A) reflexive
B) symmetric
C) transitive
D) symmetric and transitive
Correct Answer: D
Solution :
Since, R is defined as \[aRb\]iff \[\left| a-b \right|>0.\] For reflexive \[aRa\]iff \[\left| a-a \right|>0\] Which is not true. So R is not reflexive. For symmetric \[aRb\] iff\[\left| a-b \right|>0\] Now, \[bRa\] iff \[\left| b-a \right|>0\] \[\Rightarrow \] \[\left| a-b \right|>0\Rightarrow aRb\] Thus, R is symmetric. For transitive \[aRb\] iff \[\left| a-b \right|>0,\] \[bRc\]iff \[\left| b-c \right|>0\] \[\Rightarrow \]\[\left| a-b+b-c \right|>0\] \[\Rightarrow \] \[\left| a-c \right|>0\] \[\Rightarrow \] \[\left| c-a \right|>0\Rightarrow aRc\] Thus, R is also transitive.
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