{"id":9686,"date":"2022-01-22T01:57:26","date_gmt":"2022-01-21T20:27:26","guid":{"rendered":"https:\/\/mathemerize.com\/?p=9686"},"modified":"2022-01-22T01:57:30","modified_gmt":"2022-01-21T20:27:30","slug":"what-is-equivalence-relation-definition-and-examples","status":"publish","type":"post","link":"https:\/\/mathemerize.com\/what-is-equivalence-relation-definition-and-examples\/","title":{"rendered":"What is Equivalence Relation – Definition and Examples"},"content":{"rendered":"

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

Let’s begin –<\/p>\n

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

Definition<\/strong> : A relation R on a set A is said to be an equivalence relation on A iff it is<\/p>\n

(i) reflexive i.e. (a, a) \\(\\in\\) R for all a \\(\\in\\) A.<\/p>\n

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

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

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

Example<\/strong><\/span> : If A = {1,2,3}, then the relation R = {(1,1),(2,2),(3,3),(2,1),(1,2),(2,3),(1,3).(3,2),(3,1)} is the equivalence relation on A,<\/p>\n

because {(1,1),(2,2),(3,3)} \\(\\in\\) R hence it is reflexive,<\/p>\n

{(2,1),(1,2),(2,3),(3,2),(1,3),(3,1)} \\(\\in\\) R  hence it is symmetric on A,<\/p>\n

{(1,2),(2,3),(1,3)} and {(1,3),(3,2),(1,2)} \\(\\in\\) R hence it is transitive.<\/p>\n

Example<\/strong><\/span> : Let R be a relation on the set of all lines in a plane defined by \\((l_1, l_2)\\) \\(\\in\\) R \\(\\iff\\) line \\(l_1\\) is parallel to line \\(l_2\\). Show that R is an equivalence relation.<\/p>\n

Solution<\/span><\/strong> : Let L be the given set of all lines in a plane. Then, we observe the following properties.<\/p>\n

Reflexive<\/strong> : For each line l \\(\\in\\) L, we have<\/p>\n

l || l \\(\\implies\\) (l, l) \\(\\in\\) R for all l \\(\\in\\) L<\/p>\n

\\(\\implies\\) R is reflexive.<\/p>\n

Symmetric<\/strong> : Let \\(l_1\\), \\(l_2\\) \\(\\in\\) L such that \\((l_1, l_2)\\) \\(\\in\\) R. Then,<\/p>\n

\\((l_1, l_2)\\) \\(\\in\\) R \\(\\implies\\) \\((l_1\\) || \\(l_2)\\) \\(\\implies\\) \\((l_2\\) || \\(l_1)\\) \\(\\in\\) R.<\/p>\n

So, R is symmetric on L.<\/p>\n

Transitive<\/strong> : Let \\(l_1\\), \\(l_2\\), \\(l_3\\) \\(\\in\\) L such that \\((l_1, l_2)\\) \\(\\in\\) R and \\((l_2, l_3)\\)  \\(\\in\\)  R. Then,<\/p>\n

\\((l_1, l_2)\\) \\(\\in\\) R and \\((l_2, l_3)\\) \\(\\in\\) R \\(\\implies\\) \\(l_1\\) || \\(l_2\\) and \\(l_2\\) || \\(l_3\\) \\(\\implies\\) \\(l_1\\) || \\(l_3\\) \\(\\implies\\) \\((l_1, l_3)\\) \\(\\in\\) R.<\/p>\n

So, R is transitive on L.<\/p>\n

Hence, R being reflexive, symmetric and transitive is an equivalence relation on L.<\/p>\n","protected":false},"excerpt":{"rendered":"

Here you will learn what is equivalence relation on a set with definition and examples. Let’s begin – What is Equivalence Relation ? Definition : A relation R on a set A is said to be an equivalence relation on A iff it is (i) reflexive i.e. (a, a) \\(\\in\\) R for all a \\(\\in\\) …<\/p>\n

What is Equivalence 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":[582,783],"yoast_head":"\nWhat is Equivalence Relation - Definition and Examples<\/title>\n<meta name=\"description\" content=\"In this post you will learn what is equivalence relation on a 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-equivalence-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 Equivalence Relation - 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