Question

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.

• A. $R$ is reflexive and symmetric but not transitive
• B. $R$ is reflexive and transitive but not symmetric
• C. $R$ is symmetric and transitive but not reflexive
• D. $R$ is an equivalence relation

Answer 1

Answer: (B)

Here, $R=\{(1,2),(2,2),(1,1),(4,4),(1,3),(3,3),(3,2)\}$
Since, $(a,a)\in R$., for every $a\in \{1,2,3,4\}$. Therefore, $R$ is reflexive.
Now, since $(1,2)\in R$ 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.