Let R be a relation on ℝ , given by R = (a, b): 3a - 3b + 7 is an...

Given:

$R = \left\{\left(a, b\right): 3a - 3b + \sqrt{7}isirrational\right\}$

Reflexive:

$3a - 3a + \sqrt{7} = \sqrt{7}$ is irrational.

So, \left(a, a\right) \in R$, hence$R$is reflexive.</p><p>Given:</p><p>$R = \left\{\left(a, b\right): 3a - 3b + \sqrt{7}isirrational\right\}$</p><p><strong>Symmetric:</strong></p><p>Let \left(a, b\right) \in R$, then

$3a - 3b + \sqrt{7}$ is irrational.

So, $3b - 3a + \sqrt{7}$ is not necessarily irrational, since if $3a = \sqrt{7}$ and $3b \in Z$, then $3a - 3b + \sqrt{7}$ is irrational., but $3b - 3a + \sqrt{7}$ is not irrational.

So, \left(a, b\right) \in R ⇒ \left(b, a\right) \in R$, hence$R$is not symmetric</p><p><strong>Transitive:</strong></p><p>Take \left(a, b\right)$ as \left(\dfrac{\sqrt{7}}{3}, 1\right)$and \left(b, c\right)$ as \left(1, \dfrac{2\sqrt{7}}{3}\right)$</p><p>So, \left(a, b\right) \in R$ and \left(b, c\right) \in R$but \left(a, c\right) \notin R$ which means relation is not transitive.