Let $$R$$ be a relation on $$\mathbb{R}$$, given by $$R = \{(a, b) : 3a - 3b + \sqrt{7} \text{ is an irrational number}\}$$. Then $$R$$ is

Testing reflexivity for $$(a, a) \in R$$:

$$3a - 3a + \sqrt{7} = \sqrt{7} \quad (\text{Irrational for all } a \in \mathbb{R}) \implies \text{Reflexive}$$

Testing symmetry using counterexample $$a = \frac{\sqrt{7}}{3}, b = 0$$:

$$(a, b): 3\left(\frac{\sqrt{7}}{3}\right) - 3(0) + \sqrt{7} = 2\sqrt{7} \implies (a, b) \in R$$

$$(b, a): 3(0) - 3\left(\frac{\sqrt{7}}{3}\right) + \sqrt{7} = 0 \quad (\text{Rational}) \implies (b, a) \notin R \implies \text{Not symmetric}$$

Testing transitivity using counterexample $$a = \frac{\sqrt{7}}{3}, b = 0, c = \frac{2\sqrt{7}}{3}$$:

$$(a, b): 3\left(\frac{\sqrt{7}}{3}\right) - 3(0) + \sqrt{7} = 2\sqrt{7} \implies (a, b) \in R$$

$$(b, c): 3(0) - 3\left(\frac{2\sqrt{7}}{3}\right) + \sqrt{7} = -\sqrt{7} \implies (b, c) \in R$$

$$(a, c): 3\left(\frac{\sqrt{7}}{3}\right) - 3\left(\frac{2\sqrt{7}}{3}\right) + \sqrt{7} = 0 \quad (\text{Rational}) \implies (a, c) \notin R \implies \text{Not transitive}$$