InterviewSolution
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InterviewSolution
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Give an example of a relation which is (i) Reflexive and transitive but not symmetric, (ii) symmetric and transitive but not symmetric, (iii) reflexive and symmetric nor transitive, (iv) symmetric but neither reflexive nor transitive, (v) transitive but neither reflexive nor symmetric. |
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Answer» Let `A={1,2,3}.` then it is easy to verify that the relation (i) `R_(1)={(1,1),(2,2),(3,3) ,(1,2)}` is reflexive and transitive . `R_(1)` is not symmetric , since `(1,2)in R and (2,1) !in R.` (ii) `R_(2)={(1,1),(2,2),(1,2),(2,1)}.` is symmetric and transitive. But,`R_(2)` is not reflexive since `(3,3) !in R_(2).` (iii)` R_(3)={(1,1),(2,2),(3,3),(1,2),(2,1),(2,3),(3,2)}.` is reflexive and symmetric But ,`R_(3) ` is not transitive , since `(1,2)in R_(3),(2,3)in R_(3) But (1,3)!in R_(3)` (iv) ` R_(4) ={(2,2),(3,3),(1,2),(2,1)}.` is symmetric . But ,`R_(4) ` is not reflexive since `(1,1) !in R_(4)` also `R_(4)` is not transitive ,as `(1,2)in R _(4) and (2,1) in R _(4) "but" (1,1) in R_(4).` (v) `R_(5) ={(2,2),(3,3),(1,2)}` is transitive . But ,`R_(5)` is not reflexive , since` (1,1) !in R.` and ,`R_(5)` is not symmetric as `(1,2) in R_(5) but (2,1) !in R_(5).` |
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