Question

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

The correct options are
C Reflexive and symmetric relation.
D Transitive relation

$(a,b)\in N\times N(a,b)R(a,b)\text{as}ab=ba$
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 $(a,b)R(c,d)⟺ad=bc\cdots \left(1\right)$
$\text{and}(c,d)R(e,f)⟺cf=de\cdots \left(2\right)$
From $\left(1\right)$ and $\left(2\right)$
Now $af=be\Rightarrow (a,b)R(e,f)$
$\therefore (a,b)R(c,d),(c,d)R(e,f)\Rightarrow (a,b)R(e,f)$
Hence $R$ is Transitive relation.