Question

Relation R on the set A={ 1,2,3 ..20] is defined as R = {(x,y):3x-y=0}.
Determine whether the given relation is reflexive, symmetric and transitive.​

Answer 1

$\textbf{Given:}$

$\mathsf{A=\{1,2,3,\;.\;.\;.20\}\;and\;R=\{(x,y)|\;3x-y=0\}}$

$\textbf{To Check:}$

$\textsf{Whether R is reflexive, symmetric and transitive}$

$\textbf{Solution:}$

$\textsf{Consider,}$

$\mathsf{_xR_y\;\iff\;3x-y=0}$

$\mathsf{3(1)-3=0\;\implies\;_1R_3}$

$\mathsf{3(2)-6=0\;\implies\;_2R_6}$

$\mathsf{3(3)-9=0\;\implies\;_3R_9}$

$\mathsf{3(4)-12=0\;\implies\;_4R_{12}}$

$\mathsf{3(5)-15=0\;\implies\;_5R_{15}}$

$\mathsf{3(6)-18=0\;\implies\;_6R_{18}}$

$\implies\mathsf{R=\{(1,3),(2,6),(3,9),(4,12),(5,15),(6,18)\}}$

$\mathsf{1.\;(1,1)\;\notin\;R}$

$\therefore\mathsf{R\;is\;not\;reflexive}$

$\mathsf{2.\;(1,3)\;\in\;R}$

$\mathsf{But\;(3,1)\;\notin\;R}$

$\therefore\mathsf{R\;is\;not\;symmetric}$

$\mathsf{3.\;(1,3)\in\,R\;and\;(3,9)\in\,R}$

$\implies\mathsf{(1,9)\,\notin\;R}$

$\therefore\mathsf{R\;is\;not\;transitive}$

$\textbf{Find more:}$

R be a relation on Z defined by R= {(a,b): a-b is an integer} show that R is an equivalence relation

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