A) Reflexive but neither symmetric nor transitive.
B) Symmetric and transitive.
C) Reflexive and symmetric.
D) Reflexive and transitive.
Correct Answer: D
Solution :
[d] \[R=\{(x,y):x,y\in N\,\,and\,\,{{x}^{2}}-4xy+3{{y}^{2}}=0\}\] Now, \[{{x}^{2}}-4xy+3{{y}^{2}}=0\Rightarrow (x-y)(x-3y)=0\] \[\therefore x=y\,\,or\,\,x=3y\] \[\therefore R=\{(1,1),(3,1),(2,2),(6,2),(3,3),(9,3)...\}\] Since \[(1,2),\,\,(2,2),\,\,(3,3),...\] are present in the relation, therefore R is reflexive. Since, (3, 1) is an element of R but (1, 3) is not the element of R, therefore R is not symmetric Here \[(3,1)\in R\] and \[(1,1)\in R\Rightarrow (3,1)\in R(6,2)\in R\] and \[(2,2)\in R\Rightarrow (6,2)\in R\] For all such \[(a,b)\in R\]and \[(b,c)\in R\Rightarrow (a,c)\in R\] Hence R is transitive.
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