Question

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

• A. Reflexive but not symmetric
• B. Symmetric but not transitive
• C. An equivalence relation
• D. None of these

Answer 1

The correct option is C An equivalence relation

Let R be a relation on A i.e. $R\subseteq A\times A$
$R=\left\{\right(a,b)|a,b\in A\}$
Also, given $R$ is equivalence relation,
Now, let $R^{-1}=\left\{\right(b,a\left)|\right(a,b)\in R\}$
We will check whether $R^{-1}$ is reflexive, symmetric, transitive or an equivalence relation.
Reflexive:
Since, $R$ is reflexive
$\Rightarrow (a,a)\in R$
$\Rightarrow (a,a)\in R^{-1}$ (by def of $R^{-1}$)
Hence, $R^{-1}$ is reflexive.
Symmetric: Let $(b,a)\in R^{-1}$
$\Rightarrow (a,b)\in R$ (by def of $R^{-1}$)
$\Rightarrow (b,a)\in R$ (Since, R is symmetric)
$\Rightarrow (a,b)\in R^{-1}$ (by def of $R^{-1}$)
Hence, $R^{-1}$ is symmetric.
Transitive : Let $(b,a),(a,c)\in R^{-1}$
$\Rightarrow (a,b),(c,a)\in R$ (by def of $R^{-1}$)
or $(c,a)(a,b)\in R$
$\Rightarrow (c,b)\in R$ (since, R is transitive.)
$\Rightarrow (b,c)\in R^{-1}$
Hence, $R^{-1}$ is transitive.
Hence, $R^{-1}$ is an equivalence relation.