const scrollAmount = 300; \/\/ Adjust based on the container's width and your needs\n                slidesContainer.scrollLeft += scrollAmount;\n                setTimeout(checkScrollPosition, 100); \/\/ Delay to allow scroll update\n            });\n\n            adjustContainerStyles();\n            checkScrollPosition();\n            slidesContainer.addEventListener(\"scroll\", checkScrollPosition);\n        });\n    <\/script><h2 class=\"wp-block-heading\" id=\"1-union-of-sets-definition\">Union of Sets: Definition<\/h2>\n\n\n\n<p><strong>The union of two sets, denoted by <\/strong>$A \\cup B$<strong>, is a set that contains elements that are in either set A or set B, or in both.<\/strong><\/p>\n\n\n\n<p>In simple words, the union of two sets contains all the distinct elements of both the sets.<\/p>\n\n\n\n<p>The union of two or more sets is the smallest set containing all elements from individual sets. The definition of union in mathematics can be explained as an operation on two sets that results in a new set formed by combining all the elements of the given two sets.<\/p>\n\n\n\n<div id=\"recommended-worksheets-container-id\" class=\"recommended-games-container\"><h4 class=\"recommended-games-container-headline\">Recommended Worksheets<\/h4><div class=\"recommended-games-container-slides\"><div class=\"worksheet-card-container-outer\">\r\n\t<a href=\"https:\/\/www.splashlearn.com\/s\/math-worksheets\/values-of-sets-of-bills\" data-vars-ga-category=\"splashlearn_vocab\" data-vars-ga-action=\"worksheets_recommendations\" data-vars-ga-label=\"post_widget\">\r\n\t    <div class=\"worksheet-card-container-inner-block\">\r\n\t        <img decoding=\"async\" class=\"worksheet-card-container-inner-img\" src=\"https:\/\/cdn.splashmath.com\/cms_assets\/s\/math-worksheets\/values-of-sets-of-bills.jpeg\" alt=\"Values of Sets of Bills Worksheet\">\r\n\t    <\/div>\r\n\t\t<div class=\"worksheet-card-container-inner\" >\r\n\t\t<\/div><\/a>\r\n<\/div><\/div><p class=\"recommended-games-container-desc\"><a href=\"https:\/\/www.splashlearn.com\/worksheets\">More Worksheets<\/a><\/p><button class=\"scroll-right-arrow\"><\/button><\/div><script type=\"text\/javascript\">\n        document.addEventListener(\"DOMContentLoaded\", function() {\n            const container = document.querySelector(\"#recommended-worksheets-container-id\");\n            const slidesContainer = container.querySelector(\".recommended-games-container-slides\");\n            const cards = slidesContainer.querySelectorAll(\".worksheet-card-container-outer\");\n            const scrollRightArrow = container.querySelector(\".scroll-right-arrow\");\n\n            function adjustContainerStyles() {\n                const numCards = cards.length;\n\n                if (numCards === 1) {\n                    container.style.maxWidth = \"30%\";\n                    container.style.textAlign = \"center\";\n                } else if (numCards === 2) {\n                    container.style.maxWidth = \"50%\";\n                    container.style.textAlign = \"center\";\n                } else if (numCards === 3) {\n                    container.style.maxWidth = \"75%\";\n                    container.style.textAlign = \"center\";\n                } else {\n                    container.style.maxWidth = \"\";\n                    container.style.textAlign = \"\";\n                }\n            }\n\n            function checkScrollPosition() {\n                const maxScrollLeft = slidesContainer.scrollWidth - slidesContainer.clientWidth;\n                if ((slidesContainer.scrollLeft + 10) >= maxScrollLeft) {\n                    scrollRightArrow.style.display = \"none\"; \/\/ Hide the arrow if fully scrolled\n                } else {\n                    scrollRightArrow.style.display = \"block\"; \/\/ Show the arrow if not fully scrolled\n                }\n            }\n\n            scrollRightArrow.addEventListener(\"click\", function() {\n                const scrollAmount = 300; \/\/ Adjust based on the container's width and your needs\n                slidesContainer.scrollLeft += scrollAmount;\n                setTimeout(checkScrollPosition, 100); \/\/ Delay to allow scroll update\n            });\n\n            adjustContainerStyles();\n            checkScrollPosition();\n            slidesContainer.addEventListener(\"scroll\", checkScrollPosition);\n        });\n    <\/script><h2 class=\"wp-block-heading\" id=\"2-venn-diagram-of-union-of-sets\">Venn Diagram of Union of Sets<\/h2>\n\n\n\n<p>Venn diagram is a visual representation or illustration that uses circles to show relationships between finite sets. Each circle represents a specific set. Any operation on sets can be represented using a venn diagram.<\/p>\n\n\n\n<p>The union of sets is illustrated by the blue shaded region. It covers all the elements from both the sets. Here, U is a universal set, a big set such that A and B are subsets of U.&nbsp;<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"620\" height=\"346\" src=\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/07\/venn-diagram-of-union-of-two-sets-a-and-b.png\" alt=\"venn diagram of union of two sets A and B. \" class=\"wp-image-31422\" title=\"venn diagram of union of two sets A and B. \" srcset=\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/07\/venn-diagram-of-union-of-two-sets-a-and-b.png 620w, https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/07\/venn-diagram-of-union-of-two-sets-a-and-b-300x167.png 300w\" sizes=\"auto, (max-width: 620px) 100vw, 620px\" \/><\/figure>\n\n\n\n<p><strong>Example: Set <\/strong>$A = \\left\\{1, 3, 5, 7, 9\\right\\}$<strong> and Set <\/strong>$B = \\left\\{2, 3, 5, 7\\right\\}$<\/p>\n\n\n\n<p>$A \\cup B = \\left\\{1, 2, 3, 5, 7, 9\\right\\}$<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"620\" height=\"327\" src=\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/07\/a-union-B-venn-diagram-example.png\" alt=\"A union B venn diagram example\" class=\"wp-image-31423\" title=\"A union B venn diagram example\" srcset=\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/07\/a-union-B-venn-diagram-example.png 620w, https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/07\/a-union-B-venn-diagram-example-300x158.png 300w\" sizes=\"auto, (max-width: 620px) 100vw, 620px\" \/><\/figure>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"3-a-union-b-notation\">A Union B Notation<\/h2>\n\n\n\n<p>The union of two sets A and B is represented by writing the symbol \u201c\u201d between the two sets.<\/p>\n\n\n\n<p><strong>A union <\/strong>$B = A \\cup B$<\/p>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"4-a-union-b-formula\">A Union B Formula<\/h2>\n\n\n\n<p>The formula of A union B or the formula for the union of two sets is given by<\/p>\n\n\n\n<p>$A \\cup B = \\left\\{ x: x \\in A \\text{or} x \\in B\\right\\}$<\/p>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"5-number-of-elements-in-a-union-b-formula\">Number of Elements in A Union B Formula<\/h2>\n\n\n\n<p>The number of elements in any set is called its cardinality. The generalized formula for the number of elements in the set \u201cA union B\u201d is:&nbsp;<\/p>\n\n\n\n<p>$n(A \\cup B) = n(A) + n(B) \\;-\\; n(A \\cap B)$<\/p>\n\n\n\n<p>Here,<\/p>\n\n\n\n<p>n(A) is the cardinality of set A,&nbsp;<\/p>\n\n\n\n<p>n(B) is the cardinality of set B, and&nbsp;<\/p>\n\n\n\n<p>n(A B) is the number of elements present in the intersection of two sets.&nbsp;<\/p>\n\n\n\n<p><strong>Example: In a group, 20 people own a white car, 10 people own a black car, and 5 people own both a white and a black car. How many people own a white car or a black car?<\/strong><\/p>\n\n\n\n<p>$n(W) =$ Number of people who own a white car $= 20$<\/p>\n\n\n\n<p>$n(B) =$ Number of people who own a black car $= 10$<\/p>\n\n\n\n<p>$n(W \\cap B) =$ Number of people who own a white and a black car $= 5$<\/p>\n\n\n\n<p>$n(W \\cup B) = n(W) + n(B) \\;-\\; n(W \\cap B)$<\/p>\n\n\n\n<p>$n(W \\cup B) = 20 + 10 \\;-\\; 5$<\/p>\n\n\n\n<p>$n(W\\cup B) = 25$<\/p>\n\n\n\n<p>25 people own a white car or a black car.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"6-properties-of-union-of-sets\">Properties of Union of Sets<\/h2>\n\n\n\n<p>The properties of the set union indicate their nature and help carry out further operations. Here are the five properties of the union of sets.<\/p>\n\n\n\n<figure class=\"wp-block-table wj-custom-table\"><table class=\"wj-table-class\"><thead><tr><th class=\"has-text-align-left\" data-align=\"left\"><strong>Properties of Union<\/strong><\/th><th class=\"has-text-align-left\" data-align=\"left\"><strong>Notation<\/strong><\/th><\/tr><\/thead><tbody><tr><td class=\"has-text-align-left\" data-align=\"left\">Commutative Property<\/td><td class=\"has-text-align-left\" data-align=\"left\">$A \\cup B = B \\cup A$<\/td><\/tr><tr><td class=\"has-text-align-left\" data-align=\"left\">Associative Property<\/td><td class=\"has-text-align-left\" data-align=\"left\">$(A\\cup B)\\cup C = A\\cup (B\\cup C)$<\/td><\/tr><tr><td class=\"has-text-align-left\" data-align=\"left\">Idempotent Property<\/td><td class=\"has-text-align-left\" data-align=\"left\">$A \\cup A = A$<\/td><\/tr><tr><td class=\"has-text-align-left\" data-align=\"left\">Property of \u2caa\/Identity Law<\/td><td class=\"has-text-align-left\" data-align=\"left\">$A\\cup \\Phi = A$<\/td><\/tr><tr><td class=\"has-text-align-left\" data-align=\"left\">Property of Universal Set<\/td><td class=\"has-text-align-left\" data-align=\"left\">$A\\cup U = U$<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<ol class=\"wp-block-list\">\n<li><strong>Commutative Property: <\/strong>$A \\cup B = B \\cup A$<\/li>\n<\/ol>\n\n\n\n<p>The union sets will be independent of the order of the sets.&nbsp;<\/p>\n\n\n\n<p><strong>Example:<\/strong> set $A = \\left\\{\\text{orange, banana, apple, mango}\\right\\}$ and set $B = \\left\\{1, 6, 2, 4\\right\\}$.&nbsp;<\/p>\n\n\n\n<p>$A\\cup B = \\left\\{\\text{orange, banana, apple, mango,} 1, 6, 2, 4\\right\\}$<\/p>\n\n\n\n<p>$B\\cup A = \\left\\{\\text{orange, banana, apple, mango,} 1, 6, 2, 4\\right\\}$<\/p>\n\n\n\n<ol class=\"wp-block-list\" start=\"2\">\n<li><strong>Associative Property: <\/strong>$(A \\cup B) \\cup C = A \\cup (B \\cup C)$<\/li>\n<\/ol>\n\n\n\n<p>The union of three or more sets does not depend on how we group the sets.&nbsp;<\/p>\n\n\n\n<p><strong>Example:<\/strong> Set $A = \\left\\{\\text{orange, banana, apple, mango}\\right\\}$<\/p>\n\n\n\n<p>Set $B = \\left\\{1, 6, 2, 4\\right\\}$<\/p>\n\n\n\n<p>Set $C = \\left\\{\\text{A, E, I, O, U}\\right\\}$.&nbsp;<\/p>\n\n\n\n<p>$A \\cup B = \\left\\{\\text{orange, banana, apple, mango,} 1, 6, 2, 4\\right\\}$<\/p>\n\n\n\n<p>$(A\\cup B)\\cup C = \\left\\{\\text{orange, banana, apple, mango,} 1, 6, 2, 4, A, \\text{E, I, O U}\\right\\}$<\/p>\n\n\n\n<p>$(B\\cup C) = \\left\\{1, 6, 2, 4, A, \\text{E, I, O U}\\right\\}$<\/p>\n\n\n\n<p>$A \\cup (B \\cup C) = \\left\\{\\text{orange, banana, apple, mango,} 1, 6, 2, 4, \\text{A, E, I, O U}\\right\\}$<\/p>\n\n\n\n<p>Thus, $(A\\cup B)\\cup C = A\\cup (B\\cup C)$&nbsp;<\/p>\n\n\n\n<ol class=\"wp-block-list\" start=\"3\">\n<li><strong>Idempotent Property: <\/strong>$A\\cup A = A$<\/li>\n<\/ol>\n\n\n\n<p>The property states that the union of the set with itself is the original set.&nbsp;<\/p>\n\n\n\n<p><strong>Example: <\/strong>&nbsp;$A = \\left\\{\\text{orange, banana, apple, mango}\\right\\}$<\/p>\n\n\n\n<p>$A\\cup A = \\left\\{\\text{orange, banana, apple, mango}\\right\\} \\left\\{\\text{orange, banana, apple, mango}\\right\\}$<\/p>\n\n\n\n<p>$A\\cup A = \\left\\{\\text{orange, banana, apple, mango}\\right\\}$<\/p>\n\n\n\n<ol class=\"wp-block-list\" start=\"4\">\n<li><strong>Identity Law: <\/strong>$A\\cup \\phi  = A$<\/li>\n<\/ol>\n\n\n\n<p>The union of any set with a null set or empty set is the set itself. The null set is represented as . Thus, for any set A, we have $A\\cup \\phi  = A$.&nbsp;<\/p>\n\n\n\n<ol class=\"wp-block-list\" start=\"5\">\n<li><strong>Domination Property\/Property of Universal Set: <\/strong>$A\\cup U = U$<\/li>\n<\/ol>\n\n\n\n<p>The union of a set with a universal set results in a universal set. The universal set contains all the elements of other sets.&nbsp;<\/p>\n\n\n\n<p><strong>Example: <\/strong>$A = \\left\\{1, 2, 3\\right\\}, U = \\left\\{1, 2, 3, 4, 5\\right\\}$<\/p>\n\n\n\n<p>$A\\cup U = \\left\\{1, 2, 3, 4, 5\\right\\} = U$<\/p>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"7-facts-about-union-of-sets\">Facts about Union of Sets<\/h2>\n\n\n\n<div style=\"color:#0369a1;background-color:#e0f2fe\" class=\"wp-block-roelmagdaleno-callout-block has-text-color has-background is-layout-flex wp-container-roelmagdaleno-callout-block-is-layout-8cf370e7 wp-block-roelmagdaleno-callout-block-is-layout-flex\"><div>\n<ul class=\"wp-block-list\">\n<li>The complement of union of two sets is the intersection of the complements of the individual sets. This is known as De Morgan\u2019s law.<br>$(A \\cup B)\u2019 = A\u2019 \\cap B\u2019$<\/li>\n\n\n\n<li>$A \\cup B$ is the event that consists of all sample points that are either in A or in B or in both A and B.<\/li>\n\n\n\n<li>Union of two sets is larger than the individual sets.<\/li>\n\n\n\n<li>The union of a set and the empty set is the original set.<\/li>\n<\/ul>\n<\/div><\/div>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"8-conclusion\">Conclusion<\/h2>\n\n\n\n<p>Sets are clusters of numbers that can be acted upon through interesting operations. There are multiple types of sets, and the union of sets is one of the possible multiple operations. It refers to the collection of all the elements in individual subsets. The union of sets has distinguishing properties, making calculations quick and easy.&nbsp;<\/p>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"9-solved-examples-of-union-of-sets\">Solved Examples of Union of Sets<\/h2>\n\n\n\n<p><strong>1. If <\/strong>$n(A) = 9, n(B) = 1, n(A\\cap B) = 2$<strong>, what is the number of elements in A union B?&nbsp;<\/strong><\/p>\n\n\n\n<p><strong>Solution:<\/strong>&nbsp;<\/p>\n\n\n\n<p>$n(A\\cup B) = n(A) + n(B) \\;-\\; n(A\\cap B)$<\/p>\n\n\n\n<p>$n(A\\cup B) = 9 + 1 \\;-\\; 2$<\/p>\n\n\n\n<p>$n(A\\cup B) = 10 \\;-\\; 2$<\/p>\n\n\n\n<p>$n(A\\cup B) = 8$<\/p>\n\n\n\n<p><strong>2. What is the number of elements in the union of A and B, where <\/strong>$A = \\left\\{\\text{x: x is a natural number and} 1 \\lt x \\le 6\\right\\},\\; B = \\left\\{\\text{x: x is a natural number and} 6 \\lt x \\lt 10\\right\\}$<strong>?<\/strong><\/p>\n\n\n\n<p><strong>Solution:<\/strong>&nbsp;<\/p>\n\n\n\n<p>$A = \\left\\{\\text{x: x is a natural number and} 1 \\lt x \\le 6 \\right\\}$<\/p>\n\n\n\n<p>Set $A = \\left\\{2, 3, 4, 5, 6\\right\\}$<\/p>\n\n\n\n<p>$B = \\left\\{\\text{x: x is a natural number and} 6 \\lt x \\lt 10\\right\\}$<\/p>\n\n\n\n<p>Set $B = \\left\\{7, 8, 9\\right\\}$<\/p>\n\n\n\n<p>Thus, $A\\cup B = \\left\\{2, 3, 4, 5, 6, 7, 8, 9\\right\\}$<\/p>\n\n\n\n<p><strong>3. What is A<\/strong> <strong><\/strong><strong>B in the following venn diagram?<\/strong><\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"299\" height=\"198\" src=\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/07\/venn-diagram-of-sets-a-and-b.png\" alt=\"venn diagram of sets A and B\" class=\"wp-image-31424\" title=\"venn diagram of sets A and B\"\/><\/figure>\n\n\n\n<p><strong>Solution:&nbsp;<\/strong><\/p>\n\n\n\n<p>$U = \\left\\{1, 2, 3, 4, 5, 6\\right\\}$<\/p>\n\n\n\n<p>$A = \\left\\{2, 4, 6\\right\\}$<\/p>\n\n\n\n<p>$B = \\left\\{3, 4, 5 ,6\\right\\}$<\/p>\n\n\n\n<p>The entire shaded region represents $A\\cup B$. Note that element 1 is not part of the union of A and B.<\/p>\n\n\n\n<p>$A \\cup B = \\left\\{2, 3, 4, 5, 6\\right\\}$<\/p>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"10-practice-problems-on-union-of-sets\">Practice Problems on Union of Sets<\/h2>\n\n\n\n<div class=\"spq_wrapper\"><h2 style=\"display:none;\">Union of Sets - Definition, Formula, Examples, Facts, FAQs<\/h2><p style=\"display:none;\">Attend this quiz & Test your knowledge.<\/p><div class=\"spq_question_wrapper\" data-answer=\"0\"><span class=\"spq_question_header\"><span class=\"sqp_question_number\">1<\/span><h3 class=\"sqp_question_text\">What is the union of set $A = \\left\\{\\text{Parrot, Peacock, Sparrow}\\right\\}$ with the null set?<\/h3><\/span><div class=\"spq_answer_block\" data-value=\"0\">A<\/div><div class=\"spq_answer_block\" data-value=\"1\">U<\/div><div class=\"spq_answer_block\" data-value=\"2\">$\\phi$<\/div><div class=\"spq_answer_block\" data-value=\"3\">$\\left\\{\\text{Sparrow}\\right\\}$<\/div><div class=\"sqp_question_hint\"><div class=\"sqp_question_hint__header\"><span class=\"spq_correct\">Correct<\/span><span class=\"spq_incorrect\">Incorrect<\/span><\/div><div class=\"sqp_question_hint__content\"><span>Correct answer is: A<br\/>Union of any set with a null set  is the set itself.<\/span><\/div><\/div><\/div><div class=\"spq_question_wrapper\" data-answer=\"0\"><span class=\"spq_question_header\"><span class=\"sqp_question_number\">2<\/span><h3 class=\"sqp_question_text\">What is $A\\cup  A$?<\/h3><\/span><div class=\"spq_answer_block\" data-value=\"0\">A<\/div><div class=\"spq_answer_block\" data-value=\"1\">U<\/div><div class=\"spq_answer_block\" data-value=\"2\">$\\phi$<\/div><div class=\"spq_answer_block\" data-value=\"3\">$\\cup$<\/div><div class=\"sqp_question_hint\"><div class=\"sqp_question_hint__header\"><span class=\"spq_correct\">Correct<\/span><span class=\"spq_incorrect\">Incorrect<\/span><\/div><div class=\"sqp_question_hint__content\"><span>Correct answer is: A<br\/>By idempotent law, $A\\cup  A = A$.<\/span><\/div><\/div><\/div><div class=\"spq_question_wrapper\" data-answer=\"3\"><span class=\"spq_question_header\"><span class=\"sqp_question_number\">3<\/span><h3 class=\"sqp_question_text\">If $A = \\left\\{0, 1\\right\\}$ and $B = \\left\\{1, 9\\right\\}$, then $A\\cup  B =$<\/h3><\/span><div class=\"spq_answer_block\" data-value=\"0\">$\\left\\{019\\right\\}$<\/div><div class=\"spq_answer_block\" data-value=\"1\">$\\left\\{1, 9\\right\\}$<\/div><div class=\"spq_answer_block\" data-value=\"2\">$\\left\\{9\\right\\}$<\/div><div class=\"spq_answer_block\" data-value=\"3\">$\\left\\{0, 1, 9\\right\\}$<\/div><div class=\"sqp_question_hint\"><div class=\"sqp_question_hint__header\"><span class=\"spq_correct\">Correct<\/span><span class=\"spq_incorrect\">Incorrect<\/span><\/div><div class=\"sqp_question_hint__content\"><span>Correct answer is: $\\left\\{0, 1, 9\\right\\}$<br\/>$\\left\\{0, 1\\right\\}\\;\\left\\{1, 9\\right\\} = \\left\\{0, 1, 9\\right\\}$<\/span><\/div><\/div><\/div><div class=\"spq_question_wrapper\" data-answer=\"2\"><span class=\"spq_question_header\"><span class=\"sqp_question_number\">4<\/span><h3 class=\"sqp_question_text\">What is the union of A and B?<\/h3><\/span><div class=\"spq_question_image\"><img decoding=\"async\" src=\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/07\/set-a-equal-ten-and-b-equal-seven.png\"><\/div><div class=\"spq_answer_block\" data-value=\"0\">$\\left\\{10\\right\\}$<\/div><div class=\"spq_answer_block\" data-value=\"1\">$\\left\\{7\\right\\}$<\/div><div class=\"spq_answer_block\" data-value=\"2\">$\\left\\{10, 7\\right\\}$<\/div><div class=\"spq_answer_block\" data-value=\"3\">$\\left\\{10, 7, \\phi \\right\\}$<\/div><div class=\"sqp_question_hint\"><div class=\"sqp_question_hint__header\"><span class=\"spq_correct\">Correct<\/span><span class=\"spq_incorrect\">Incorrect<\/span><\/div><div class=\"sqp_question_hint__content\"><span>Correct answer is: $\\left\\{10, 7\\right\\}$<br\/>$A\\cup  B = \\left\\{10, 7\\right\\}$<\/span><\/div><\/div><\/div><div class=\"spq_question_wrapper\" data-answer=\"1\"><span class=\"spq_question_header\"><span class=\"sqp_question_number\">5<\/span><h3 class=\"sqp_question_text\">The union of the set of natural numbers and the set {0} gives the set of _________.<\/h3><\/span><div class=\"spq_answer_block\" data-value=\"0\">integers<\/div><div class=\"spq_answer_block\" data-value=\"1\">whole numbers<\/div><div class=\"spq_answer_block\" data-value=\"2\">rational numbers<\/div><div class=\"spq_answer_block\" data-value=\"3\">real numbers<\/div><div class=\"sqp_question_hint\"><div class=\"sqp_question_hint__header\"><span class=\"spq_correct\">Correct<\/span><span class=\"spq_incorrect\">Incorrect<\/span><\/div><div class=\"sqp_question_hint__content\"><span>Correct answer is: whole numbers<br\/>The set of whole numbers include the set of natural numbers and 0.<\/span><\/div><\/div><\/div><\/div>  <script type=\"application\/ld+json\">{\n        \"@context\": \"https:\/\/schema.org\/\", \n        \"@type\": \"Quiz\", \n        \"typicalAgeRange\": \"3-11\",\n        \"educationalLevel\":  \"beginner\",\n        \"assesses\" : \"Attend this quiz & Test your knowledge.\",\n        \"educationalAlignment\": [\n              {\n                \"@type\": \"AlignmentObject\",\n                \"alignmentType\": \"educationalSubject\",\n                \"targetName\": \"Math\"\n              }] ,\n        \"name\": \"Union of Sets - Definition, Formula, Examples, Facts, FAQs\",        \n        \"about\": {\n                \"@type\": \"Thing\",\n                \"name\": \"Union of Sets\"\n        },  \n        \"hasPart\": [{\n                    \"@type\": \"Question\",   \n                    \"eduQuestionType\": \"Multiple choice\",\n                    \"learningResourceType\": \"Practice problem\",\n                    \"name\": \"What is 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\"learningResourceType\": \"Practice problem\",\n                    \"name\": \"What is $$A\\\\cup  A$$?\",\n                    \"text\": \"What is $$A\\\\cup  A$$?\",\n                    \"comment\": {\n                      \"@type\": \"Comment\",\n                      \"text\": \"By idempotent law, $$A\\\\cup  A = A$$.\"\n                    },\n                    \"encodingFormat\": \"text\/html\",\n                    \"suggestedAnswer\": [ {\n                                \"@type\": \"Answer\",\n                                \"position\": 1,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"U\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"By idempotent law, $$A\\\\cup  A = A$$.\"\n                                    }\n                                }, {\n                                \"@type\": \"Answer\",\n                                \"position\": 2,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"$$\\\\phi$$\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"By idempotent law, $$A\\\\cup  A = A$$.\"\n                                    }\n                                }, {\n                                \"@type\": \"Answer\",\n                                \"position\": 3,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"$$\\\\cup$$\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"By idempotent law, $$A\\\\cup  A = A$$.\"\n                                    }\n                                }],\n          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\"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"$$\\\\left\\\\{0, 1\\\\right\\\\}\\\\;\\\\left\\\\{1, 9\\\\right\\\\} = \\\\left\\\\{0, 1, 9\\\\right\\\\}$$\"\n                                    }\n                                }],\n                    \"acceptedAnswer\": {\n                      \"@type\": \"Answer\",\n                      \"position\": 3,\n                      \"encodingFormat\": \"text\/html\",\n                      \"text\": \"$$\\\\left\\\\{0, 1, 9\\\\right\\\\}$$\",\n                      \"comment\": {\n                          \"@type\": \"Comment\",\n                          \"text\": \"$$\\\\left\\\\{0, 1\\\\right\\\\}\\\\;\\\\left\\\\{1, 9\\\\right\\\\} = \\\\left\\\\{0, 1, 9\\\\right\\\\}$$\"\n                        },\n                      \"answerExplanation\": {\n                        \"@type\": \"Comment\",\n                        \"text\": \"$$\\\\left\\\\{0, 1\\\\right\\\\}\\\\;\\\\left\\\\{1, 9\\\\right\\\\} = \\\\left\\\\{0, 1, 9\\\\right\\\\}$$\"\n                      }\n                    } \n\n                    },{\n                    \"@type\": \"Question\",   \n                    \"eduQuestionType\": \"Multiple choice\",\n                    \"learningResourceType\": \"Practice problem\",\n                    \"name\": \"What is the union of A and B?\",\n                    \"text\": \"What is the union of A and B? <img src=\\\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/07\/set-a-equal-ten-and-b-equal-seven.png\\\"\/>\",\n                    \"comment\": {\n                      \"@type\": \"Comment\",\n                      \"text\": \"$$A\\\\cup  B = \\\\left\\\\{10, 7\\\\right\\\\}$$\"\n                    },\n                    \"encodingFormat\": \"text\/html\",\n                    \"suggestedAnswer\": [ {\n                                \"@type\": \"Answer\",\n                                \"position\": 0,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"$$\\\\left\\\\{10\\\\right\\\\}$$\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"$$A\\\\cup  B = \\\\left\\\\{10, 7\\\\right\\\\}$$\"\n                                    }\n                                }, {\n                                \"@type\": \"Answer\",\n                                \"position\": 1,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"$$\\\\left\\\\{7\\\\right\\\\}$$\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"$$A\\\\cup  B = \\\\left\\\\{10, 7\\\\right\\\\}$$\"\n                                    }\n                                }, {\n                                \"@type\": \"Answer\",\n                                \"position\": 3,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"$$\\\\left\\\\{10, 7, \\\\phi \\\\right\\\\}$$\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"$$A\\\\cup  B = \\\\left\\\\{10, 7\\\\right\\\\}$$\"\n                                    }\n                                }],\n                    \"acceptedAnswer\": {\n                      \"@type\": \"Answer\",\n                      \"position\": 2,\n                      \"encodingFormat\": \"text\/html\",\n                      \"text\": \"$$\\\\left\\\\{10, 7\\\\right\\\\}$$\",\n                      \"comment\": {\n                          \"@type\": \"Comment\",\n                          \"text\": \"$$A\\\\cup  B = \\\\left\\\\{10, 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set of whole numbers include the set of natural numbers and 0.\"\n                                    }\n                                }, {\n                                \"@type\": \"Answer\",\n                                \"position\": 3,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"real numbers\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"The set of whole numbers include the set of natural numbers and 0.\"\n                                    }\n                                }],\n                    \"acceptedAnswer\": {\n                      \"@type\": \"Answer\",\n                      \"position\": 1,\n                      \"encodingFormat\": \"text\/html\",\n                      \"text\": \"whole numbers\",\n                      \"comment\": {\n                          \"@type\": \"Comment\",\n                          \"text\": \"The set of whole numbers include the set of natural numbers and 0.\"\n                        },\n                      \"answerExplanation\": {\n                        \"@type\": \"Comment\",\n                        \"text\": \"The set of whole numbers include the set of natural numbers and 0.\"\n                      }\n                    } \n\n                    }]}<\/script>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"11-frequently-asked-questions-about-union-of-sets\">Frequently Asked Questions about Union of Sets<\/h2>\n\n\n<div class=\"wp-block-ub-content-toggle\" id=\"ub-content-toggle-d2dbf97a-c54b-4ac1-8cc6-76543d0ca93e\" data-mobilecollapse=\"true\" data-desktopcollapse=\"true\">\n<div class=\"wp-block-ub-content-toggle-accordion\">\n                <div class=\"wp-block-ub-content-toggle-accordion-title-wrap\" aria-expanded=\"false\" aria-controls=\"ub-content-toggle-panel-0-d2dbf97a-c54b-4ac1-8cc6-76543d0ca93e\" tabindex=\"0\">\n                    <p class=\"wp-block-ub-content-toggle-accordion-title ub-content-toggle-title-d2dbf97a-c54b-4ac1-8cc6-76543d0ca93e\"><strong>What is the difference between union of sets and intersection of sets?<\/strong><\/p><div class=\"wp-block-ub-content-toggle-accordion-toggle-wrap right\"><span class=\"wp-block-ub-content-toggle-accordion-state-indicator wp-block-ub-chevron-down\"><\/span>\n                    <\/div><\/div><div role=\"region\" class=\"wp-block-ub-content-toggle-accordion-content-wrap ub-hide\" id=\"ub-content-toggle-panel-0-d2dbf97a-c54b-4ac1-8cc6-76543d0ca93e\">\n\n<p>The union contains all the elements from each set, while the intersection contains only the common elements between the sets.<\/p>\n\n<\/div><\/div>\n\n<div class=\"wp-block-ub-content-toggle-accordion\">\n                <div class=\"wp-block-ub-content-toggle-accordion-title-wrap\" aria-expanded=\"false\" aria-controls=\"ub-content-toggle-panel-1-d2dbf97a-c54b-4ac1-8cc6-76543d0ca93e\" tabindex=\"0\">\n                    <p class=\"wp-block-ub-content-toggle-accordion-title ub-content-toggle-title-d2dbf97a-c54b-4ac1-8cc6-76543d0ca93e\"><strong>What is a complement of a set?<\/strong><\/p><div class=\"wp-block-ub-content-toggle-accordion-toggle-wrap right\"><span class=\"wp-block-ub-content-toggle-accordion-state-indicator wp-block-ub-chevron-down\"><\/span>\n                    <\/div><\/div><div role=\"region\" class=\"wp-block-ub-content-toggle-accordion-content-wrap ub-hide\" id=\"ub-content-toggle-panel-1-d2dbf97a-c54b-4ac1-8cc6-76543d0ca93e\">\n\n<p>The complement of a set is a set containing all the elements in the universal set that do not belong to the given set.<\/p>\n\n<\/div><\/div>\n\n<div class=\"wp-block-ub-content-toggle-accordion\">\n                <div class=\"wp-block-ub-content-toggle-accordion-title-wrap\" aria-expanded=\"false\" aria-controls=\"ub-content-toggle-panel-2-d2dbf97a-c54b-4ac1-8cc6-76543d0ca93e\" tabindex=\"0\">\n                    <p class=\"wp-block-ub-content-toggle-accordion-title ub-content-toggle-title-d2dbf97a-c54b-4ac1-8cc6-76543d0ca93e\"><strong>What are disjoint sets?<\/strong><\/p><div class=\"wp-block-ub-content-toggle-accordion-toggle-wrap right\"><span class=\"wp-block-ub-content-toggle-accordion-state-indicator wp-block-ub-chevron-down\"><\/span>\n                    <\/div><\/div><div role=\"region\" class=\"wp-block-ub-content-toggle-accordion-content-wrap ub-hide\" id=\"ub-content-toggle-panel-2-d2dbf97a-c54b-4ac1-8cc6-76543d0ca93e\">\n\n<p>Disjoint sets do not have any common elements. The union of such sets contains all the elements of individual sets, while intersection leads to empty or null sets.<\/p>\n\n<\/div><\/div>\n\n<div class=\"wp-block-ub-content-toggle-accordion\">\n                <div class=\"wp-block-ub-content-toggle-accordion-title-wrap\" aria-expanded=\"false\" aria-controls=\"ub-content-toggle-panel-3-d2dbf97a-c54b-4ac1-8cc6-76543d0ca93e\" tabindex=\"0\">\n                    <p class=\"wp-block-ub-content-toggle-accordion-title ub-content-toggle-title-d2dbf97a-c54b-4ac1-8cc6-76543d0ca93e\"><strong>Can the union of two sets be an empty set?<\/strong><\/p><div class=\"wp-block-ub-content-toggle-accordion-toggle-wrap right\"><span class=\"wp-block-ub-content-toggle-accordion-state-indicator wp-block-ub-chevron-down\"><\/span>\n                    <\/div><\/div><div role=\"region\" class=\"wp-block-ub-content-toggle-accordion-content-wrap ub-hide\" id=\"ub-content-toggle-panel-3-d2dbf97a-c54b-4ac1-8cc6-76543d0ca93e\">\n\n<p>The union of two sets is empty when both sets are empty sets.<\/p>\n\n<\/div><\/div>\n<\/div>","protected":false},"excerpt":{"rendered":"<p>What Is the Union of Sets? The union of a collection of sets is a set containing all the elements of individual sets. The union of two sets A and B is a new set denoted by $A\\cup B$, which contains all the elements of the set A and all the elements of set B &#8230; <a title=\"Union of Sets &#8211; Definition, Formula, Examples, Facts, FAQs\" class=\"read-more\" href=\"https:\/\/www.splashlearn.com\/math-vocabulary\/union-of-sets\" aria-label=\"More on Union of Sets &#8211; Definition, Formula, Examples, Facts, FAQs\">Read more<\/a><\/p>\n","protected":false},"author":2,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"ub_ctt_via":"","footnotes":""},"categories":[1],"tags":[],"class_list":["post-31416","post","type-post","status-publish","format-standard","hentry","category-all"],"featured_image_src":null,"author_info":{"display_name":"Mithun Jhawar","author_link":"https:\/\/www.splashlearn.com\/math-vocabulary\/author\/mithun-jhawarsplashlearn-com\/"},"acf":[],"aioseo_notices":[],"_links":{"self":[{"href":"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-json\/wp\/v2\/posts\/31416","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-json\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-json\/wp\/v2\/comments?post=31416"}],"version-history":[{"count":11,"href":"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-json\/wp\/v2\/posts\/31416\/revisions"}],"predecessor-version":[{"id":31432,"href":"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-json\/wp\/v2\/posts\/31416\/revisions\/31432"}],"wp:attachment":[{"href":"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-json\/wp\/v2\/media?parent=31416"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-json\/wp\/v2\/categories?post=31416"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-json\/wp\/v2\/tags?post=31416"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}