The relation R = \ (a, b): (a, b) = 1, 2a ≠ b, a, b ∈ ℤ\ is:

Given relation is

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

Now,

Reflexive:

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

Symmetric:

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

Also, $2a = 4 \ne b$

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

But, $2a = 2 = b$

Hence, $a = 2b$

$⇒ R$ 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.