Question

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

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

Answer 1

Answer: (C), (D)

Transitive relation

$(a,b)\in N\times N$
$(a,b)R(a,b)$ as $ab=ba$
Hence, $R$ is reflexive relation.

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

Let $(a,b)R(c,d)⟺ad=bc\cdots \left(1\right)$
and $(c,d)R(e,f)⟺cf=de\cdots \left(2\right)$
From $\left(1\right)$ and $\left(2\right),$
$af=be\Rightarrow (a,b)R(e,f)$
$\therefore (a,b)R(c,d),(c,d)R(e,f)\Rightarrow (a,b)R(e,f)$
Hence, $R$ is transitive relation.