Question

Let $R$ be the relation in the set $\{1,2,3,4\}$ given by $R=\left\{\right(1,2),(2,2),(1,1),(4,4),(1,3),(3,3),(3,2\left)\right\}$. 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)

$R$ is reflexive and transitive but not symmetric.

$R=\left\{\right(1,2),(2,2),(1,1),(4,4),(1,3),(3,3),(3,2\left)\right\}$
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.