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Example 3 Let L be the set of all lines in a plane and R be the relation in L defined as R = {(L1, L2) : L1 is perpendicular to L2}. Show that R is symmetric but neither reflexive nor transitive. R = {(L1, L2) : L1 is perpendicular to L2} Check reflexive If R is reflexive, then (L, L) ∈ R Line L cannot be perpendicular to itself So, line L is not perpendicular to line L So, (L, L) ∉ R. ∴ R is not reflexive Check symmetric If L1 is perpendicular to L2 , then L2 is perpendicular to L1 So, if (L1, L2) ∈ R , then (L2, L1) ∈ R. ∴ R is symmetric Check transitive If L1 is perpendicular to L2 & L2 is perpendicular to L3 , then L1 is not perpendicular to L3 , it is parallel to L3 So, if (L1, L2) ∈ R, (L2, L3) ∈ R then , (L1, L3) ∉ R. ∴ R is not transitive

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Davneet Singh

Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 13 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.