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Let R1 and R2 be two relations defined on \(R\) by αR1 b ⇔ ab ≥ 0 and α R2b ⇔ a ≥ b , then(A) R1 is an equivalence relation but not R2(B) R2 is an equivalence relation but not R1(C) both R1 and R2 are equivalence relations(D) neither R1 nor R2 is an equivalence relation |
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Answer» Correct option is (D) neither R1 nor R2 is an equivalence relation \(R_1 = \{xy\ge 0 , x, y\in R\}\) For reflexive x \(\times\) x ≥ 0 which is true. For symmetric If xy ≥ 0 ⇒ yx ≥ 0 If x = 2, y = 0 and z = -2 Then x.y ≥ 0 & y.z ≥ 0 but x.z ≥ 0 is not true ⇒ not transitive relation. ⇒ R1 is not equivalence R2 if a ≥ b it does not implies b ≥ a ⇒ R2 is not equivalence relation ⇒ D |
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