Thank you for reporting, we will resolve it shortly
Q.
If R is an equivalence relation on a set A, then $R^{-1}$ is
Relations and Functions - Part 2
Solution:
An equivalence relation is one which is reflexive, symmetric and transitive.
R is an equivalence relation on set A.
Let the element of set A be $a_1, a_2, a_3$
So, $a_1Ra_1$ - since it is reflexive, this is also true for $R^{-1}$
It is symmetric hence, $a_1Ra_2 \, \Rightarrow \, a_2Ra_1$ and this is also true for $R^{-1}$
Also, R is transitive i.e., $a_1Ra_2$ and $a_2Ra_3$
$\Rightarrow \, a_1 \, Ra_3$
For $R^{-1}: a_2 \, R^{-1} a_1$ and $a_3R^{-1}a_2$
$\Rightarrow \, a_3 \, R^{-1} \, a_1$ or $a_1 \, R^{- 1 } a_3$
Thus $R^{-1}$ is symmetric, reflexive and transitive.
i.e. $R^{-1}$ is equivalence Relation.