Question

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

• A. reflexive only
• B. symmetric but not transitive
• C. equivalence
• D. none of the above

Answer 1

Answer: (C)

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, $a_{1}R{a}_1$ - since it is reflexive, this is also true for $R^{-1}$
It is symmetric hence, $a_{1}R{a}_2\Rightarrow a_{2}R{a}_1$ and this is also true for $R^{-1}$
Also, R is transitive i.e., $a_{1}R{a}_2$ and $a_{2}R{a}_3$
$\Rightarrow a_{1}R{a}_3$
For $R^{-1}:a_2{R}^{-1}{a}_1$ and $a_3{R}^{-1}{a}_2$
$\Rightarrow a_3{R}^{-1}{a}_1$ or $a_1{R}^{-1}{a}_3$
Thus $R^{-1}$ is symmetric, reflexive and transitive.
i.e. $R^{-1}$ is equivalence Relation.