Question

Let $A=\left\{p,q,r\right\}$. which of the following is an equivalence relation on $A$ ?

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

Answer 1

The correct option is D

None of the above

Explanation For The Correct Option:

Determining the correct option

For equivalence relation, a relation should be reflexive, symmetric, and transitive.

Checking equivalence relation criterion Option A:

Given relation $R_1=\left\{(p,q),(q,r),(p,r),(p,p)\right\}$

Since $(p,q)\in R_1$ but $(p,q)\notin R_1$.

Therefore, it is not symmetric

Thus, $R_1$ is not an equivalence relation.

Hence, option A is an incorrect answer.

Checking equivalence relation criterion Option B:

Given relation $R_2=\left\{(r,p),(r,p),(r,r),(q,q)\right\}$

Since $(r,q)\in R_2$ but $(q,r)\notin R_2.$.

Therefore, it is not symmetric

Thus, $R_2$ is not an equivalence relation.

Hence, option B is an incorrect answer.

Checking equivalence relation criterion Option C:

Given relation $R_3=\left\{(p,p),(q,q),(r,r),(p,q)\right\}$

Since $(p,q)\in R_3$ but $(q,p)\notin R_3$.

Therefore, it is not symmetric

Thus, $R_3$ is not an equivalence relation.

Hence, option C is an incorrect answer.

Since non of the given options satisfies the necessary conditions for an equivalence relation, none of the above is the correct answer.

Hence, The correct answer is Option (D).