{"id":9528,"date":"2022-01-16T01:30:28","date_gmt":"2022-01-15T20:00:28","guid":{"rendered":"https:\/\/mathemerize.com\/?p=9528"},"modified":"2022-01-16T17:16:30","modified_gmt":"2022-01-16T11:46:30","slug":"what-is-power-set-definition-formula-examples","status":"publish","type":"post","link":"https:\/\/mathemerize.com\/what-is-power-set-definition-formula-examples\/","title":{"rendered":"What is Power Set – Definition, Formula and Examples"},"content":{"rendered":"

Here you will learn what is power set and its definition with examples.<\/p>\n

Let’s begin –<\/p>\n

What is Power Set ?<\/h2>\n

Definition<\/strong> : Let A be a set. Then the collection or family of all subsets of A is called the power set of A and is denoted by P(A).<\/p>\n

That is,  P(A) = { S : S \\(\\subset\\) A}<\/p>\n

Since the empty set and the set A itself are subsets of A and are therefore elements of P(A). Thus, the power set of a given set is always non-empty.<\/p>\n

Power Set of Empty Set<\/strong><\/h4>\n

If A is the void set \\(\\phi\\), then P(A) has just one element \\(\\phi\\) i.e \\(P(\\phi)\\) = {\\(\\phi\\)}<\/p>\n

Example 1<\/strong><\/span> : Let A = {1, 2. 3}. Then, the subsets of A are :<\/p>\n

\\(\\phi\\), {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3} and {1, 2, 3}<\/p>\n

Hence, P(A) = { \\(\\phi\\), {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3} and {1, 2, 3} }<\/p>\n

Example 2<\/strong><\/span> : Show that n{P[P.(P(\\(\\phi\\)))]} = 4.<\/p>\n

Solution<\/strong><\/span> : We have,  \\(P(\\phi)\\) = {\\(\\phi\\)}<\/p>\n

\\(\\therefore\\)  \\(P(P(\\phi))\\) = {\\(\\phi\\), {\\(\\phi\\)}}<\/p>\n

\\(\\implies\\)  \\(P[P.(P(\\phi))]\\) = {\\(\\phi\\), {\\(\\phi\\)}, {{\\(\\phi\\)}}, {\\(\\phi\\). {\\(\\phi\\)}}}<\/p>\n

Hence, \\(P[P.(P(\\phi))]\\)  consists of 4 elements i.e  n{P[P.(P(\\(\\phi\\)))]} = 4<\/p>\n

We know that a set having n elements has \\(2^n\\) subsets<\/a>. Therefore, if A is a finite set having n elements, then P(A) has \\(2^n\\) elements.<\/p>\n

Example 3<\/strong><\/span> : If A = {a, {b}}, find P(A).<\/p>\n

Solution<\/strong><\/span> : Let B = {b}. Then, A = {a, B}.<\/p>\n

\\(\\therefore\\)  P(A) = {\\(\\phi\\), {a}, {B}, {a, B}}<\/p>\n

P(A) = {\\(\\phi\\), {a}, {{b}}, {a, {b}}}.<\/p>\n","protected":false},"excerpt":{"rendered":"

Here you will learn what is power set and its definition with examples. Let’s begin – What is Power Set ? Definition : Let A be a set. Then the collection or family of all subsets of A is called the power set of A and is denoted by P(A). That is,  P(A) = { …<\/p>\n

What is Power Set – Definition, Formula and Examples<\/span> Read More »<\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"closed","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":[16],"tags":[755,757,756,758],"yoast_head":"\nWhat is Power Set - Definition, Formula and Examples<\/title>\n<meta name=\"description\" content=\"In this post you will learn what is power set, its definition and formula and examples based on power set.\" \/>\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\/what-is-power-set-definition-formula-examples\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"What is Power Set - 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