The relation R = a , b : g c d a , b = 1 ,   2 a ≠ b ,   a ,   b ∈ ℤ is:

Given relation is

$R=\left\{\left(a,b\right):gcd\left(a\mathit{,}b\right)\mathit{=}1, 2a\ne b, a, b\in \mathrm{\mathbb{Z}}\right\}$

Now,

Reflexive: 

$\gcd \left(a,a\right)\ne 1 \forall a\in Z$, hence $R$ is not reflexive.

Symmetric:

$a=2, b=1\Rightarrow \gcd \left(2,1\right)=1$

Also, $2a=4\ne b$

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

But, $2a=2=b$

Hence, $a=2b$

$\Rightarrow R$ is not Symmetric

Transitive:

Let $a=14, b=19, c=21$

$gcd(a, b)=1$

$gcd(b, c)=1$

$gcd(a, c)=7$

Hence, not transitive

So, $R$ is neither symmetric nor transitive.