Question

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

• A. Reflexive only
• B. Symmetric but not transitive
• C. An equivalence
• D. None of the above

Answer 1

The correct option is C

An equivalence

Checking the relation for $R^{-1}$:

A relation is said to be an equivalence relation if it is reflexive, symmetric and transitive.

R is reflexive when each element is related to itself.
R is symmetric when any one element is related to any other element, then the second element is related to the first.
R is transitive when any one element is related to a second and that second element is related to a third, then the first element is related to the third.

Let us assume three elements $x,y,z$ belongs to a relation $R$ on a set $A$ that is $A=\left\{x,y,z\right\}$

Given that $R$ is an equivalence relation on a set $A$, then relation $R$ can be defined as

$R=\left\{\left(x,x\right),\left(y,y\right),\left(z,z\right),\left(x,y\right),\left(y,x\right),\left(x,z\right),\left(z,x\right),\left(y,z\right),\left(z,y\right)\right\}$

Therefore, $R^{-1}$ can be written as

$R^{-1}=\left\{\left(x,x\right),\left(y,y\right),\left(z,z\right),\left(y,x\right),\left(x,y\right),\left(z,x\right),\left(x,z\right),\left(z,y\right),\left(y,z\right)\right\}$

which shows that $R^{-1}$ is also an equivalence relation on set $A$.

$\left\{\left(x,x\right),\left(y,y\right),\left(z,z\right)\right\}\in R^{-1}$ shows reflexive.

$\left\{\left(y,x\right),\left(z,x\right),\left(z,y\right)\right\}\in R^{-1}$ also $\left\{\left(x,y\right),\left(x,z\right),\left(y,z\right)\right\}\in R^{-1}$ shows symmetric.

$\left\{\left(y,x\right),\left(x,z\right)\right\}\in R^{-1}$ then $\left\{\left(y,z\right)\right\}\in R^{-1}$ shows transitive.

Therefore, $R^{-1}$ is equivalence.

Hence, the correct option is (C).