{"id":9682,"date":"2022-01-22T01:57:11","date_gmt":"2022-01-21T20:27:11","guid":{"rendered":"https:\/\/mathemerize.com\/?p=9682"},"modified":"2022-01-22T01:57:14","modified_gmt":"2022-01-21T20:27:14","slug":"what-is-transitive-relation-definition-and-examples","status":"publish","type":"post","link":"https:\/\/mathemerize.com\/what-is-transitive-relation-definition-and-examples\/","title":{"rendered":"What is Transitive Relation – Definition and Examples"},"content":{"rendered":"

Here you will learn what is transitive relation on set with definition and examples based on it.<\/p>\n

Let’s begin –<\/p>\n

What is Transitive Relation ?<\/h2>\n

Definition<\/strong> : Let A be any set. A relation R on A is said to be a transitive relation iff<\/p>\n

(a, b) \\(\\in\\) R and  (b, c) \\(\\in\\) R  \\(\\implies\\)  (a, c) \\(\\in\\) R for all a, b, c  \\(\\in\\) A.<\/p>\n

i.e. a R b  and  b R c  \\(\\implies\\)  a R c for all a, b, c  \\(\\in\\) A.<\/p><\/blockquote>\n

Note<\/strong> : The identity and the universal relations on a void set are transitive.<\/p>\n

Also Read<\/strong> : Types of Relations in Math<\/a><\/p>\n

Given below are some transitive relation examples.<\/p>\n

Example<\/span><\/strong> : If A = {1,2,3}, then the relation R = {(1,2),(2,3),(1,3),(2,2)} is the transitive relation on A, But \\(R_1\\) = {(1,2),(2,3),(2,2),(1,1)}  is not a transitive relation on A, because (1,3) and (2,3) \\(\\in\\) \\(R_1\\) but (1,3) \\(\\notin\\) \\(R_1\\).<\/p>\n

Example<\/strong><\/span> : The relation R on the set N of all natural numbers defined by<\/p>\n

(x, y)  \\(\\in\\)  R  \\(\\iff\\)  x divides y, for all x, y \\(\\in\\) N is transitive.<\/p>\n

Solution<\/span><\/strong> : Let x, y, z \\(\\in\\) N be such that (x, y) \\(\\in\\) R and (y, z) \\(\\in\\) R. Then,<\/p>\n

(x, y) \\(\\in\\) R and (y, z) \\(\\in\\) R<\/p>\n

\\(\\implies\\) x divides y and, y divides z.<\/p>\n

\\(\\implies\\) There exist p, q \\(\\in\\) N such that y = xp and z = yq<\/p>\n

\\(\\implies\\) z = (xp) q<\/p>\n

\\(\\implies\\) z = x (pq)<\/p>\n

\\(\\implies\\) x divides z<\/p>\n

(\\implies\\) (x, z) \\(\\in\\) R<\/p>\n

Thus, (x, y) \\(\\in\\) R, (y, z) \\(\\in\\) R \\(\\implies\\) (x, z) \\(\\in\\) R for all x, y, z \\(\\in\\) N.<\/p>\n

Hence, R is a transitive relation on N.<\/p>\n","protected":false},"excerpt":{"rendered":"

Here you will learn what is transitive relation on set with definition and examples based on it. Let’s begin – What is Transitive Relation ? Definition : Let A be any set. A relation R on A is said to be a transitive relation iff (a, b) \\(\\in\\) R and  (b, c) \\(\\in\\) R  \\(\\implies\\)  …<\/p>\n

What is Transitive Relation – Definition and Examples<\/span> Read More »<\/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":[17],"tags":[581,780,779],"yoast_head":"\nWhat is Transitive Relation - Definition and Examples<\/title>\n<meta name=\"description\" content=\"In this post you will learn what is transitive relation on set with definition and examples based on it.\" \/>\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-transitive-relation-definition-and-examples\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"What is Transitive Relation - 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