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1.Relation and Function
normal
Let $I$ be the set of positve integers. $R$ is a relation on the set $I$ given by $R =\left\{ {\left( {a,b} \right) \in I \times I\,|\,\,{{\log }_2}\left( {\frac{a}{b}} \right)} \right.$ is a non-negative integer$\}$, then $R$ is
A
neither symmetric not transitive but reflexive.
B
reflexive, transitive but not symmetric
C
neither reflexive non transitive but symmetric
D
equivalence relation.
Solution
$\mathrm{a} \mathrm{R} \mathrm{b} \Leftrightarrow \mathrm{a}=2^{\mathrm{k}} \cdot \mathrm{a}$ it is true for $\mathrm{k}=0$
$\therefore$ reflexive
$(2,1) \in \mathrm{R}$ but $(1,2) \notin \mathrm{R} \Rightarrow$ it is not symmetric
if $a=2^{k_{1}} b$ and $b=2^{k_{2}} c,$ then $a=2^{k_{1}+k_{2}} c$
$\Rightarrow$ it is transistive.
Standard 12
Mathematics
