Question

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

Answer 1

To determine nation are whether reflective & symmetric & transective

Set $A=\{1,2,3,....13,14\}$ define as

$R=\left\{\right(x,y)3x-,=0\}$

$=3x-y=0$

$3x=y;xyϵA$

If $x=1y=\left(3\right(1\left)\right)=3;x,y\epsilon A$

$x=2y3\left(2\right)=6;x,y\epsilon A$

$x=3y3\left(3\right)=9;x,y\epsilon A$

$x=4y3\left(4\right)=12;x,y\epsilon A$

$x=5y3\left(5\right)=15;x,y\epsilon A$

$R=\left\{\right(1,3),(2,6),(3,9),(4,12\left)\right\}$

Reflective condition:

If reaction is reflective

then $(k,k)\epsilon A$ & $\forall k\epsilon A$ i.e., $\{1,2,3,....14\}$

Since $(1,1)\epsilon R,(2,2)\epsilon R,(3,3)\epsilon R,....(14,14)\epsilon R$

$\therefore R$ is not reflective

Symmetricity Condition:

If $(K,E)\epsilon R$ then $(l,k)\epsilon R$

then relation is symmetric

Since $(1,3)\epsilon R$, but $(3,1)\epsilon R$

$\therefore R$ is not symmetric

Transective Condition:

If $(a,b)\epsilon R$ & $(b,c)\epsilon R$ then $(a.c)\epsilon R$

then re;lation in transective

Since $(1,3)\epsilon R$ & $(3,9)\epsilon R$ but $(1,9)\epsilon R$

$\therefore R$ is not transective

$\to$ So. $R$ is nither reflective, nor symmetric, nor transective