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

a, b$\in$z, GCD(a, b) = 1, b $\ne$ 2a
GCD(a, a)$\ne$1$\Rightarrow$Not Reflexive
GCD(3, 7) = 1 & GCD(7, 15) = 1
But GCD(3, 15) $\ne$ 1$\Rightarrow$Not Transitive
$\text{ Take }a=2,b=1\Rightarrow \gcd (2,1)=1$

$\text{ Also }2\mathrm{a}=4\ne b$

$\text{ Now when }a=1,b=2\Rightarrow \gcd (1,2)=1$

$\text{ Also now }2\mathrm{a}=2=\mathrm{b}$

$\text{ Hence }a=2b$

$\Rightarrow \mathrm{R}\text{ is not Symmetric }$