Question

Let $R$ be a symmetric and transitive relation on a set $A$. Then

• A. $R$ is reflexive and hence an equivalence relation
• B. $R$ is reflexive and hence a partial order
• C. $R$ is reflexive and hence not an equivalence relation
• D. None of the above

Answer 1

The correct option is D None of the above

A relation which is symmetric and transitive, need not be reflexive relation.
(i) $R=\left\{\right\}:$ on the set $A=\{a,b\}$. The relation $R$ is symmetric and transitive but not reflexive.
(ii) $R=\left\{\right(a,a).(b,b\left)\right\}$; on the set $A=\{a,b\}$
The relation $R$ is symmetric, transitive and also reflexive.
$\therefore$ A relation is transitive and symmetric relation but need not be reflexive relation.