{"id":32484,"date":"2023-08-02T16:44:19","date_gmt":"2023-08-02T16:44:19","guid":{"rendered":"https:\/\/www.splashlearn.com\/math-vocabulary\/?page_id=32484"},"modified":"2023-08-04T02:08:25","modified_gmt":"2023-08-04T02:08:25","slug":"power-set-definition-formula-cardinality-properties-examples","status":"publish","type":"post","link":"https:\/\/www.splashlearn.com\/math-vocabulary\/power-set","title":{"rendered":"Power Set &#8211; Definition, Formula, Cardinality, Properties, Examples"},"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-b297fca2-c671-45cb-b3e8-dd724b364581\" 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\/power-set#0-what-is-a-power-set-in-math>What Is a Power Set in Math?<\/a><\/li><li><a href=https:\/\/www.splashlearn.com\/math-vocabulary\/power-set#3-how-to-find-the-power-set>How to Find the Power Set<\/a><\/li><li><a href=https:\/\/www.splashlearn.com\/math-vocabulary\/power-set#5-power-set-properties>Power Set Properties<\/a><\/li><li><a href=https:\/\/www.splashlearn.com\/math-vocabulary\/power-set#9-solved-examples-on-power-set>Solved Examples on Power Set<\/a><\/li><li><a href=https:\/\/www.splashlearn.com\/math-vocabulary\/power-set#10-practice-problems-on-power-set>Practice Problems on Power Set<\/a><\/li><li><a href=https:\/\/www.splashlearn.com\/math-vocabulary\/power-set#11-frequently-asked-questions-about-power-set>Frequently Asked Questions about Power Set<\/a><\/li><\/ul><\/div><\/div><\/div>\n\n\n<h2 class=\"wp-block-heading\" id=\"0-what-is-a-power-set-in-math\">What Is a Power Set in Math?<\/h2>\n\n\n\n<p><strong>The power set in set theory is a set of all subsets of a given set.<\/strong> Thus, the empty set and the set itself are always included in the power set. We denote the power set of set A as P(A) or \u2118(A).<\/p>\n\n\n\n<p>To define the power set of a set, we write the <a href=\"https:\/\/www.splashlearn.com\/math-vocabulary\/sets\">set<\/a> of all subsets of the given set.<\/p>\n\n\n\n<p><strong>Example:&nbsp;<\/strong><\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"620\" height=\"429\" src=\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/08\/power-set-visual-example.png\" alt=\"Power set visual example\" class=\"wp-image-32497\" title=\"Power set visual example\" srcset=\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/08\/power-set-visual-example.png 620w, https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/08\/power-set-visual-example-300x208.png 300w\" sizes=\"auto, (max-width: 620px) 100vw, 620px\" \/><\/figure>\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\/count-out-the-objects-from-a-set\" 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_count_out_w10_pt.png\" alt=\"Count Out the Objects from a Set 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\">Count Out the Objects from a Set 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\/divide-the-decimal-by-power-of-10\" 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\/dec_mul_div_div_power_10_1_pt.png\" alt=\"Divide the Decimal by Power of 10 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\">Divide the Decimal by Power of 10 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 class=\"game-card-container-outer\">\r\n\t<a href=\"https:\/\/www.splashlearn.com\/s\/math-games\/multiply-by-power-of-10\" 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\/dec_mul_div_multiply_by_power_10_2_pt.png\" alt=\"Multiply by Power of 10 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\">Multiply by Power of 10 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\/multiply-the-decimal-numbers-by-a-power-of-10\" 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\/dec_mul_div_multiply_by_power_10_1_pt.png\" alt=\"Multiply the Decimal Numbers by a Power of 10 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\">Multiply the Decimal Numbers by a Power of 10 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\/perform-division-with-powers-of-10-as-divisors\" 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\/dec_mul_div_div_power_10_2_pt.png\" alt=\"Perform Division With Powers of 10 as Divisors 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\">Perform Division With Powers of 10 as Divisors 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\/rearrange-groups-to-make-a-set-bigger\" 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_rearrange_grps_pt.png\" alt=\"Rearrange Groups to Make a Set Bigger 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\">Rearrange Groups to Make a Set Bigger 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\/set-quarter-hours-on-an-analog-clock\" 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\/time_qrtr_hrs_set_time_words_pt.png\" alt=\"Set Quarter Hours on an Analog Clock 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\">Set Quarter Hours on an Analog Clock 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\/set-the-time-after-an-interval\" 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\/time_set_time_interval_pt.png\" alt=\"Set the Time after an Interval 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\">Set the Time after an Interval 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 = 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     checkScrollPosition();\n            slidesContainer.addEventListener(\"scroll\", checkScrollPosition);\n        });\n    <\/script><h2 class=\"wp-block-heading\" id=\"1-power-set-definition\">Power Set: Definition<\/h2>\n\n\n\n<p>The set of all the subsets of a given set is called the <strong>power set in math<\/strong>. The original set and an empty set are always the members of a power set.&nbsp;<\/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\/decode-the-code-powers-of-10\" 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\/decode-the-code-powers-of-10.jpeg\" alt=\"Decode the Code\u2014 Powers of 10 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 class=\"worksheet-card-container-outer\">\r\n\t<a href=\"https:\/\/www.splashlearn.com\/s\/math-worksheets\/divide-decimals-by-powers-of-10-horizontal-division\" 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\/divide-decimals-by-powers-of-10-horizontal-division.jpeg\" alt=\"Divide Decimals by Powers of 10: Horizontal Division 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 class=\"worksheet-card-container-outer\">\r\n\t<a href=\"https:\/\/www.splashlearn.com\/s\/math-worksheets\/divide-decimals-by-powers-of-10-horizontal-timed-practice\" 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\/divide-decimals-by-powers-of-10-horizontal-timed-practice.jpeg\" alt=\"Divide Decimals by Powers of 10: Horizontal Timed Practice\">\r\n\t    <\/div>\r\n\t\t<div class=\"worksheet-card-container-inner\" >\r\n\t\t<\/div><\/a>\r\n<\/div><div class=\"worksheet-card-container-outer\">\r\n\t<a href=\"https:\/\/www.splashlearn.com\/s\/math-worksheets\/divide-decimals-by-powers-of-10-missing-numbers\" 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\/divide-decimals-by-powers-of-10-missing-numbers.jpeg\" alt=\"Divide Decimals by Powers of 10: Missing Numbers 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 class=\"worksheet-card-container-outer\">\r\n\t<a href=\"https:\/\/www.splashlearn.com\/s\/math-worksheets\/divide-hundredths-by-powers-of-10-horizontal-division\" 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\/divide-hundredths-by-powers-of-10-horizontal-division.jpeg\" alt=\"Divide Hundredths by Powers of 10: Horizontal Division 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 class=\"worksheet-card-container-outer\">\r\n\t<a href=\"https:\/\/www.splashlearn.com\/s\/math-worksheets\/divide-hundredths-by-powers-of-10-missing-numbers\" 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\/divide-hundredths-by-powers-of-10-missing-numbers.jpeg\" alt=\"Divide Hundredths by Powers of 10: Missing Numbers 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 class=\"worksheet-card-container-outer\">\r\n\t<a href=\"https:\/\/www.splashlearn.com\/s\/math-worksheets\/divide-tenths-by-powers-of-10-horizontal-division\" 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\/divide-tenths-by-powers-of-10-horizontal-division.jpeg\" alt=\"Divide Tenths by Powers of 10: Horizontal Division 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 class=\"worksheet-card-container-outer\">\r\n\t<a href=\"https:\/\/www.splashlearn.com\/s\/math-worksheets\/divide-tenths-by-powers-of-10-missing-numbers\" 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\/divide-tenths-by-powers-of-10-missing-numbers.jpeg\" alt=\"Divide Tenths by Powers of 10: Missing Numbers 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 class=\"worksheet-card-container-outer\">\r\n\t<a href=\"https:\/\/www.splashlearn.com\/s\/math-worksheets\/find-fraction-of-a-set-of-objects\" 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\/find-fraction-of-a-set-of-objects.jpeg\" alt=\"Find Fraction of a Set of Objects 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 class=\"worksheet-card-container-outer\">\r\n\t<a href=\"https:\/\/www.splashlearn.com\/s\/math-worksheets\/find-fraction-of-a-set\" 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\/find-fraction-of-a-set.jpeg\" alt=\"Find Fraction of a Set 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 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     checkScrollPosition();\n            slidesContainer.addEventListener(\"scroll\", checkScrollPosition);\n        });\n    <\/script><h2 class=\"wp-block-heading\" id=\"2-cardinality-of-a-power-set\">Cardinality of a Power Set<\/h2>\n\n\n\n<p>The number of subsets of a set with cardinality n can be determined by the formula $2^{n}$. Thus, the cardinality of the power set of a set having \u201cn\u201d elements is $2^{n}$.<\/p>\n\n\n\n<p><strong>Example:<\/strong> If a set has 2 elements, its power set has $2^{2} = 4$ elements.<\/p>\n\n\n\n<p>$A = \\left\\{1,\\; 2\\right\\}$&nbsp;<\/p>\n\n\n\n<p>$n(A) = | A | = 2$<\/p>\n\n\n\n<p>$P(A) = \\left\\{ \\left\\{1\\right\\}, \\left\\{2\\right\\}, \\left\\{1, 2\\right\\}, \u2205 \\right\\}$<\/p>\n\n\n\n<p>$n( P(A) ) = | P(A) | = 4$<\/p>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"3-how-to-find-the-power-set\">How to Find the Power Set<\/h2>\n\n\n\n<p>Power set symbol is P. The <strong>power set<\/strong> of a set A is denoted by P(A). To define the power set, start listing down the subsets of the given set.&nbsp;<\/p>\n\n\n\n<p><strong>Example:<\/strong> $S = \\left\\{x, y, z\\right\\}$<\/p>\n\n\n\n<p>Let P(S) be the powerset of S. It contains all the subsets of S.<\/p>\n\n\n\n<p>Cardinality of $S = 3$<\/p>\n\n\n\n<p>The power set will have $2^{n} = 2^{3} =&nbsp; 8$ elements.<\/p>\n\n\n\n<p>Let\u2019s list down the subsets of S to define P(S). The subsets that can be formed using the elements x, y, z are listed below.<\/p>\n\n\n\n<ol class=\"wp-block-list\">\n<li>$\\left\\{\\right\\}$<\/li>\n\n\n\n<li>$\\left\\{x\\right\\}$<\/li>\n\n\n\n<li>$\\left\\{y\\right\\}$<\/li>\n\n\n\n<li>$\\left\\{z\\right\\}$<\/li>\n\n\n\n<li>$\\left\\{x, y\\right\\}$<\/li>\n\n\n\n<li>$\\left\\{y, z\\right\\}$<\/li>\n\n\n\n<li>$\\left\\{x, z\\right\\}$<\/li>\n\n\n\n<li>$\\left\\{x, y, z\\right\\}$<\/li>\n<\/ol>\n\n\n\n<p>Therefore, the power set of set S will be written as<\/p>\n\n\n\n<p>$P(S) = \\left\\{ \u2205, \\left\\{x\\right\\}, \\left\\{y\\right\\}, \\left\\{z\\right\\}, \\left\\{x, y\\right\\}, \\left\\{y, z\\right\\}, \\left\\{x, z\\right\\}, \\left\\{x, y, z\\right\\}\\right\\}$<\/p>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"4-power-set-of-an-empty-set\">Power Set of an Empty Set<\/h2>\n\n\n\n<p>The only subset of an empty set is the empty set itself. Thus, the power set of an empty set has only one element.<\/p>\n\n\n\n<p>Empty set has no elements (zero number of elements).&nbsp;<\/p>\n\n\n\n<p>So, the power set of the empty set has $2^{0} = 1$ element.<\/p>\n\n\n\n<p>$P(\u2205) = \\left\\{ \u2205 \\right\\}$<\/p>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"5-power-set-properties\">Power Set Properties<\/h2>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Power sets are larger sets compared to the original set.<\/li>\n\n\n\n<li>The power set has $2^{n}$ elements, where n is the number of members in a set.<\/li>\n\n\n\n<li>The power set of a countable finite set is countable.<\/li>\n\n\n\n<li>\u200b\u200bThe power set of an empty set has only one element.<\/li>\n\n\n\n<li>Empty set and the set itself are the fixed elements of any power set.<\/li>\n<\/ul>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"6-examples-of-power-set\">Examples of Power Set<\/h2>\n\n\n\n<figure class=\"wp-block-table\"><table class=\"wj-table-class\"><thead><tr><th><strong>Set<\/strong><\/th><th class=\"has-text-align-left\" data-align=\"left\"><strong>Power Set<\/strong><\/th><\/tr><\/thead><tbody><tr><td>$A = \\left\\{1\\right\\}$<\/td><td class=\"has-text-align-left\" data-align=\"left\">$P(A) = \\left\\{ \u2205, \\left\\{1\\right\\} \\right\\}$<\/td><\/tr><tr><td>$B = \\left\\{ X, Y \\right\\}$<\/td><td class=\"has-text-align-left\" data-align=\"left\">$P(B) = \\left\\{ \u2205, \\left\\{X\\right\\}, \\left\\{Y\\right\\}, \\left\\{X, Y\\right\\} \\right\\}$<\/td><\/tr><tr><td>$C = \\left\\{0\\right\\}$<\/td><td class=\"has-text-align-left\" data-align=\"left\">$P(C) = \\left\\{ \u2205, \\left\\{0\\right\\} \\right\\}$<\/td><\/tr><tr><td><br>$D = \\left\\{1, 2, 3. 4, 5\\right\\}$<\/td><td class=\"has-text-align-left\" data-align=\"left\">$P(D) = \\left\\{\u2205, \\{1\\right\\}, \\left\\{2\\right\\}, \\left\\{3\\right\\}, \\left\\{4\\right\\}, \\left\\{5\\right\\}, \\left\\{1, 2\\right\\}, \\left\\{1, 3\\right\\}, \\left\\{2, 3\\right\\}, \\left\\{1, 4\\right\\}, \\left\\{2, 4\\right\\}, \\left\\{3, 4\\right\\}, \\left\\{1, 5\\right\\}, \\left\\{2, 5\\right\\}, \\left\\{3, 5\\right\\}, \\left\\{4, 5\\right\\}, \\left\\{1, 2, 3\\right\\}, \\left\\{1, 2, 4\\right\\}, \\left\\{1, 3, 4\\right\\}, \\left\\{2, 3, 4\\right\\}, \\left\\{1, 2, 3, 4\\right\\}, \\left\\{1, 2, 5\\right\\}, \\left\\{1, 3, 5\\right\\}, \\left\\{2, 3, 5\\right\\}, \\left\\{3, 4, 5\\right\\}, \\left\\{1, 2, 3, 5\\right\\}, \\left\\{1, 4, 5\\right\\}, \\left\\{2, 4, 5\\right\\}, \\left\\{1, 2, 4, 5\\right\\}, \\left\\{1, 3, 4, 5\\right\\}, \\left\\{2, 3, 4, 5\\right\\}, \\left\\{1, 2, 3, 4, 5\\right\\}$<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"7-facts-about-power-set\">Facts about Power Set<\/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 power set of a set is a set that contains all possible subsets of the original set, including the empty set and the set itself.<\/li>\n\n\n\n<li>The power set of a set with n elements will have $2^{n}$ elements.<\/li>\n\n\n\n<li>Different Power Set Notations for Set A:&nbsp; P(A) or \u2118(A) or 2<sup>A<\/sup><\/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>In this article, we learned about the power set, how to find the power set, its cardinality, properties, and examples. Let\u2019s solve a few more examples and practice problems for better understanding.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"9-solved-examples-on-power-set\">Solved Examples on Power Set<\/h2>\n\n\n\n<p><strong>1. Determine the power set of set <\/strong>$X = \\left\\{\\text{june, july, august}\\right\\}$<strong>.<\/strong><\/p>\n\n\n\n<p><strong>Solution:&nbsp;<\/strong><\/p>\n\n\n\n<p>$X = \\left\\{\\text{june, july, august}\\right\\}$<\/p>\n\n\n\n<p>There are three elements in set X.&nbsp;<\/p>\n\n\n\n<p>$N(X) = 3$<\/p>\n\n\n\n<p>The power set of set X will have $2^{n} = 2^{3} = 222 = 8$ elements.<\/p>\n\n\n\n<p>We need to list 8 subsets of the set X.<\/p>\n\n\n\n<p>Subset of $X = \\left\\{\\right\\}, \\left\\{\\text{june}\\right\\}, \\left\\{\\text{july}\\right\\}$,<\/p>\n\n\n\n<p>$\\left\\{\\text{august}\\right\\}, \\left\\{\\text{june, july}\\right\\}, \\left\\{\\text{june, august}\\right\\}$,<\/p>\n\n\n\n<p>$\\left\\{\\text{july, august}\\right\\}, \\left\\{\\text{june, july, august}\\right\\}$<\/p>\n\n\n\n<p>The power set of subset X of a set will be written as<\/p>\n\n\n\n<p>$P(X) = \\left\\{\\{\\right\\}, \\left\\{\\text{june}\\right\\}, \\left\\{\\text{july}\\right\\}$,<\/p>\n\n\n\n<p>$\\left\\{\\text{august}\\right\\}, \\left\\{\\text{june, july}\\right\\}, \\left\\{\\text{june, august}\\right\\}$,<\/p>\n\n\n\n<p>$\\left\\{\\text{july, august}\\right\\}, \\left\\{\\text{june, july, august}\\right\\}$<\/p>\n\n\n\n<p><strong>2. Find the power set of set <\/strong>$A = \\left\\{0, 1, 2, 3\\right\\}$<strong>.<\/strong><\/p>\n\n\n\n<p><strong>Solution:&nbsp;<\/strong><\/p>\n\n\n\n<p>$A= \\left\\{0, 1, 2, 3\\right\\}$<\/p>\n\n\n\n<p>Here $n (A) = 4$,<\/p>\n\n\n\n<p>P(A) has $2^{4} = 2 \\times 2 \\times 2 \\times 2 = 16$<\/p>\n\n\n\n<p>Thus, it is clear that we will have 16 subsets of set A.<\/p>\n\n\n\n<p>Subsets of $A = \\left\\{\\right\\}, \\left\\{0\\right\\}, \\left\\{1\\right\\}, \\left\\{2\\right\\}$,<\/p>\n\n\n\n<p>$\\left\\{3\\right\\}, \\left\\{0, 1\\right\\}, \\left\\{0, 2\\right\\}, \\left\\{0, 3\\right\\}, \\left\\{1, 2\\right\\}$,<\/p>\n\n\n\n<p>$\\left\\{1, 3\\right\\}, \\left\\{2, 3\\right\\}, \\left\\{0, 1, 2\\right], \\left\\{0, 1, 3\\right\\}$,<\/p>\n\n\n\n<p>$\\left\\{0, 2, 3\\right\\}, \\left\\{1, 2, 3\\right\\}, \\left\\{0, 1, 2, 3\\right\\}$<\/p>\n\n\n\n<p>Now, the power set subset A will be written as<\/p>\n\n\n\n<p>$P(A) = \\left\\{\\{\\right\\}, \\left\\{0\\right\\}, \\left\\{1\\right\\}, \\left\\{2\\right\\}, \\left\\{3\\right\\}$,<\/p>\n\n\n\n<p>$\\left\\{0, 1\\right\\}, \\left\\{0, 2\\right\\}, \\left\\{0, 3\\right\\}, \\left\\{1, 2\\right\\}, \\left\\{1, 3\\right\\}$,<\/p>\n\n\n\n<p>$\\left\\{2, 3\\right\\}, \\left\\{0, 1, 2\\right], \\left\\{0, 1, 3\\right\\}, \\left\\{0, 2, 3\\right\\}, \\left\\{1, 2, 3\\right\\}$,<\/p>\n\n\n\n<p>$\\left\\{0, 1, 2, 3\\right\\}\\}$<\/p>\n\n\n\n<p><strong>3. Find the power set of <\/strong>$M = \\left\\{\\text{Black, White}\\right\\}$<strong>. Determine the total number of elements.<\/strong><\/p>\n\n\n\n<p><strong>Solution:<\/strong>&nbsp;<\/p>\n\n\n\n<p>Set $M = \\left\\{\\text{Black, White}\\right\\}$<\/p>\n\n\n\n<p>Here $n(M) = 2$<\/p>\n\n\n\n<p>Therefore, applying the power set formula, we can calculate the following:<\/p>\n\n\n\n<p>$P(M) = 2^{2} = 4$<\/p>\n\n\n\n<p>Thus, we will have 4 subsets of set M, i.e.,<\/p>\n\n\n\n<p>Subsets of $M = \\left\\{\\right\\}, \\left\\{\\text{Black}\\right\\}, \\left\\{\\text{White}\\right\\}, \\left\\{\\text{Black, White}\\right\\}$<\/p>\n\n\n\n<p>Now, the power set of subsets in set A will be determined as<\/p>\n\n\n\n<p>$P(M) = \\left\\{ \\left\\{\\right\\}, \\left\\{\\text{Black}\\right\\}, \\left\\{\\text{White}\\right\\}, \\left\\{\\text{Black, White}\\right\\}\\right\\}$<\/p>\n\n\n\n<p><strong>4. How many subsets does a set with 6 elements have?<\/strong><\/p>\n\n\n\n<p><strong>Solution:<\/strong><\/p>\n\n\n\n<p>A set has 6 elements.<\/p>\n\n\n\n<p>Thus, its power set will have $2^{6} = 64$ elements.<\/p>\n\n\n\n<p>It means that a set with 6 elements has 64 subsets.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"10-practice-problems-on-power-set\">Practice Problems on Power Set<\/h2>\n\n\n\n<div class=\"spq_wrapper\"><h2 style=\"display:none;\">Power Set - Definition, Formula, Cardinality, Properties, Examples<\/h2><p style=\"display:none;\">Attend this quiz & Test your knowledge.<\/p><div class=\"spq_question_wrapper\" data-answer=\"2\"><span class=\"spq_question_header\"><span class=\"sqp_question_number\">1<\/span><h3 class=\"sqp_question_text\">Power set of a set with n elements has ____ elements.<\/h3><\/span><div class=\"spq_answer_block\" data-value=\"0\">2n<\/div><div class=\"spq_answer_block\" data-value=\"1\">$3^{n}$<\/div><div class=\"spq_answer_block\" data-value=\"2\">$2^{n}$<\/div><div class=\"spq_answer_block\" data-value=\"3\">$n^{2}$<\/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: $2^{n}$<br\/>If a set has n elements, then the power set has $2^{n}$  elements.<\/span><\/div><\/div><\/div><div class=\"spq_question_wrapper\" data-answer=\"2\"><span class=\"spq_question_header\"><span class=\"sqp_question_number\">2<\/span><h3 class=\"sqp_question_text\">How many subsets does the set $A = \\left\\{a,\\; e,\\; i,\\; o,\\; u\\right\\}$ have?<\/h3><\/span><div class=\"spq_answer_block\" data-value=\"0\">10<\/div><div class=\"spq_answer_block\" data-value=\"1\">16<\/div><div class=\"spq_answer_block\" data-value=\"2\">32<\/div><div class=\"spq_answer_block\" data-value=\"3\">5<\/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: 32<br\/>$| A | = 5$<br>\r\n$| P(A) | = 2^{5} = 32$<\/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\">A power set is a set of<\/h3><\/span><div class=\"spq_answer_block\" data-value=\"0\">supersets<\/div><div class=\"spq_answer_block\" data-value=\"1\">singleton subsets of a set<\/div><div class=\"spq_answer_block\" data-value=\"2\">empty sets<\/div><div class=\"spq_answer_block\" data-value=\"3\">all subsets of a set<\/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: all subsets of a set<br\/>A power set is the set of all subsets of a given set.<\/span><\/div><\/div><\/div><div class=\"spq_question_wrapper\" data-answer=\"0\"><span class=\"spq_question_header\"><span class=\"sqp_question_number\">4<\/span><h3 class=\"sqp_question_text\">What is the notation of a power set for set Q?<\/h3><\/span><div class=\"spq_answer_block\" data-value=\"0\">P(Q)<\/div><div class=\"spq_answer_block\" data-value=\"1\">P\u2019(Q)<\/div><div class=\"spq_answer_block\" data-value=\"2\">N(Q)<\/div><div class=\"spq_answer_block\" data-value=\"3\">$P\\left\\{Q\\right\\}$<\/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: P(Q)<br\/>P(Q) is the correct power set notation used to determine the power sets of a set.<\/span><\/div><\/div><\/div><div class=\"spq_question_wrapper\" data-answer=\"3\"><span class=\"spq_question_header\"><span class=\"sqp_question_number\">5<\/span><h3 class=\"sqp_question_text\">The power set of empty set can be given as<\/h3><\/span><div class=\"spq_answer_block\" data-value=\"0\">$\\left\\{ \u2205, \\{\u2205\\right\\} \\}$<\/div><div class=\"spq_answer_block\" data-value=\"1\">$\\left\\{ 0, \u2205 \\right\\}$<\/div><div class=\"spq_answer_block\" data-value=\"2\">$\\left\\{0\\right\\}$<\/div><div class=\"spq_answer_block\" data-value=\"3\">$\\left\\{\u2205\\right\\}$<\/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: $\\left\\{\u2205\\right\\}$<br\/>$P(\u2205) = \\left\\{\u2205\\right\\}$<br>\r\nIt is a set containing the null set.<\/span><\/div><\/div><\/div><\/div>  <script type=\"application\/ld+json\">{\n  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     \"position\": 1,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"$$3^{n}$$\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"If a set has n elements, then the power set has $$2^{n}$$  elements.\"\n                                    }\n                                }, {\n                                \"@type\": \"Answer\",\n                                \"position\": 3,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"$$n^{2}$$\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"If a set has n elements, then the power set has $$2^{n}$$  elements.\"\n                                    }\n                                }],\n       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\\\\left\\\\{a,\\\\; e,\\\\; i,\\\\; o,\\\\; u\\\\right\\\\}$$ have?\",\n                    \"text\": \"How many subsets does the set $$A = \\\\left\\\\{a,\\\\; e,\\\\; i,\\\\; o,\\\\; u\\\\right\\\\}$$ have?\",\n                    \"comment\": {\n                      \"@type\": \"Comment\",\n                      \"text\": \"$$| A | = 5$$<br>\r\n$$| P(A) | = 2^{5} = 32$$\"\n                    },\n                    \"encodingFormat\": \"text\/html\",\n                    \"suggestedAnswer\": [ {\n                                \"@type\": \"Answer\",\n                                \"position\": 0,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"10\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"$$| A | = 5$$<br>\r\n$$| P(A) | = 2^{5} = 32$$\"\n                                    }\n                                }, {\n                                \"@type\": \"Answer\",\n                                \"position\": 1,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"16\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"$$| A | = 5$$<br>\r\n$$| P(A) | = 2^{5} = 32$$\"\n                                    }\n                                }, {\n                                \"@type\": \"Answer\",\n                                \"position\": 3,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"5\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"$$| A | = 5$$<br>\r\n$$| P(A) | = 2^{5} = 32$$\"\n                                    }\n                                }],\n                    \"acceptedAnswer\": {\n                      \"@type\": \"Answer\",\n                      \"position\": 2,\n                      \"encodingFormat\": \"text\/html\",\n                      \"text\": \"32\",\n                      \"comment\": {\n                          \"@type\": \"Comment\",\n                          \"text\": \"$$| A | = 5$$<br>\r\n$$| P(A) | = 2^{5} = 32$$\"\n                        },\n                      \"answerExplanation\": {\n                        \"@type\": \"Comment\",\n                        \"text\": \"$$| A | = 5$$<br>\r\n$$| P(A) | = 2^{5} = 32$$\"\n                      }\n                    } \n\n                    },{\n                    \"@type\": \"Question\",   \n                    \"eduQuestionType\": \"Multiple choice\",\n                    \"learningResourceType\": \"Practice problem\",\n                    \"name\": \"A power set is a set of\",\n                    \"text\": \"A power set is a set of\",\n                    \"comment\": {\n                      \"@type\": \"Comment\",\n                      \"text\": \"A power set is the set of all subsets of a given set.\"\n                    },\n                    \"encodingFormat\": \"text\/html\",\n                    \"suggestedAnswer\": [ {\n                                \"@type\": \"Answer\",\n                                \"position\": 0,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"supersets\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"A power set is the set of all subsets of a given set.\"\n                                    }\n                                }, {\n                                \"@type\": \"Answer\",\n                                \"position\": 1,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"singleton subsets of a set\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"A power set is the set of all subsets of a given set.\"\n                                    }\n                                }, {\n                                \"@type\": \"Answer\",\n                                \"position\": 2,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"empty sets\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"A power set is the set of all subsets of a given set.\"\n                                    }\n                                }],\n                    \"acceptedAnswer\": {\n                      \"@type\": \"Answer\",\n                      \"position\": 3,\n                      \"encodingFormat\": \"text\/html\",\n                      \"text\": \"all subsets of a set\",\n                      \"comment\": {\n                          \"@type\": \"Comment\",\n                          \"text\": \"A power set is the set of all subsets of a given set.\"\n                        },\n                      \"answerExplanation\": {\n                        \"@type\": \"Comment\",\n                        \"text\": \"A power set is the set of all subsets of a given set.\"\n                      }\n                    } \n\n                    },{\n                    \"@type\": \"Question\",   \n                    \"eduQuestionType\": \"Multiple choice\",\n                    \"learningResourceType\": \"Practice problem\",\n                    \"name\": \"What is the notation of a power set for set Q?\",\n                    \"text\": \"What is the notation of a power set for set Q?\",\n                    \"comment\": {\n                      \"@type\": \"Comment\",\n                      \"text\": \"P(Q) is the correct power set notation used to determine the power sets of a set.\"\n                    },\n                    \"encodingFormat\": \"text\/html\",\n                    \"suggestedAnswer\": [ {\n                                \"@type\": \"Answer\",\n                                \"position\": 1,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"P\u2019(Q)\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"P(Q) is the correct power set notation used to determine the power sets of a set.\"\n                                    }\n                                }, {\n                                \"@type\": \"Answer\",\n                                \"position\": 2,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"N(Q)\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"P(Q) is the correct power set notation used to determine the power sets of a set.\"\n                                    }\n                                }, {\n                                \"@type\": \"Answer\",\n                                \"position\": 3,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"$$P\\\\left\\\\{Q\\\\right\\\\}$$\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"P(Q) is the correct power set notation used to determine the power sets of a set.\"\n                                    }\n                                }],\n                    \"acceptedAnswer\": {\n                      \"@type\": \"Answer\",\n                      \"position\": 0,\n                      \"encodingFormat\": \"text\/html\",\n                      \"text\": \"P(Q)\",\n                      \"comment\": {\n                          \"@type\": \"Comment\",\n                          \"text\": \"P(Q) is the correct power set notation used to determine the power sets of a set.\"\n                        },\n                      \"answerExplanation\": {\n                        \"@type\": \"Comment\",\n                        \"text\": \"P(Q) is the correct power set notation used to determine the power sets of a set.\"\n                      }\n                    } \n\n                    },{\n                    \"@type\": \"Question\",   \n                    \"eduQuestionType\": \"Multiple choice\",\n                    \"learningResourceType\": \"Practice problem\",\n                    \"name\": \"The power set of empty set can be given as\",\n                    \"text\": \"The power set of empty set can be given as\",\n                    \"comment\": {\n                      \"@type\": \"Comment\",\n                      \"text\": \"$$P(\u2205) = \\\\left\\\\{\u2205\\\\right\\\\}$$<br>\r\nIt is a set containing the null set.\"\n                    },\n                    \"encodingFormat\": \"text\/html\",\n                    \"suggestedAnswer\": [ {\n                                \"@type\": \"Answer\",\n                                \"position\": 0,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"$$\\\\left\\\\{ \u2205, \\\\{\u2205\\\\right\\\\} \\\\}$$\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"$$P(\u2205) = \\\\left\\\\{\u2205\\\\right\\\\}$$<br>\r\nIt is a set containing the null set.\"\n                                    }\n                                }, {\n                                \"@type\": \"Answer\",\n                                \"position\": 1,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"$$\\\\left\\\\{ 0, \u2205 \\\\right\\\\}$$\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"$$P(\u2205) = \\\\left\\\\{\u2205\\\\right\\\\}$$<br>\r\nIt is a set containing the null set.\"\n                                    }\n                                }, {\n                                \"@type\": \"Answer\",\n                                \"position\": 2,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"$$\\\\left\\\\{0\\\\right\\\\}$$\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"$$P(\u2205) = \\\\left\\\\{\u2205\\\\right\\\\}$$<br>\r\nIt is a set containing the null set.\"\n                                    }\n                                }],\n                    \"acceptedAnswer\": {\n                      \"@type\": \"Answer\",\n                      \"position\": 3,\n                      \"encodingFormat\": \"text\/html\",\n                      \"text\": \"$$\\\\left\\\\{\u2205\\\\right\\\\}$$\",\n                      \"comment\": {\n                          \"@type\": \"Comment\",\n                          \"text\": \"$$P(\u2205) = \\\\left\\\\{\u2205\\\\right\\\\}$$<br>\r\nIt is a set containing the null set.\"\n                        },\n                      \"answerExplanation\": {\n                        \"@type\": \"Comment\",\n                        \"text\": \"$$P(\u2205) = \\\\left\\\\{\u2205\\\\right\\\\}$$<br>\r\nIt is a set containing the null set.\"\n                      }\n                    } \n\n                    }]}<\/script>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"11-frequently-asked-questions-about-power-set\">Frequently Asked Questions about Power Set<\/h2>\n\n\n<div class=\"wp-block-ub-content-toggle\" id=\"ub-content-toggle-ce684a9d-9296-423b-b4eb-4505ec00612d\" 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-ce684a9d-9296-423b-b4eb-4505ec00612d\" tabindex=\"0\">\n                    <p class=\"wp-block-ub-content-toggle-accordion-title ub-content-toggle-title-ce684a9d-9296-423b-b4eb-4505ec00612d\"><strong>What is a power set of the 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-0-ce684a9d-9296-423b-b4eb-4505ec00612d\">\n\n<p>The <strong>power set of the empty set <\/strong>is the empty set itself. It has only one element. It can be written as:&nbsp;$P\\left(\u2205\\right) = \\left\\{ \u2205 \\right\\} = \\left\\{ \\left\\{\\right\\} \\right\\}$<\/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-ce684a9d-9296-423b-b4eb-4505ec00612d\" tabindex=\"0\">\n                    <p class=\"wp-block-ub-content-toggle-accordion-title ub-content-toggle-title-ce684a9d-9296-423b-b4eb-4505ec00612d\"><strong>Is the null set a proper subset of every 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-ce684a9d-9296-423b-b4eb-4505ec00612d\">\n\n<p>Yes, the null set or \u2205 is a proper subset of every set (except the null 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-ce684a9d-9296-423b-b4eb-4505ec00612d\" tabindex=\"0\">\n                    <p class=\"wp-block-ub-content-toggle-accordion-title ub-content-toggle-title-ce684a9d-9296-423b-b4eb-4505ec00612d\"><strong>How many proper subsets can be found in a power 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-2-ce684a9d-9296-423b-b4eb-4505ec00612d\">\n\n<p>If there are n elements in a set, then the set has $2^{n}$ subsets and $2^{n} \\;\u2212\\; 1$ proper subsets.<\/p>\n\n<\/div><\/div>\n<\/div>","protected":false},"excerpt":{"rendered":"<p>What Is a Power Set in Math? The power set in set theory is a set of all subsets of a given set. Thus, the empty set and the set itself are always included in the power set. We denote the power set of set A as P(A) or \u2118(A). To define the power set &#8230; <a title=\"Power Set &#8211; Definition, Formula, Cardinality, Properties, Examples\" class=\"read-more\" href=\"https:\/\/www.splashlearn.com\/math-vocabulary\/power-set\" aria-label=\"More on Power Set &#8211; Definition, Formula, Cardinality, Properties, Examples\">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-32484","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\/32484","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=32484"}],"version-history":[{"count":39,"href":"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-json\/wp\/v2\/posts\/32484\/revisions"}],"predecessor-version":[{"id":32641,"href":"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-json\/wp\/v2\/posts\/32484\/revisions\/32641"}],"wp:attachment":[{"href":"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-json\/wp\/v2\/media?parent=32484"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-json\/wp\/v2\/categories?post=32484"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-json\/wp\/v2\/tags?post=32484"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}