{"id":9522,"date":"2022-01-15T23:41:49","date_gmt":"2022-01-15T18:11:49","guid":{"rendered":"https:\/\/mathemerize.com\/?p=9522"},"modified":"2022-01-15T23:41:52","modified_gmt":"2022-01-15T18:11:52","slug":"prove-that-the-total-number-of-subsets-of-a-finite-set-containing-n-elements-is-2n","status":"publish","type":"post","link":"https:\/\/mathemerize.com\/prove-that-the-total-number-of-subsets-of-a-finite-set-containing-n-elements-is-2n\/","title":{"rendered":"Prove that the total number of subsets of a finite set containing n elements is \\(2^n\\)."},"content":{"rendered":"<h2>Solution :<\/h2>\n<p>Let A be a finite set containing n elements. Let 0 \\(\\le\\) r \\(\\le\\) n.<\/p>\n<p>Consider those subsets of A that have r elements each. We know that the number of ways in which r elements can be chosen out of n elements is \\(^nC_r\\).<\/p>\n<p>Therefore, the number of subsets of A having r elements each is \\(^nC_r\\).<\/p>\n<p>Hence, the total number of subsets of A<\/p>\n<p>= \\(^nC_0\\) + \\(^nC_1\\) + \\(^nC_2\\) + &#8230;. + \\(^nC_n\\) = \\((1 + 1)^n\\) = \\(2^n\\).<\/p>\n<p>[ Using binomial theorem ]<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Solution : Let A be a finite set containing n elements. Let 0 \\(\\le\\) r \\(\\le\\) n. Consider those subsets of A that have r elements each. We know that the number of ways in which r elements can be chosen out of n elements is \\(^nC_r\\). Therefore, the number of subsets of A having &hellip;<\/p>\n<p class=\"read-more\"> <a class=\"\" href=\"https:\/\/mathemerize.com\/prove-that-the-total-number-of-subsets-of-a-finite-set-containing-n-elements-is-2n\/\"> <span class=\"screen-reader-text\">Prove that the total number of subsets of a finite set containing n elements is \\(2^n\\).<\/span> Read More &raquo;<\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"default","ast-global-header-display":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":""},"categories":[43,56],"tags":[],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v20.12 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>Prove that the total number of subsets of a finite set containing n elements is \\(2^n\\).<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/mathemerize.com\/prove-that-the-total-number-of-subsets-of-a-finite-set-containing-n-elements-is-2n\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Prove that the total number of subsets of a finite set containing n elements is \\(2^n\\).\" \/>\n<meta property=\"og:description\" content=\"Solution : Let A be a finite set containing n elements. Let 0 (le) r (le) n. Consider those subsets of A that have r elements each. We know that the number of ways in which r elements can be chosen out of n elements is (^nC_r). Therefore, the number of subsets of A having &hellip; Prove that the total number of subsets of a finite set containing n elements is (2^n). Read More &raquo;\" \/>\n<meta property=\"og:url\" content=\"https:\/\/mathemerize.com\/prove-that-the-total-number-of-subsets-of-a-finite-set-containing-n-elements-is-2n\/\" \/>\n<meta property=\"og:site_name\" content=\"Mathemerize\" \/>\n<meta property=\"article:published_time\" content=\"2022-01-15T18:11:49+00:00\" \/>\n<meta property=\"article:modified_time\" content=\"2022-01-15T18:11:52+00:00\" \/>\n<meta name=\"author\" content=\"mathemerize\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:label1\" content=\"Written by\" \/>\n\t<meta name=\"twitter:data1\" content=\"mathemerize\" \/>\n\t<meta name=\"twitter:label2\" content=\"Est. reading time\" \/>\n\t<meta name=\"twitter:data2\" content=\"1 minute\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\/\/schema.org\",\"@graph\":[{\"@type\":\"Article\",\"@id\":\"https:\/\/mathemerize.com\/prove-that-the-total-number-of-subsets-of-a-finite-set-containing-n-elements-is-2n\/#article\",\"isPartOf\":{\"@id\":\"https:\/\/mathemerize.com\/prove-that-the-total-number-of-subsets-of-a-finite-set-containing-n-elements-is-2n\/\"},\"author\":{\"name\":\"mathemerize\",\"@id\":\"https:\/\/mathemerize.com\/#\/schema\/person\/104c8bc54f90618130a6665299bc55df\"},\"headline\":\"Prove that the total number of subsets of a finite set containing n elements is \\\\(2^n\\\\).\",\"datePublished\":\"2022-01-15T18:11:49+00:00\",\"dateModified\":\"2022-01-15T18:11:52+00:00\",\"mainEntityOfPage\":{\"@id\":\"https:\/\/mathemerize.com\/prove-that-the-total-number-of-subsets-of-a-finite-set-containing-n-elements-is-2n\/\"},\"wordCount\":94,\"commentCount\":0,\"publisher\":{\"@id\":\"https:\/\/mathemerize.com\/#organization\"},\"articleSection\":[\"Maths Questions\",\"Sets Questions\"],\"inLanguage\":\"en-US\",\"potentialAction\":[{\"@type\":\"CommentAction\",\"name\":\"Comment\",\"target\":[\"https:\/\/mathemerize.com\/prove-that-the-total-number-of-subsets-of-a-finite-set-containing-n-elements-is-2n\/#respond\"]}]},{\"@type\":\"WebPage\",\"@id\":\"https:\/\/mathemerize.com\/prove-that-the-total-number-of-subsets-of-a-finite-set-containing-n-elements-is-2n\/\",\"url\":\"https:\/\/mathemerize.com\/prove-that-the-total-number-of-subsets-of-a-finite-set-containing-n-elements-is-2n\/\",\"name\":\"Prove that the total number of subsets of a finite set containing n elements is \\\\(2^n\\\\).\",\"isPartOf\":{\"@id\":\"https:\/\/mathemerize.com\/#website\"},\"datePublished\":\"2022-01-15T18:11:49+00:00\",\"dateModified\":\"2022-01-15T18:11:52+00:00\",\"breadcrumb\":{\"@id\":\"https:\/\/mathemerize.com\/prove-that-the-total-number-of-subsets-of-a-finite-set-containing-n-elements-is-2n\/#breadcrumb\"},\"inLanguage\":\"en-US\",\"potentialAction\":[{\"@type\":\"ReadAction\",\"target\":[\"https:\/\/mathemerize.com\/prove-that-the-total-number-of-subsets-of-a-finite-set-containing-n-elements-is-2n\/\"]}]},{\"@type\":\"BreadcrumbList\",\"@id\":\"https:\/\/mathemerize.com\/prove-that-the-total-number-of-subsets-of-a-finite-set-containing-n-elements-is-2n\/#breadcrumb\",\"itemListElement\":[{\"@type\":\"ListItem\",\"position\":1,\"name\":\"Home\",\"item\":\"https:\/\/mathemerize.com\/\"},{\"@type\":\"ListItem\",\"position\":2,\"name\":\"Prove that the total number of subsets of a finite set containing n elements is \\\\(2^n\\\\).\"}]},{\"@type\":\"WebSite\",\"@id\":\"https:\/\/mathemerize.com\/#website\",\"url\":\"https:\/\/mathemerize.com\/\",\"name\":\"Mathemerize\",\"description\":\"Maths Tutorials - 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Let 0 (le) r (le) n. Consider those subsets of A that have r elements each. We know that the number of ways in which r elements can be chosen out of n elements is (^nC_r). Therefore, the number of subsets of A having &hellip; Prove that the total number of subsets of a finite set containing n elements is (2^n). 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