Question

Let $N$ denote the set of all natural numbers and $R$ be the relation on $NxN$ defined by $(a,b)R(c,d)\iff ad(b+c)=bc(a+d)$, them $R$ is

• A. reflexive only
• B. symmetric only
• C. transitive only
• D. equivalence

Answer 1

The correct option is D equivalence

For Reflexive

$\begin{array}{c}\left(a,b\right)\in N\\ \Rightarrow ab=ba\in N\\ \Rightarrow \left(a,b\right)R\left(a,b\right)\\ \Rightarrow RisReflexive\end{array}$

Symmetric

$\begin{array}{c}\left(a,b\right)R\left(c,d\right)\\ \Rightarrow ad=bc\\ \Rightarrow bc=ad\\ \Rightarrow cb=ad\Rightarrow cb=da\\ \left(c,d\right)R\left(a,b\right)\\ So,RisSymmetric\end{array}$

For Transitive

$\begin{array}{cc}If\left(a,b\right)R\left(c,d\right)& \left(c,d\right)R\left(e,f\right)\\ \Rightarrow ad=bc& \Rightarrow cf=de\\ \Rightarrow \frac{a}{b}=\frac{c}{d}\to \left(i\right)& \Rightarrow \frac{c}{d}=\frac{e}{f}\to \left(ii\right)\\ compareequation\left(i\right)and\left(ii\right)& \\ \frac{a}{b}=\frac{e}{f}\Rightarrow af=eb\Rightarrow af=be& \\ \Rightarrow \left(a,b\right)R\left(e,f\right)& \\ So,RisTransitive& \end{array}$

So, Relation of R is equivalence.