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Question

Let N be the set of all natural numbers and let R be a relation on N×N defined by (a,b)R(c,d)ad=bc (a,b),(c,d)N×N. Then R is

A
Reflexive but not transitive relation.
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B
Reflexive but not symmetric relation.
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C
Reflexive and symmetric relation.
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D
Transitive relation
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Solution

The correct option is D Transitive relation
(a,b)N×N
(a,b)R(a,b) as ab=ba
Hence, R is reflexive relation.

If (a,b)R(c,d)ad=bc
cb=da
(c,d)R(a,b)
Hence, R is symmetric relation.

Let (a,b)R(c,d)ad=bc (1)
and (c,d)R(e,f)cf=de (2)
From (1) and (2),
af=be(a,b)R(e,f)
(a,b)R(c,d),(c,d)R(e,f)(a,b)R(e,f)
Hence, R is transitive relation.

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