The relation $\mathrm{R = \{ (a,b):\gcd (a,b) = 1,2a \ne b,a,b \in \mathbb{Z}\}}$ is :

<p>Given,</p> <p>(a, b) belongs to relation R if $\gcd (a,b) = 1, 2a \ne b$.</p> <p>Here $\gcd$ means greatest common divisor. $\gcd$ of two numbers is the largest number that divides both of them.</p> <p>(1) For Reflexive,</p> <p>In $aRa,\,\gcd (a,a) = a$</p> <p>$\therefore$ This relation is not reflexive.</p> (2) For Symmetric:<br/><br/> Take $a=2, b=1 \Rightarrow \operatorname{gcd}(2,1)=1$ Also $2 a=4 \neq b$ <br/><br/>Now $\gcd (b,a) = 1$ $ \Rightarrow \operatorname{gcd}(1,2)=1$<br/><br/> and 2b should not be equal to a<br/><br/> But here, $2 b=2=a$<br/><br/> $\Rightarrow \mathrm{R}$ is not Symmetric<br/><br/> (3) For Transitive:<br/><br/> Let $\mathrm{a}=14, \mathrm{~b}=19, \mathrm{c}=21$<br/><br/> $\operatorname{gcd}(\mathrm{a}, \mathrm{b})=1, 2a \ne b$<br/><br/> $\operatorname{gcd}(\mathrm{b}, \mathrm{c})=1, 2b \ne c$<br/><br/> $\operatorname{gcd}(\mathrm{a}, \mathrm{c})=7, 2a \ne c$<br/><br/> Hence not transitive<br/><br/> $\Rightarrow R$ is neither symmetric nor transitive.