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. $R_2=\left\{\right(r,q),(r,p),(r,r\left)\right\}$
• 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

$R_1$ is not an equivalence because $R_1$ is not reflexive

[since $(q,q)\notin R$ and $(r,r)\notin R$]

$R_1$ is not symmetric also

$R_2$ is also not reflexive (Since $(p,p)\notin R$)

$R_3$ is not symmetric

( since $(p,q)\in R_3$ and $(q,p)\notin R_3$)