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On R, a relation ρ is defined by xρy if and only if x – y is zero or irrational. Then
(A) ρ is equivalence relation
(B) ρ is reflexive but neither symmetric nor transitive
(C) ρ is reflexive & symmetric but not transitive
(D) ρ is symmetric & transitive but not reflexive
On R, a relation ρ is defined by xρy if and only if x – y is zero or irrational. Then
(A) ρ is equivalence relation
(B) ρ is reflexive but neither symmetric nor transitive
(C) ρ is reflexive & symmetric but not transitive
(D) ρ is symmetric & transitive but not reflexive
The correct option (C) ρ is reflexive & symmetric but not transitive
Explanation:
xRy ⇒ x – y is zero or irrational
xRx ⇒ 0 ∴ reflective
if xRy ⇒ x – y is zero or irrational
⇒ y – x is zero or irrational
∴ yRx symmetric xRy
⇒ x – y is 0 or irrational yRz
⇒ y – z is 0 or irrational then (x – y) + (y – z) = x – z may be rational
∴ it is not transitive