InterviewSolution
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InterviewSolution
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`R_(1)` on Z defined by `(a,b)inR_(1) " iff "|a-b|le7, R_(2)` on Q defined by `(a,b)inR_(2) " iff "ab=4 and R_(3)` on R defined by `(a, b)inR_(3)" iff "a^(2)-4ab+3b^(2)=0` Relation `R_(3)` isA. reflexiveB. symmetricC. transitiveD. equivalence |
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Answer» Correct Answer - A We have, (a, b) `in R_(3)` iff `a^(2) - 4ab + 3b^(2) = 0` where a, b `in R` Reflexivity `therefore a^(2)-4a.a+3d^(2)=4a^(2)-4a^(2)=0` `therefore (a, a) in R_(3)` `therefore` The relation `R_(3)` is reflexive. Symmetry `(a, b) in R_(3)` `implies a^(2)-4ab+3b^(2)=0`, we get a = b and a = 3b and `(b, a) in R_(3)` implies `b^(2) - 4ab + 3a^(2) = 0` we get b = a and b = 3a `therefore (a, b) in R_(3) cancelimplies(b, a)in R_(3)` `therefore` The relation `R_(3)` is not symmetric. Transitivity We have `(3, 1), (1, (1)/(3))inR_(3)` because `(3)^(2)-4(3)(1)+3(1)^(2)=9-12+3=0` and `(1)^(2)-4(1)((1)/(3))+3((1)/(3))^(2)=1-(4)/(3)+(1)/(3)=0` Also, `(3, (1)/(3))cancelinR_(3)`, because `(3)^(2)-4.(3)((1)/(3))+3((1)/(3))^(2)=9-4+(1)/(3)=(16)/(3)ne0` `therefore` The relation `R_(3)` is not transitive. |
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