A) an equivalence relation
B) reflexive but not symmetric
C) reflexive and transitive but not symmetric
D) reflexive and symmetric but not transitive
E) symmetric and transitive but not reflexive
Correct Answer: A
Solution :
\[aRb\Leftrightarrow a={{2}^{k}}.b\] for some integer Reflexive \[\therefore \]\[aRa\,for\,k=0\] Symmetric\[aRa\Leftrightarrow a={{2}^{k}}b\Rightarrow b={{2}^{-k}}a\] \[\Leftrightarrow \] \[bRa\] Transitive \[aRb\Leftrightarrow a={{2}^{{{k}_{1}}}}b\] \[bRc\Leftrightarrow b={{2}^{{{k}_{2}}}}c\] \[\Rightarrow \] \[a={{2}^{{{k}_{1}}}}{{.2}^{{{k}_{2}}}}c\] \[\Rightarrow \] \[a={{2}^{{{k}_{1}}+}}^{{{k}_{2}}}c\] \[\Leftrightarrow \] \[aRc\] \[\Rightarrow \] \[aRb,bRc\]\[\Rightarrow \]\[aRc\] \[\therefore \]R is an equivalence relation.
You need to login to perform this action.
You will be redirected in
3 sec
