Question

Determine whether the relation $R$ in the set $A=\{1,2,3,.....,13,14\}$ defined as $R=\left\{\right(x,y):3x-y=0\}$, is reflexive, symmetric and transitive.

Answer 1

Relation $R$ on the set $A=\{1,2,3,.....,13,14\}$ is defined as $R=\left\{\right(x,y):3x-y=0\}$

Let $a\in A$

$R$ is reflexive if $(a,a)\in R$

if $3a-a=0$

if $3a=a$ i.e. if $3=1$ which is not true

Thus, $R$ is not reflexive ...... $\left(1\right)$

Let $a,b\in A$ such that $(a,b)\in R$

$⟹3a-b=0$

$⟹3a=b$

This does not imply $3b=a$ i.e $3b-a=0$

$\therefore (b,a)$ does not belong to $R$

$\therefore$ For $a,b\in A$, $(a,b)\in R⟹(b,a)$ is not in $R$.

Thus, $R$ is not symmetric ....... $\left(2\right)$

Let $a,b,c\in A$ such that $(a,b),(b,c)\in R$

$⟹3a-b=0,3b-c=0$

$⟹3a=b,3b=c$

$⟹3a=\frac{c}{3}$

$⟹9a=c$

This does not imply $(a,c)\in R$

$\therefore$ For $a,b,c\in A,$ $(a,b),(b,c)\in R$ does not imply $(a,c)\in R$

Hence, $R$ is non-reflexive, non-symmetric and non-transitive.