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Question

Let the relation ρ be defined on R as a ρ b iff 1+ab>0. Then

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

The correct option is D ρ is reflexive and symmetric but not transitive
For reflexive:
(a,a)ρ, aR
Since, 1+a2>0
ρ is reflexive.

For symmetric:
(a,b)ρ, then (b,a)ρ
Since, 1+ab>0, then 1+ba>0
ρ is symmetric.

For transitive:
(a,b)ρ, (b,c)ρ(a,c)ρ
Now, (2,0)ρ and (0,1)ρ
but (2,1)ρ
Hence, ρ is not transitive.

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