Let $X = \mathbb{R} \times \mathbb{R}$. Define a relation $R$ on $X$ as: $(a_1, b_1) R (a_2, b_2) \iff b_1 = b_2$.

Statement I: $R$ is an equivalence relation.

Statement II: For some $(a, b) \in X$, the set $S = \{(x, y) \in X : (x, y) R (a, b)\}$ represents a line parallel to $y = x$.

In the light of the above statements, choose the correct answer from the options given below: