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 a f=eb$
$\Rightarrow (a$, $b) R (e$, $f) \,\forall\, a$, $b$, $c$, $d$, $e$ $f \in S$.
Hence, $R$ is an equivalence relation.