A) \[{{R}_{2}}\] is symmetric but it is not transitive
B) Both \[{{R}_{1}}\] and \[{{R}_{2}}\] are transitive
C) Both \[{{R}_{1}}\] and \[{{R}_{2}}\] are not symmetric
D) \[{{R}_{1}}\] is not symmetric but it is transitive
Correct Answer: A
Solution :
Both \[{{R}_{1}}\]and \[{{R}_{2}}\] are symmetric as for any \[({{a}_{1}},{{a}_{2}})\in {{R}_{1}}\], we have \[({{a}_{2}},{{a}_{1}})\in {{R}_{1}}\] and same thing can be verified for \[{{R}_{2}}\] as well \[({{a}_{1}}\ne {{a}_{2}})\]. For checking transitivity, we observe for \[{{R}_{2}}\] that, \[(b,a)\in {{R}_{2}},(a,c)\in {{R}_{2}}\] but \[(b,c){{R}_{2}}\]. Similarly, for \[{{R}_{1}},(b,c)\in {{R}_{1}},(c,a)\in {{R}_{1}}\], but \[(b,a){{R}_{1}}\]. So neither \[{{R}_{1}}\]nor \[{{R}_{2}}\] is transitive. So, the correct answer is option A.
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