Question

On R, the set of real numbers, a relation ρ is defined as 'aρb if and only if 1+ab>0'. Then

A
ρ is an equivalence relation
B
ρ is reflexive and transitive but not symmetric
C
ρ is reflexive and symmetric but not transitive
D
ρ is only symmetric

Solution

The correct option is C ρ is reflexive and symmetric but not transitiveFor every a∈R, 1+a2>0 ∴ρ is reflexive. Let (a,b)∈ρ ⇒1+ab>0 ⇒1+ba>0 ⇒(b,a)∈ρ ∴ρ is symmetric. Let a=−1,b=12,c=1 Then 1+ab>0 and 1+bc>0 But 1+ac≯0 Hence, ρ is not transitive.

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