Question

Let $A=\{p,q,r\}$. Which of the following is an equivalence relation in $A$?

• A. $R_1=\left\{\right(p,q),(q,r),(p,r),(p,p\left)\right\}$
• B. $R_2=\left\{\right(r,q),(r,p),(r,r),(q,q\left)\right\}$
• C. $R_3=\left\{\right(p,p),(q,q),(r,r),(p,q\left)\right\}$
• D. None of these

Answer 1

The correct option is D None of these

Given $A=\{p,q,r\}$
$R_1=\left\{\right(p,q),(q,r),(p,r),(p,p\left)\right\}$
$R_1$ is not reflexive as it does not have $(q,q),(r,r)$ . So, $R_1$ is not an equivalence relation on A.
$R_2=\left\{\right(r,q),(r,p),(r,r),(q,q\left)\right\}$
$R_2$ is not reflexive as it does not have $(p,p)$ . So, $R_2$ is not an equivalence relation on A.
$R_3=\left\{\right(p,p),(q,q),(r,r),(p,q\left)\right\}$
$R_3$ is reflexive but not symmetric as $(p,q)$ is there ,but not $(q,p)$