A) \[{{R}_{1}}=\{(p,q),(q,r),(p,r),(p,q)\}\]
B) \[{{R}_{2}}=\{(r,q),(r,p),(r,r),(q,q)\}\]
C) \[{{R}_{3}}=\{(p,p),(q,q),(r,r),(p,q)\}\]
D) None of these
Correct Answer: D
Solution :
[d] \[{{R}_{1}}\] is not reflexive, because \[(q,q)\,\,(r,r)\notin {{R}_{1}}.\] \[\therefore {{R}_{1}}\] is not an equivalence relation \[{{R}_{2}}\] is not reflexive, because \[(p,p)\notin {{R}_{2}}.\] \[\therefore {{R}_{2}}\] is not an equivalence relation. \[{{R}_{3}}\] is reflexive, because \[(p,p),(q,q),(r,r)\in {{R}_{3}}.\] \[{{R}_{3}}\] is not symmetric, because \[(p,\,\,q)\in {{R}_{3}}\] but \[(q,p)\notin {{R}_{3}}.\]
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