Question

Let $S=\{1$, $2$, $3$, $4$, $5\}$ and let $A=S\times S$. Define the relation $R$ on $A$ as follows : $(a$, $b)R(c$, $d)$ iff $ad=cb$. Then, $R$ is

• A. Reflexive only
• B. Symmetric only
• C. Transitive only
• D. Equivalence relation

Answer 1

Answer: (D)

Given that $S=\{1$, $2$, $3$, $4$, $5\}$ and $A=S\times S$
$(i)$ Reflexive : $ab=ba$
$\Rightarrow (a$, $b)R(a$, $b)R(a$, $b)\forall a$, $b\in S$.
$(ii)$ Symmetric : $(a$, $b)R(c$, $d)$
$\Rightarrow ad=cb$
$\Rightarrow cb=ad$
$\Rightarrow (c$, $d)R(a$, $b)\forall a$, $b$, $c$, $d\in S$.
$(iii)$ Transitive : $(a$, $b)R(c$, $d)$ and $(c$, $d)R(e$, $f)$
$\Rightarrow ad=cb$ and $cf=ed$
$\Rightarrow adcf=cbed$
$\Rightarrow cd\left(af\right)=cd\left(be\right)$
$\Rightarrow af=eb$
$\Rightarrow (a$, $b)R(e$, $f)\forall a$, $b$, $c$, $d$, $e$ $f\in S$.
Hence, $R$ is an equivalence relation.