Question

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

• A. an equivalence relation
• B. reflexive but not transitive
• C. symmetric but not reflexive
• D. both reflexive and symmetric but not transitive

Answer 1

The correct option is A an equivalence relation

$(a,b)R(a,b)$ as $ab=ba$
So, $R$ is a reflexive relation.

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

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