Question

If $R$ is an equivalence relation on a set $A$, then $R^{-1}$ is

• A. Reflexive relation only
• B. Symmetric but not transitive relation
• C. Reflexive, symmetric and transitive relation
• D. reflexive and transitive but not symmetric relation

Answer 1

The correct option is C Reflexive, symmetric and transitive relation

Let $(a,a)\in R,a\in A$
as $R$ is reflexive
Hence $(a,a)\in R^{-1}$
$\therefore R^{-1}$ is also reflexive relation.

Let $(a,b)\in R,a,b\in A$
$\Rightarrow (b,a)\in R$, as $R$ is symmetric
Hence $(b,a),(a,b)\in R^{-1}$
$\therefore R^{-1}$ is also symmetric relation

Let $(a,b),(b,c)\in R,a,b,c\in A$
$\Rightarrow (a,c)\in R$ as $R$ is transitive
$\Rightarrow (c,b),(b,a)\in R^{-1}\text{and}(c,a)\in R^{-1}$
$\therefore R^{-1}$ is also transitive relation.

Hence $R^{-1}$ is equivalence relation.