{"id":3727,"date":"2021-08-07T21:17:04","date_gmt":"2021-08-07T21:17:04","guid":{"rendered":"https:\/\/mathemerize.com\/?p=3727"},"modified":"2021-11-28T02:08:03","modified_gmt":"2021-11-27T20:38:03","slug":"operation-of-sets","status":"publish","type":"post","link":"https:\/\/mathemerize.com\/operation-of-sets\/","title":{"rendered":"Operation of Sets – Formula of Sets in Math"},"content":{"rendered":"

Here, you will learn operation of sets i.e intersection and union of two sets with examples.<\/p>\n

Let’s begin –<\/p>\n

Basic Operation of Sets – Intersection and Union of Two Sets<\/h2>\n

(i) Union of two sets :<\/strong><\/p>\n

A \\(\\cup\\) B = {x : x \\(\\in\\) A or x \\(\\in\\) B}<\/p>\n

Example<\/span> : A = {1, 2, 3} B = {2, 3, 4} then A \\(\\cup\\) B ={1, 2, 3, 4}<\/p>\n

(ii) Intersection of two sets :<\/strong><\/p>\n

A \\(\\cap\\) B = {x : x \\(\\in\\) A and x \\(\\in\\) B}<\/p>\n

Example<\/span> : A = {1, 2, 3} B = {2, 3, 4} then A \\(\\cap\\) B ={2, 3}<\/p>\n

(iii) Difference of two sets :<\/strong><\/p>\n

A – B = {x : x \\(\\in\\) A and x \\(\\notin\\) B}<\/p>\n

(iv) Complement of a set :<\/strong><\/p>\n

A’ = {x : x \\(\\notin\\) A but x \\(\\in\\) U} = U – A<\/p>\n

(v) De-Morgan Laws :<\/strong><\/p>\n

(A \\(\\cup\\) B)’ = A’ \\(\\cap\\) B’ ; (A \\(\\cap\\) B)’ = A’ \\(\\cup\\) B’<\/p>\n

(vi) Distributive Laws :<\/strong><\/p>\n

A \\(\\cup\\) (B \\(\\cap\\) C) = (A \\(\\cup\\) B)\\(\\cap\\) (A \\(\\cup\\) C) ; A \\(\\cap\\) (B \\(\\cup\\) C) = (A \\(\\cap\\) B)\\(\\cup\\) (A \\(\\cap\\) C)<\/p>\n

(vii) Commutative Laws :<\/strong><\/p>\n

A \\(\\cup\\) B = B \\(\\cup\\) A ; A \\(\\cap\\) B = B \\(\\cap\\) A<\/p>\n

(viii) Associative Laws :<\/strong><\/p>\n

(A \\(\\cup\\) B) \\(\\cup\\) C = A \\(\\cup\\) (B \\(\\cup\\) C) ; (A \\(\\cap\\) B) \\(\\cap\\) C = A \\(\\cap\\) (B \\(\\cap\\) C)<\/p>\n

Disjoint Sets<\/h2>\n

If A \\(\\cap\\) B = \\(\\phi\\), then A, B are disjoint.<\/p>\n

e.g. if A = {1, 2, 3}, B = {7, 8, 9} then A \\(\\cap\\) B = \\(\\phi\\)<\/p>\n

A \\(\\cap\\) A’ = \\(\\phi\\) ]    \\(\\therefore\\)   A, A’ are disjoint.<\/p>\n

Formula of Sets<\/h2>\n

If A, B and C are finite sets, and U be the finite universal set, then<\/p>\n

(i) n(A \\(\\cup\\) B) = n(A) + n(B) – n(A \\(\\cap\\) B)<\/p>\n

(ii) n(A \\(\\cup\\) B) = n(A) + n(B) \\(\\implies\\) A, B are disjoint non-void sets.<\/p>\n

(iii) n(A – B) = n(A) – n(A \\(\\cap\\) B)<\/p>\n

(iv) n(A \\(\\triangle\\) B) = No. of elements which belong to exactly one of A or B<\/p>\n

= n((A – B) \\(\\cup\\) (B – A))<\/p>\n

n(A \\(\\triangle\\) B) = n(A – B) + n(B – A)<\/p>\n

= n(A) – n(A \\(\\cap\\) B) + n(B) – n(A \\(\\cap\\) B)<\/p>\n

n(A \\(\\triangle\\) B)  = n(A) + n(B) – 2n(A \\(\\cap\\) B)<\/p>\n

(v) \\(n(A \\cup B \\cup C)\\) = n(A) + n(B) – n(A \\(\\cap\\) B) – n(B \\(\\cap\\) C) – n(A \\(\\cap\\) C) + \\(n(A \\cap B \\cap C)\\)<\/p>\n

(vi) Number of elements in exactly two of the sets A, B, C<\/p>\n

= n(A \\(\\cap\\) B) + n(B \\(\\cap\\) C) + n(C \\(\\cap\\) A) – \\(3n(A \\cap B \\cap C)\\)<\/p>\n

(vii) Number of elements in exactly one of the sets A, B, C<\/p>\n

= n(A) + n(B) + n(C) – 2n(A \\(\\cap\\) B) – 2n(B \\(\\cap\\) C) – 2n(A \\(\\cap\\) C) + \\(3n(A \\cap B \\cap C)\\)<\/p>\n

(viii) n(A’ \\(\\cup\\) B’) = n((A \\(\\cap\\) B)’) = n(U) – n(A \\(\\cap\\) B)<\/p>\n

(ix) n(A’ \\(\\cap\\) B’) = n((A \\(\\cup\\) B)’) = n(U) – n(A \\(\\cup\\) B)<\/p>\n\n\n

Example : <\/span>In a group of 1000 people, there are 750 who can speak Hindi and 400 who can speak Bengali. How many can speak Hindi only? and How many can speak Bengali only? How many can speak both Hindi and Bengali?<\/p>\n

Solution : <\/span>Let A and B be two sets of person who can speak Hindi and Bengali respectively.

\n\t\t then n(A \\(\\cup\\) B) = 1000, n(A) = 750, n(B) = 400

\n\t\t Number of person who can speak both Hindi and Bengali

\n\t\t = n(A \\(\\cap\\) B) = n(A) + n(B) – n(A \\(\\cup\\) B)

\n\t\t = 750 + 400 – 1000 = 150

\n\t\t and Number of persons who can speak hindi only

\n\t\t = n(A – B) = n(A) – n(A \\(\\cap\\) B) = 750 – 150 = 600

\n\t\t Number of persons who can speak Bengali only

\n\t\t = n(B – A) = n(B) – n(A \\(\\cap\\) B) = 400 – 150 = 250<\/p>\n\n\n

Hope you learn operation of sets i.e. intersection and union of sets and formula of sets.<\/p>\n\n\n

\n
Previous – Types of Sets in Mathematics<\/a><\/div>\n<\/div>\n\n\n\n

<\/p>\n","protected":false},"excerpt":{"rendered":"

Here, you will learn operation of sets i.e intersection and union of two sets with examples. Let’s begin – Basic Operation of Sets – Intersection and Union of Two Sets (i) Union of two sets : A \\(\\cup\\) B = {x : x \\(\\in\\) A or x \\(\\in\\) B} Example : A = {1, 2, …<\/p>\n

Operation of Sets – Formula of Sets in Math<\/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":"default","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":""},"categories":[16],"tags":[590,591,592,593,589,587,588],"yoast_head":"\nOperation of Sets - Formula of Sets in Math<\/title>\n<meta name=\"description\" content=\"In this post, you will learn operation of sets i.e intersection and union of two sets and formula of sets in math with examples.\" \/>\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\/operation-of-sets\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Operation of Sets - 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