Question

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

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

Answer 1

Answer: (A), (B)

Reflexive only
symmetric only

Given tha $S=1,2,3,4,5$ and $A=S\times S$
A relation $R$ on $A$ is defined as follows $:"(a,b)R(c,d)"$ if and only if $ad=cd$

$\left(i\right)$ $R$ is reflexive, since $ab=ab$, therefore, $(a,a)R(b,b)\forall a,b\in S$

$\left(ii\right)$ $R$ is symmetric, since $(a,b)\in R$ $\Rightarrow (b,a)\in R\forall a,b\in S$

$\left(iii\right)$ $R$ is not transitive $(a,b),(b,c)\in R$ doesnot necessarily imply $(a,c)$ belongs to $R$