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Let $R$ be the relation in the set $\{1,2,3,4\}$ given by $R =\{(1,2),\,(2,2),\,(1,1),\,(4,4)$ $(1,3),\,(3,3),\,(3,2)\}$. Choose the correct answer.
$R$ is reflexive and symmetric but not transitive.
$R$ is reflexive and transitive but not symmetric.
$R$ is symmetric and transitive but not reflexive.
$R$ is an equivalence relation
Solution
$R=\{(1,2),\,(2,2),\,(1,1),\,(4,4),\,(1,3),\,(3,3),\,(3,2)\}$
It is seen that $(a, \,a) \in R,$ for every $a \in\{1,\,2,\,3,\,4\}$
$\therefore R$ is reflexive.
It is seen that $(1,\,2) \in R ,$ but $(2,\,1)\notin R$
$\therefore R$ is not symmetric.
Also, it is observed that $(a, \,b),\,(b, \,c) \in R \Rightarrow(a,\, c) \in R$ for all $a, \,b, \,c \in\{1,\,2,\,3,\,4\}$
$\therefore R$ is transitive.
Hence, $R$ is reflexive and transitive but not symmetric.
The correct answer is $B$.
