Question
Let $$A = \left\{ {p,\,q,\,r} \right\}.$$ Which of the following is an equivalence relation in $$A\,?$$
A.
$${R_1} = \left\{ {\left( {p,\,q} \right),\left( {q,\,r} \right),\left( {p,\,r} \right),\left( {p,\,q} \right)} \right\}$$
B.
$${R_2} = \left\{ {\left( {r,\,q} \right),\left( {r,\,p} \right),\left( {r,\,r} \right),\left( {q,\,q} \right)} \right\}$$
C.
$${R_3} = \left\{ {\left( {p,\,p} \right),\left( {q,\,q} \right),\left( {r,\,r} \right),\left( {p,\,q} \right)} \right\}$$
D.
None of these
Answer :
None of these
Solution :
$${R_1}$$ is not reflexive, because $$\left( {q,\,q} \right)\left( {r,\,r} \right)\, \notin \,{R_1}$$
$$\therefore \,{R_1}$$ is not equivalence relation
$${R_2}$$ is not reflexive, because $$\left( {p,\,p} \right)\, \notin \,{R_2}$$
$$\therefore \,{R_2}$$ is not equivalence relation
$${R_3}$$ is reflexive, because $$\left( {p,\,p} \right),\,\left( {q,\,q} \right),\,\left( {r,\,r} \right)\, \in \,{R_3}$$
$${R_3}$$ is not symmetric, because $$\left( {p,\,q} \right)\, \in \,{R_3}$$ but $$\left( {q,\,p} \right)\, \notin \,{R_3}.$$